abstract.py

"""
.. module:: abstract
    :platform: Unix, Windows
    :synopsis: Provides abstract base classes for representing the geometries

.. moduleauthor:: Onur Rauf Bingol <orbingol@gmail.com>

"""

import copy
import abc
import warnings
import math
from . import vis, helpers, knotvector, voxelize, utilities
from . import tessellate
from .evaluators import AbstractEvaluator
from .exceptions import GeomdlException
from . import _utilities as utl


@utl.add_metaclass(abc.ABCMeta)
class GeomdlBase(object):
    """ Abstract base class for defining geomdl objects.

    This class provides the following properties:

    * :py:attr:`type`
    * :py:attr:`id`
    * :py:attr:`name`
    * :py:attr:`dimension`
    * :py:attr:`opt`

    **Keyword Arguments:**

    * ``id``: object ID (as integer)
    * ``precision``: number of decimal places to round to. *Default: 18*
    """
    # __slots__ = ('_precision', '_id', '_dimension', '_geometry_type', '_name', '_opt_data', '_cache')

    def __init__(self, **kwargs):
        self._dimension = 0 if not hasattr(self, '_dimension') else self._dimension  # spatial dimension
        self._geometry_type = "none" if not hasattr(self, '_geometry_type') else self._geometry_type  # geometry type
        self._name = "base object" if not hasattr(self, '_name') else self._name  # object name
        self._opt_data = dict() if not hasattr(self, '_opt_data') else self._opt_data  # custom data dict
        self._cache = dict() if not hasattr(self, '_cache') else self._cache  # cache dict
        self._precision = int(kwargs.get('precision', 18))  # number of decimal places to round to
        self._id = int(kwargs.get('id', 0))  # object ID

    def __copy__(self):
        cls = self.__class__
        result = cls.__new__(cls)
        result.__dict__.update(self.__dict__)
        return result

    def __deepcopy__(self, memo):
        # Don't copy self reference
        cls = self.__class__
        result = cls.__new__(cls)
        memo[id(self)] = result
        # Don't copy the cache
        memo[id(self._cache)] = self._cache.__new__(dict)
        # Copy all other attributes
        for k, v in self.__dict__.items():
            setattr(result, k, copy.deepcopy(v, memo))
        return result

    def __str__(self):
        return self.name

    __repr__ = __str__

    @property
    def dimension(self):
        """ Spatial dimension.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the spatial dimension, e.g. 2D, 3D, etc.
        :type: int
        """
        return self._dimension

    @property
    def type(self):
        """ Geometry type

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the geometry type
        :type: str
        """
        return self._geometry_type

    @property
    def id(self):
        """ Object ID (as an integer).

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the object ID
        :setter: Sets the object ID
        :type: int
        """
        return self._id

    @id.setter
    def id(self, value):
        if not isinstance(value, int):
            raise GeomdlException("Identifier value must be an integer")
        self._id = value

    @id.deleter
    def id(self):
        self._id = 0

    @property
    def name(self):
        """ Object name (as a string)

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the object name
        :setter: Sets the object name
        :type: str
        """
        return self._name

    @name.setter
    def name(self, value):
        self._name = str(value)

    @name.deleter
    def name(self):
        self._name = ""

    @property
    def opt(self):
        """ Dictionary for storing custom data in the current geometry object.

        ``opt`` is a wrapper to a dict in *key => value* format, where *key* is string, *value* is any Python object.
        You can use ``opt`` property to store custom data inside the geometry object. For instance:

        .. code-block:: python

            geom.opt = ["face_id", 4]  # creates "face_id" key and sets its value to an integer
            geom.opt = ["contents", "data values"]  # creates "face_id" key and sets its value to a string
            print(geom.opt)  # will print: {'face_id': 4, 'contents': 'data values'}

            del geom.opt  # deletes the contents of the hash map
            print(geom.opt)  # will print: {}

            geom.opt = ["body_id", 1]  # creates "body_id" key  and sets its value to 1
            geom.opt = ["body_id", 12]  # changes the value of "body_id" to 12
            print(geom.opt)  # will print: {'body_id': 12}

            geom.opt = ["body_id", None]  # deletes "body_id"
            print(geom.opt)  # will print: {}

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the dict
        :setter: Adds key and value pair to the dict
        :deleter: Deletes the contents of the dict
        """
        return self._opt_data

    @opt.setter
    def opt(self, key_value):
        if not isinstance(key_value, (list, tuple)):
            raise GeomdlException("opt input must be a list or a tuple")
        if len(key_value) != 2:
            raise GeomdlException("opt input must have a size of 2, corresponding to [0:key] => [1:value]")
        if not isinstance(key_value[0], str):
            raise GeomdlException("key must be string")

        if key_value[1] is None:
            self._opt_data.pop(*key_value)
        else:
            self._opt_data[key_value[0]] = key_value[1]

    @opt.deleter
    def opt(self):
        self._opt_data = dict()

    def opt_get(self, value):
        """ Safely query for the value from the :py:attr:`opt` property.

        :param value: a key in the :py:attr:`opt` property
        :type value: str
        :return: the corresponding value, if the key exists. ``None``, otherwise.
        """
        try:
            return self._opt_data[value]
        except KeyError:
            return None


@utl.add_metaclass(abc.ABCMeta)
class Geometry(GeomdlBase):
    """ Abstract base class for defining geometry objects.

    This class provides the following properties:

    * :py:attr:`type`
    * :py:attr:`id`
    * :py:attr:`name`
    * :py:attr:`dimension`
    * :py:attr:`evalpts`
    * :py:attr:`opt`

    **Keyword Arguments:**

    * ``id``: object ID (as integer)
    * ``precision``: number of decimal places to round to. *Default: 18*
    """
    # __slots__ = ('_iter_index', '_array_type', '_eval_points')

    def __init__(self, **kwargs):
        self._geometry_type = "default" if not hasattr(self, '_geometry_type') else self._geometry_type  # geometry type
        super(Geometry, self).__init__(**kwargs)
        self._array_type = list if not hasattr(self, '_array_type') else self._array_type  # array storage type
        self._eval_points = self._init_array()  # evaluated points

    def __iter__(self):
        self._iter_index = 0
        return self

    def next(self):
        return self.__next__()

    def __next__(self):
        if self._iter_index > 0:
            raise StopIteration
        self._iter_index += 1
        return self

    def __len__(self):
        return 1

    def __getitem__(self, index):
        return self

    def _init_array(self):
        """ Initializes the arrays. """
        if callable(self._array_type):
            return self._array_type()
        return list()

    @property
    def evalpts(self):
        """ Evaluated points.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the coordinates of the evaluated points
        :type: list
        """
        if self._eval_points is None or len(self._eval_points) == 0:
            self.evaluate()
        return self._eval_points

    @abc.abstractmethod
    def evaluate(self, **kwargs):
        """ Abstract method for the implementation of evaluation algorithm.

        .. note::

            This is an abstract method and it must be implemented in the subclass.
        """
        pass


@utl.add_metaclass(abc.ABCMeta)
class SplineGeometry(Geometry):
    """ Abstract base class for defining spline geometry objects.

    This class provides the following properties:

    * :py:attr:`type` = spline
    * :py:attr:`id`
    * :py:attr:`name`
    * :py:attr:`rational`
    * :py:attr:`dimension`
    * :py:attr:`pdimension`
    * :py:attr:`degree`
    * :py:attr:`knotvector`
    * :py:attr:`ctrlpts`
    * :py:attr:`ctrlpts_size`
    * :py:attr:`weights` (for completeness with the rational spline implementations)
    * :py:attr:`evalpts`
    * :py:attr:`bbox`
    * :py:attr:`evaluator`
    * :py:attr:`vis`
    * :py:attr:`opt`

    **Keyword Arguments:**

    * ``id``: object ID (as integer)
    * ``precision``: number of decimal places to round to. *Default: 18*
    * ``normalize_kv``: if True, knot vector(s) will be normalized to [0,1] domain. *Default: True*
    * ``find_span_func``: default knot span finding algorithm. *Default:* :func:`.helpers.find_span_linear`
    """
    # __slots__ = (
    #     '_pdim', '_dinit', '_rational', '_degree', '_knot_vector', '_control_points', '_control_points_size',
    #     '_delta', '_bounding_box', '_evaluator', '_vis_component', '_span_func', '_kv_normalize'
    # )

    def __init__(self, **kwargs):
        self._geometry_type = "spline" if not hasattr(self, '_geometry_type') else self._geometry_type  # geometry type
        super(SplineGeometry, self).__init__(**kwargs)
        self._pdim = 0 if not hasattr(self, '_pdim') else self._pdim  # parametric dimension
        self._dinit = 0.1 if not hasattr(self, '_dinit') else self._dinit  # evaluation delta init value
        self._rational = False  # defines whether the B-spline object is rational or not
        self._degree = [0 for _ in range(self._pdim)]  # degree
        self._knot_vector = [self._init_array() for _ in range(self._pdim)]  # knot vector
        self._control_points = self._init_array()  # control points
        self._control_points_size = [0 for _ in range(self._pdim)]  # control points length
        self._delta = [self._dinit for _ in range(self._pdim)]  # evaluation delta
        self._bounding_box = self._init_array()  # bounding box
        self._evaluator = None  # evaluator instance
        self._vis_component = None  # visualization component
        self._span_func = kwargs.get('find_span_func', helpers.find_span_linear)  # default "find_span" function
        self._kv_normalize = kwargs.get('normalize_kv', True)  # flag to control knot vector normalization

    def __eq__(self, other):
        if not hasattr(other, '_pdim'):
            return False
        if not hasattr(other, '_degree') or not hasattr(other, '_knot_vector') or not hasattr(other, '_control_points'):
            return False
        if self.pdimension != other.pdimension:
            return False
        if self.rational != other.rational:
            return False
        try:
            for s, o in zip(self._control_points_size, other._control_points_size):
                if s != o:
                    return False
            chk_degree = []
            for s, o in zip(self._degree, other._degree):
                tmp = True if s == o else False
                chk_degree.append(tmp)
            if not all(chk_degree):
                return False
            chk_kv = []
            for sk, ok in zip(self._knot_vector, other._knot_vector):
                if len(sk) != len(ok):
                    return False
                chk = []
                for s, o in zip(sk, ok):
                    tmp = True if abs(s - o) < self._precision else False
                    chk.append(tmp)
                chk_kv.append(all(chk))
            if not all(chk_kv):
                return False
            chk_ctrlpts = []
            for sk, ok in zip(self._control_points, other._control_points):
                if len(sk) != len(ok):
                    return False
                chk = []
                for s, o in zip(sk, ok):
                    tmp = True if abs(s - o) < self._precision else False
                    chk.append(tmp)
                chk_ctrlpts.append(all(chk))
            if not all(chk_kv):
                return False
        except Exception:
            return False
        return True

    def __ne__(self, other):
        return not self.__eq__(other)

    @property
    def rational(self):
        """ Defines the rational and non-rational B-spline shapes.

        Rational shapes use homogeneous coordinates which includes a weight alongside with the Cartesian coordinates.
        Rational B-splines are also named as NURBS (Non-uniform rational basis spline) and non-rational B-splines are
        sometimes named as NUBS (Non-uniform basis spline) or directly as B-splines.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Returns True is the B-spline object is rational (NURBS)
        :type: bool
        """
        return self._rational

    @property
    def dimension(self):
        """ Spatial dimension.

        Spatial dimension will be automatically estimated from the first element of the control points array.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the spatial dimension, e.g. 2D, 3D, etc.
        :type: int
        """
        if self._rational:
            return self._dimension - 1
        return self._dimension

    @property
    def pdimension(self):
        """ Parametric dimension.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the parametric dimension
        :type: int
        """
        return self._pdim

    @property
    def degree(self):
        """ Degree

        .. note::

            This is an expert property for getting and setting the degree(s) of the geometry.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the degree
        :setter: Sets the degree
        :type: list
        """
        return self._degree

    @degree.setter
    def degree(self, value):
        self._degree = value

    @property
    def knotvector(self):
        """ Knot vector

        .. note::

            This is an expert property for getting and setting the knot vector(s) of the geometry.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the knot vector
        :setter: Sets the knot vector
        :type: list
        """
        return self._knot_vector

    @knotvector.setter
    def knotvector(self, value):
        self._knot_vector = value

    @property
    def ctrlpts(self):
        """ Control points.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the control points
        :setter: Sets the control points
        :type: list
        """
        return self._control_points

    @ctrlpts.setter
    def ctrlpts(self, value):
        self._control_points = value

    @property
    def weights(self):
        """ Weights.

        .. note::

            Only available for rational spline geometries. Getter return ``None`` otherwise.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the weights
        :setter: Sets the weights
        """
        return None

    @weights.setter
    def weights(self, value):
        pass

    @property
    def cpsize(self):
        """ Number of control points in all parametric directions.

        .. note::

            This is an expert property for getting and setting control point size(s) of the geometry.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the number of control points
        :setter: Sets the number of control points
        :type: list
        """
        return self._control_points_size

    @cpsize.setter
    def cpsize(self, value):
        self._control_points_size = value

    @property
    def ctrlpts_size(self):
        """ Total number of control points.

        :getter: Gets the total number of control points
        :type: int
        """
        res = 1
        for sz in self._control_points_size:
            res *= sz
        return res

    @property
    def domain(self):
        """ Domain.

        Domain is determined using the knot vector(s).

        :getter: Gets the domain
        """
        retval = []
        for idx, kv in enumerate(self._knot_vector):
            retval.append((kv[self._degree[idx]], kv[-(self._degree[idx] + 1)]))
        return retval[0] if self._pdim == 1 else retval

    @property
    def range(self):
        """ Domain range.

        :getter: Gets the range
        """
        retval = []
        for idx, kv in enumerate(self._knot_vector):
            retval.append(kv[-(self._degree[idx]) + 1] - kv[self._degree[idx]])
        return retval[0] if self._pdim == 1 else retval

    @property
    def bbox(self):
        """ Bounding box.

        Evaluates the bounding box and returns the minimum and maximum coordinates.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the bounding box
        :type: tuple
        """
        if self._bounding_box is None or len(self._bounding_box) == 0:
            self._bounding_box = utilities.evaluate_bounding_box(self.ctrlpts)
        return self._bounding_box

    @property
    def evaluator(self):
        """ Evaluator instance.

        Evaluators allow users to use different algorithms for B-Spline and NURBS evaluations. Please see the
        documentation on ``Evaluator`` classes.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the current Evaluator instance
        :setter: Sets the Evaluator instance
        :type: evaluators.AbstractEvaluator
        """
        return self._evaluator

    @evaluator.setter
    def evaluator(self, value):
        if not isinstance(value, AbstractEvaluator):
            raise TypeError("The evaluator must be an instance of AbstractEvaluator")
        value._span_func = self._span_func
        self._evaluator = value

    @property
    def vis(self):
        """ Visualization component.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the visualization component
        :setter: Sets the visualization component
        :type: vis.VisAbstract
        """
        return self._vis_component

    @vis.setter
    def vis(self, value):
        if not isinstance(value, vis.VisAbstract):
            warnings.warn("Visualization component is NOT an instance of VisAbstract class")
            return
        self._vis_component = value

    def set_ctrlpts(self, ctrlpts, *args, **kwargs):
        """ Sets control points and checks if the data is consistent.

        This method is designed to provide a consistent way to set control points whether they are weighted or not.
        It directly sets the control points member of the class, and therefore it doesn't return any values.
        The input will be an array of coordinates. If you are working in the 3-dimensional space, then your coordinates
        will be an array of 3 elements representing *(x, y, z)* coordinates.

        Keyword Arguments:
            * ``array_init``: initializes the control points array in the instance
            * ``array_check_for``: defines the types for input validation
            * ``callback``: defines the callback function for processing input points
            * ``dimension``: defines the spatial dimension of the input points

        :param ctrlpts: input control points as a list of coordinates
        :type ctrlpts: list
        :param args: number of control points corresponding to each parametric dimension
        :type args: tuple
        """
        def validate_and_clean(pts_in, check_for, dimension, pts_out, **kws):
            for idx, cpt in enumerate(pts_in):
                if not isinstance(cpt, check_for):
                    raise ValueError("Element number " + str(idx) + " is not a valid input")
                if len(cpt) != dimension:
                    raise ValueError("The input must be " + str(self._dimension) + " dimensional list - " + str(cpt) +
                                     " is not a valid control point")
                # Convert to list of floats
                pts_out[idx] = [float(coord) for coord in cpt]
            return pts_out

        # Argument validation
        if len(args) == 0:
            args = [len(ctrlpts)]
        if len(args) != self._pdim:
            raise ValueError("Number of arguments after ctrlpts must be " + str(self._pdim))

        # Keyword arguments
        array_init = kwargs.get('array_init', [[] for _ in range(len(ctrlpts))])
        array_check_for = kwargs.get('array_check_for', (list, tuple))
        callback_func = kwargs.get('callback', validate_and_clean)
        self._dimension = kwargs.get('dimension', len(ctrlpts[0]))

        # Pop existing keywords from kwargs dict
        existing_kws = ['array_init', 'array_check_for', 'callback', 'dimension']
        for ekw in existing_kws:
            if ekw in kwargs:
                kwargs.pop(ekw)

        # Set control points and sizes
        self._control_points = callback_func(ctrlpts, array_check_for, self._dimension, array_init, **kwargs)
        self._control_points_size = [int(arg) for arg in args]

    @abc.abstractmethod
    def render(self, **kwargs):
        """ Abstract method for spline rendering and visualization.

        .. note::

            This is an abstract method and it must be implemented in the subclass.
        """
        pass


@utl.add_metaclass(abc.ABCMeta)
class Curve(SplineGeometry):
    """ Abstract base class for defining spline curves.

    Curve ABC is inherited from abc.ABCMeta class which is included in Python standard library by default. Due to
    differences between Python 2 and 3 on defining a metaclass, the compatibility module ``six`` is employed. Using
    ``six`` to set metaclass allows users to use the abstract classes in a correct way.

    The abstract base classes in this module are implemented using a feature called Python Properties. This feature
    allows users to use some of the functions as if they are class fields. You can also consider properties as a
    pythonic way to set getters and setters. You will see "getter" and "setter" descriptions on the documentation of
    these properties.

    The Curve ABC allows users to set the *FindSpan* function to be used in evaluations with ``find_span_func`` keyword
    as an input to the class constructor. NURBS-Python includes a binary and a linear search variation of the FindSpan
    function in the ``helpers`` module.
    You may also implement and use your own *FindSpan* function. Please see the ``helpers`` module for details.

    Code segment below illustrates a possible implementation of Curve abstract base class:

    .. code-block:: python
        :linenos:

        from geomdl import abstract

        class MyCurveClass(abstract.Curve):
            def __init__(self, **kwargs):
            super(MyCurveClass, self).__init__(**kwargs)
            # Add your constructor code here

            def evaluate(self, **kwargs):
                # Implement this function
                pass

            def evaluate_single(self, uv):
                # Implement this function
                pass

            def evaluate_list(self, uv_list):
                # Implement this function
                pass

            def derivatives(self, u, v, order, **kwargs):
                # Implement this function
                pass

    The properties and functions defined in the abstract base class will be automatically available in the subclasses.

    **Keyword Arguments:**

    * ``id``: object ID (as integer)
    * ``precision``: number of decimal places to round to. *Default: 18*
    * ``normalize_kv``: if True, knot vector(s) will be normalized to [0,1] domain. *Default: True*
    * ``find_span_func``: default knot span finding algorithm. *Default:* :func:`.helpers.find_span_linear`
    """

    def __init__(self, **kwargs):
        self._pdim = 1  # number of parametric directions
        self._dinit = 0.01  # evaluation delta init value
        self._name = "curve"  # object name
        super(Curve, self).__init__(**kwargs)  # Call parent function

    @property
    def order(self):
        """ Order.

        Defined as ``order = degree + 1``

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the order
        :setter: Sets the order
        :type: int
        """
        return self.degree + 1

    @order.setter
    def order(self, value):
        self.degree = value - 1

    @property
    def degree(self):
        """ Degree.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the degree
        :setter: Sets the degree
        :type: int
        """
        return self._degree[0]

    @degree.setter
    def degree(self, value):
        val = int(value)
        if val < 0:
            raise ValueError("Degree cannot be less than zero")

        # Clean up the curve points list
        self.reset(evalpts=True)

        # Set degree
        self._degree[0] = val

    @property
    def knotvector(self):
        """ Knot vector.

        The knot vector will be normalized to [0, 1] domain if the class is initialized with ``normalize_kv=True``
        argument.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the knot vector
        :setter: Sets the knot vector
        :type: list
        """
        return self._knot_vector[0]

    @knotvector.setter
    def knotvector(self, value):
        if self.degree == 0 or self._control_points is None or len(self._control_points) == 0:
            raise ValueError("Set degree and control points first")

        # Check knot vector validity
        if not knotvector.check(self.degree, value, len(self._control_points)):
            raise ValueError("Input is not a valid knot vector")

        # Clean up the curve points lists
        self.reset(evalpts=True)

        # Set knot vector
        self._knot_vector[0] = knotvector.normalize(value, decimals=self._precision) if self._kv_normalize else value

    @property
    def ctrlpts(self):
        """ Control points.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the control points
        :setter: Sets the control points
        :type: list
        """
        return self._control_points

    @ctrlpts.setter
    def ctrlpts(self, value):
        self.set_ctrlpts(value)

    @property
    def sample_size(self):
        """ Sample size.

        Sample size defines the number of evaluated points to generate. It also sets the ``delta`` property.

        The following figure illustrates the working principles of sample size property:

        .. math::

            \\underbrace {\\left[ {{u_{start}}, \\ldots ,{u_{end}}} \\right]}_{{n_{sample}}}

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets sample size
        :setter: Sets sample size
        :type: int
        """
        ss = math.floor((1.0 / self.delta) + 0.5)
        return int(ss)

    @sample_size.setter
    def sample_size(self, value):
        if not isinstance(value, int):
            raise ValueError("Sample size must be an integer value")

        if self.knotvector is None or len(self.knotvector) == 0 or self.degree == 0:
            warnings.warn("Cannot determine the delta value. Please set knot vector and degree before sample size.")
            return

        # To make it operate like linspace, we have to know the starting and ending points.
        start = self.knotvector[self.degree]
        stop = self.knotvector[-(self.degree+1)]

        # Set delta value
        self.delta = (stop - start) / float(value)

    @property
    def delta(self):
        """ Evaluation delta.

        Evaluation delta corresponds to the *step size* while ``evaluate`` function iterates on the knot vector to
        generate curve points. Decreasing step size results in generation of more curve points.
        Therefore; smaller the delta value, smoother the curve.

        The following figure illustrates the working principles of the delta property:

        .. math::

            \\left[{{u_{start}},{u_{start}} + \\delta ,({u_{start}} + \\delta ) + \\delta , \\ldots ,{u_{end}}} \\right]

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the delta value
        :setter: Sets the delta value
        :type: float
        """
        return self._delta[0]

    @delta.setter
    def delta(self, value):
        # Delta value for surface evaluation should be between 0 and 1
        if float(value) <= 0 or float(value) >= 1:
            raise ValueError("Curve evaluation delta should be between 0.0 and 1.0")

        # Clean up the curve points list
        self.reset(evalpts=True)

        # Set new delta value
        self._delta[0] = float(value)

    @property
    def data(self):
        """ Returns a dict which contains the geometry data.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.
        """
        return dict(
            type=self.type,
            rational=self.rational,
            dimension=self.dimension,
            pdimension=self.pdimension,
            delta=tuple(self._delta),
            sample_size=(self.sample_size,),
            precision=self._precision,
            degree=tuple(self._degree),
            knotvector=tuple(self._knot_vector),
            size=(self.ctrlpts_size,),
            control_points=tuple(self._control_points)
        )

    def reverse(self):
        """ Reverses the curve """
        self._control_points = list(reversed(self._control_points))
        max_k = self.knotvector[-1]
        new_kv = [max_k - k for k in self.knotvector]
        self._knot_vector[0] = list(reversed(new_kv))
        self.reset(evalpts=True)

    def set_ctrlpts(self, ctrlpts, *args, **kwargs):
        """ Sets control points and checks if the data is consistent.

        This method is designed to provide a consistent way to set control points whether they are weighted or not.
        It directly sets the control points member of the class, and therefore it doesn't return any values.
        The input will be an array of coordinates. If you are working in the 3-dimensional space, then your coordinates
        will be an array of 3 elements representing *(x, y, z)* coordinates.

        :param ctrlpts: input control points as a list of coordinates
        :type ctrlpts: list
        """
        # It is not necessary to input args for curves
        if not args:
            args = [len(ctrlpts)]

        # Validate input
        for arg, degree in zip(args, self._degree):
            if degree <= 0:
                raise GeomdlException("Set the degree first")
            if arg < degree + 1:
                raise GeomdlException("Number of control points should be at least degree + 1")

        if len(ctrlpts[0]) < 2:
            raise GeomdlException("A curve should be at least 2-dimensional")

        if self.rational and len(ctrlpts[0]) < 3:
            raise GeomdlException("Rational curves expect weighted control points, e.g. (x * w, y * w, w)")

        # Clean up the curve and control points lists
        self.reset(ctrlpts=True, evalpts=True)

        # Call parent function
        super(Curve, self).set_ctrlpts(ctrlpts, **kwargs)

    def render(self, **kwargs):
        """ Renders the curve using the visualization component

        The visualization component must be set using :py:attr:`~vis` property before calling this method.

        Keyword Arguments:
            * ``cpcolor``: sets the color of the control points polygon
            * ``evalcolor``: sets the color of the curve
            * ``bboxcolor``: sets the color of the bounding box
            * ``filename``: saves the plot with the input name
            * ``plot``: controls plot window visibility. *Default: True*
            * ``animate``: activates animation (if supported). *Default: False*
            * ``extras``: adds line plots to the figure. *Default: None*

        ``plot`` argument is useful when you would like to work on the command line without any window context.
        If ``plot`` flag is False, this method saves the plot as an image file (.png file where possible) and disables
        plot window popping out. If you don't provide a file name, the name of the image file will be pulled from the
        configuration class.

        ``extras`` argument can be used to add extra line plots to the figure. This argument expects a list of dicts
        in the format described below:

        .. code-block:: python
            :linenos:

            [
                dict(  # line plot 1
                    points=[[1, 2, 3], [4, 5, 6]],  # list of points
                    name="My line Plot 1",  # name displayed on the legend
                    color="red",   # color of the line plot
                    size=6.5  # size of the line plot
                ),
                dict(  # line plot 2
                    points=[[7, 8, 9], [10, 11, 12]],  # list of points
                    name="My line Plot 2",  # name displayed on the legend
                    color="navy",   # color of the line plot
                    size=12.5  # size of the line plot
                )
            ]

        :return: the figure object
        """
        if not self._vis_component:
            warnings.warn("No visualization component has been set")
            return

        cpcolor = kwargs.pop('cpcolor', 'blue')
        evalcolor = kwargs.pop('evalcolor', 'black')
        bboxcolor = kwargs.pop('bboxcolor', 'darkorange')
        filename = kwargs.pop('filename', None)
        plot_visible = kwargs.pop('plot', True)
        extra_plots = kwargs.pop('extras', None)
        animate_plot = kwargs.pop('animate', False)

        # Check all parameters are set
        self._check_variables()

        # Check if the curve has been evaluated
        if self._eval_points is None or len(self._eval_points) == 0:
            self.evaluate()

        # Clear the visualization component
        self._vis_component.clear()

        # Control points
        self._vis_component.add(ptsarr=self.ctrlpts, name="control points", color=cpcolor, plot_type='ctrlpts')

        # Evaluated points
        self._vis_component.add(ptsarr=self.evalpts, name=self.name, color=evalcolor, plot_type='evalpts')

        # Bounding box
        self._vis_component.add(ptsarr=self.bbox, name="Bounding Box", color=bboxcolor, plot_type='bbox')

        # User-defined plots
        if extra_plots is not None:
            for ep in extra_plots:
                self._vis_component.add(ptsarr=ep['points'], name=ep['name'],
                                        color=(ep['color'], ep['size']), plot_type='extras')

        # Data requested by the visualization module
        if self._vis_component.mconf['others']:
            vis_other = self._vis_component.mconf['others'].split(",")
            for vo in vis_other:
                vo_clean = vo.strip()
                # Send center point of the parametric space to the visualization module
                if vo_clean == "midpt":
                    midprm = (max(self.knotvector) + min(self.knotvector)) / 2.0
                    midpt = self.evaluate_single(midprm)
                    self._vis_component.add(ptsarr=[midpt], plot_type=vo_clean)

        # Display the figure
        if animate_plot:
            return self._vis_component.animate(fig_save_as=filename, display_plot=plot_visible)
        else:
            return self._vis_component.render(fig_save_as=filename, display_plot=plot_visible)

    def reset(self, **kwargs):
        """ Resets control points and/or evaluated points.

        Keyword Arguments:
            * ``evalpts``: if True, then resets evaluated points
            * ``ctrlpts`` if True, then resets control points

        """
        reset_ctrlpts = kwargs.get('ctrlpts', False)
        reset_evalpts = kwargs.get('evalpts', False)

        if reset_ctrlpts:
            self._control_points = self._init_array()
            self._bounding_box = self._init_array()

        if reset_evalpts:
            self._eval_points = self._init_array()

    # Checks whether the curve evaluation is possible or not
    def _check_variables(self):
        works = True
        param_list = []
        if self.degree == 0:
            works = False
            param_list.append('degree')
        if self._control_points is None or len(self._control_points) == 0:
            works = False
            param_list.append('ctrlpts')
        if self.knotvector is None or len(self.knotvector) == 0:
            works = False
            param_list.append('knotvector')
        if not works:
            raise ValueError("Please set the following variables before evaluation: " + ",".join(param_list))

    @abc.abstractmethod
    def evaluate(self, **kwargs):
        """ Evaluates the curve.

        .. note::

            This is an abstract method and it must be implemented in the subclass.

        """
        # Check all parameters are set before the curve evaluation
        self._check_variables()

    @abc.abstractmethod
    def evaluate_single(self, param):
        """ Evaluates the curve at the given parameter.

        .. note::

            This is an abstract method and it must be implemented in the subclass.

        :param param: parameter (u)
        """
        # Check all variables are set before the evaluation
        self._check_variables()

        if isinstance(param, (int, float)):
            param = [float(param) for _ in range(self.pdimension)]

        # Check parameters
        if self._kv_normalize:
            if not utilities.check_params(param):
                raise GeomdlException("Parameters should be between 0 and 1")

    @abc.abstractmethod
    def evaluate_list(self, param_list):
        """ Evaluates the curve for an input range of parameters.

        .. note::

            This is an abstract method and it must be implemented in the subclass.

        :param param_list: array of parameters
        """
        # Check all parameters are set before the evaluation
        self._check_variables()

    @abc.abstractmethod
    def derivatives(self, u, order, **kwargs):
        """ Evaluates the derivatives of the curve at parameter u.

        .. note::

            This is an abstract method and it must be implemented in the subclass.

        :param u: parameter (u)
        :type u: float
        :param order: derivative order
        :type order: int
        """
        # Check all variables are set before the curve evaluation
        self._check_variables()

        # Check parameters
        if self._kv_normalize:
            if not utilities.check_params([u]):
                raise GeomdlException("Parameters should be between 0 and 1")


@utl.add_metaclass(abc.ABCMeta)
class Surface(SplineGeometry):
    """ Abstract base class for defining spline surfaces.

    Surface ABC is inherited from abc.ABCMeta class which is included in Python standard library by default. Due to
    differences between Python 2 and 3 on defining a metaclass, the compatibility module ``six`` is employed. Using
    ``six`` to set metaclass allows users to use the abstract classes in a correct way.

    The abstract base classes in this module are implemented using a feature called Python Properties. This feature
    allows users to use some of the functions as if they are class fields. You can also consider properties as a
    pythonic way to set getters and setters. You will see "getter" and "setter" descriptions on the documentation of
    these properties.

    The Surface ABC allows users to set the *FindSpan* function to be used in evaluations with ``find_span_func``
    keyword as an input to the class constructor. NURBS-Python includes a binary and a linear search variation of the
    FindSpan function in the ``helpers`` module.
    You may also implement and use your own *FindSpan* function. Please see the ``helpers`` module for details.

    Code segment below illustrates a possible implementation of Surface abstract base class:

    .. code-block:: python
        :linenos:

        from geomdl import abstract

        class MySurfaceClass(abstract.Surface):
            def __init__(self, **kwargs):
            super(MySurfaceClass, self).__init__(**kwargs)
            # Add your constructor code here

            def evaluate(self, **kwargs):
                # Implement this function
                pass

            def evaluate_single(self, uv):
                # Implement this function
                pass

            def evaluate_list(self, uv_list):
                # Implement this function
                pass

            def derivatives(self, u, v, order, **kwargs):
                # Implement this function
                pass

    The properties and functions defined in the abstract base class will be automatically available in the subclasses.

    **Keyword Arguments:**

    * ``id``: object ID (as integer)
    * ``precision``: number of decimal places to round to. *Default: 18*
    * ``normalize_kv``: if True, knot vector(s) will be normalized to [0,1] domain. *Default: True*
    * ``find_span_func``: default knot span finding algorithm. *Default:* :func:`.helpers.find_span_linear`
    """
    # __slots__ = ('_tsl_component', '_trims')

    def __init__(self, **kwargs):
        self._pdim = 2  # number of parametric directions
        self._dinit = 0.05  # evaluation delta init value
        self._name = "surface"  # object name
        super(Surface, self).__init__(**kwargs)
        self._tsl_component = None  # tessellation component
        self._trims = self._init_array()  # trimming curves

    @property
    def order_u(self):
        """ Order for the u-direction.

        Defined as ``order = degree + 1``

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets order for the u-direction
        :setter: Sets order for the u-direction
        :type: int
        """
        return self.degree_u + 1

    @order_u.setter
    def order_u(self, value):
        self.degree_u = value - 1

    @property
    def order_v(self):
        """ Order for the v-direction.

        Defined as ``order = degree + 1``

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets surface order for the v-direction
        :setter: Sets surface order for the v-direction
        :type: int
        """
        return self.degree_v + 1

    @order_v.setter
    def order_v(self, value):
        self.degree_v = value - 1

    @property
    def degree(self):
        """ Degree for u- and v-directions

        :getter: Gets the degree
        :setter: Sets the degree
        :type: list
        """
        return self._degree

    @degree.setter
    def degree(self, value):
        if not isinstance(value, (list, tuple)):
            raise ValueError("Please input a list with a length of " + str(self.pdimension))
        self.degree_u = value[0]
        self.degree_v = value[1]

    @property
    def degree_u(self):
        """ Degree for the u-direction.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets degree for the u-direction
        :setter: Sets degree for the u-direction
        :type: int
        """
        return self._degree[0]

    @degree_u.setter
    def degree_u(self, value):
        val = int(value)
        if val <= 0:
            raise ValueError("Degree cannot be less than zero")
        # Clean up the surface points
        self.reset(evalpts=True)
        # Set degree u
        self._degree[0] = int(value)

    @property
    def degree_v(self):
        """ Degree for the v-direction.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets degree for the v-direction
        :setter: Sets degree for the v-direction
        :type: int
        """
        return self._degree[1]

    @degree_v.setter
    def degree_v(self, value):
        val = int(value)
        if val <= 0:
            raise ValueError("Degree cannot be less than zero")
        # Clean up the surface points
        self.reset(evalpts=True)
        # Set degree v
        self._degree[1] = val

    @property
    def knotvector(self):
        """ Knot vector for u- and v-directions

        :getter: Gets the knot vector
        :setter: Sets the knot vector
        :type: list
        """
        return self._knot_vector

    @knotvector.setter
    def knotvector(self, value):
        if not isinstance(value, (list, tuple)):
            raise ValueError("Please input a list with a length of " + str(self.pdimension))
        self.knotvector_u = value[0]
        self.knotvector_v = value[1]

    @property
    def knotvector_u(self):
        """ Knot vector for the u-direction.

        The knot vector will be normalized to [0, 1] domain if the class is initialized with ``normalize_kv=True``
        argument.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets knot vector for the u-direction
        :setter: Sets knot vector for the u-direction
        :type: list
        """
        return self._knot_vector[0]

    @knotvector_u.setter
    def knotvector_u(self, value):
        if self.degree_u == 0 or self.ctrlpts_size_u == 0:
            raise ValueError("Set degree and control points first for the u-direction")

        # Check knot vector validity
        if not knotvector.check(self.degree_u, value, self.ctrlpts_size_u):
            raise ValueError("Input is not a valid knot vector for the u-direction")

        # Clean up the surface points
        self.reset(evalpts=True)

        # Set knot vector
        self._knot_vector[0] = knotvector.normalize(value, decimals=self._precision) if self._kv_normalize else value

    @property
    def knotvector_v(self):
        """ Knot vector for the v-direction.

        The knot vector will be normalized to [0, 1] domain if the class is initialized with ``normalize_kv=True``
        argument.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets knot vector for the v-direction
        :setter: Sets knot vector for the v-direction
        :type: list
        """
        return self._knot_vector[1]

    @knotvector_v.setter
    def knotvector_v(self, value):
        if self.degree_v == 0 or self.ctrlpts_size_v == 0:
            raise ValueError("Set degree and control points first for the v-direction")

        # Check knot vector validity
        if not knotvector.check(self.degree_v, value, self.ctrlpts_size_v):
            raise ValueError("Input is not a valid knot vector for the v-direction")

        # Clean up the surface points
        self.reset(evalpts=True)

        # Set knot vector
        self._knot_vector[1] = knotvector.normalize(value, decimals=self._precision) if self._kv_normalize else value

    @property
    def ctrlpts(self):
        """ 1-dimensional array of control points.

        .. note::

            The v index varies first. That is, a row of v control points for the first u value is found first.
            Then, the row of v control points for the next u value.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the control points
        :setter: Sets the control points
        :type: list
        """
        return self._control_points

    @ctrlpts.setter
    def ctrlpts(self, value):
        if self.ctrlpts_size_u <= 0 or self.ctrlpts_size_v <= 0:
            raise ValueError("Please set the number of control points on the u- and v-directions")
        self.set_ctrlpts(value, self.ctrlpts_size_u, self.ctrlpts_size_v)

    @property
    def ctrlpts_size_u(self):
        """ Number of control points for the u-direction.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets number of control points for the u-direction
        :setter: Sets number of control points for the u-direction
        """
        return self._control_points_size[0]

    @ctrlpts_size_u.setter
    def ctrlpts_size_u(self, value):
        if not isinstance(value, int):
            raise TypeError("Number of control points for the u-direction must be an integer number")
        if value <= 0:
            raise ValueError("Control points size cannot be less than and equal to zero")

        # Assume that user is doing this right
        self._control_points_size[0] = value

    @property
    def ctrlpts_size_v(self):
        """ Number of control points for the v-direction.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets number of control points on the v-direction
        :setter: Sets number of control points on the v-direction
        """
        return self._control_points_size[1]

    @ctrlpts_size_v.setter
    def ctrlpts_size_v(self, value):
        if not isinstance(value, int):
            raise TypeError("Number of control points on the v-direction must be an integer number")
        if value <= 0:
            raise ValueError("Control points size cannot be less than and equal to zero")

        # Assume that user is doing this right
        self._control_points_size[1] = value

    @property
    def sample_size_u(self):
        """ Sample size for the u-direction.

        Sample size defines the number of surface points to generate. It also sets the ``delta_u`` property.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets sample size for the u-direction
        :setter: Sets sample size for the u-direction
        :type: int
        """
        ss = math.floor((1.0 / self.delta_u) + 0.5)
        return int(ss)

    @sample_size_u.setter
    def sample_size_u(self, value):
        if not isinstance(value, int):
            raise ValueError("Sample size must be an integer value")

        if (self.knotvector_u is None or len(self.knotvector_u) == 0) or self.degree_u == 0:
            warnings.warn("Cannot determine 'delta_u' value. Please set knot vectors and degrees before sample size.")
            return

        # To make it operate like linspace, we have to know the starting and ending points.
        start_u = self.knotvector_u[self.degree_u]
        stop_u = self.knotvector_u[-(self.degree_u+1)]

        # Set delta values
        self.delta_u = (stop_u - start_u) / float(value)

    @property
    def sample_size_v(self):
        """ Sample size for the v-direction.

        Sample size defines the number of surface points to generate. It also sets the ``delta_v`` property.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets sample size for the v-direction
        :setter: Sets sample size for the v-direction
        :type: int
        """
        ss = math.floor((1.0 / self.delta_v) + 0.5)
        return int(ss)

    @sample_size_v.setter
    def sample_size_v(self, value):
        if not isinstance(value, int):
            raise ValueError("Sample size must be an integer value")

        if (self.knotvector_v is None or len(self.knotvector_v) == 0) or self.degree_v == 0:
            warnings.warn("Cannot determine 'delta_v' value. Please set knot vectors and degrees before sample size.")
            return

        # To make it operate like linspace, we have to know the starting and ending points.
        start_v = self.knotvector_v[self.degree_v]
        stop_v = self.knotvector_v[-(self.degree_v+1)]

        # Set delta values
        self.delta_v = (stop_v - start_v) / float(value)

    @property
    def sample_size(self):
        """ Sample size for both u- and v-directions.

        Sample size defines the number of surface points to generate. It also sets the ``delta`` property.

        The following figure illustrates the working principles of sample size property:

        .. math::

            \\underbrace {\\left[ {{u_{start}}, \\ldots ,{u_{end}}} \\right]}_{{n_{sample}}}

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets sample size as a tuple of values corresponding to u- and v-directions
        :setter: Sets sample size for both u- and v-directions
        :type: int
        """
        sample_size_u = math.floor((1.0 / self.delta_u) + 0.5)
        sample_size_v = math.floor((1.0 / self.delta_v) + 0.5)
        return int(sample_size_u), int(sample_size_v)

    @sample_size.setter
    def sample_size(self, value):
        if (self.knotvector_u is None or len(self.knotvector_u) == 0) or self.degree_u == 0 or\
                (self.knotvector_v is None or len(self.knotvector_v) == 0 or self.degree_v == 0):
            warnings.warn("Cannot determine 'delta' value. Please set knot vectors and degrees before sample size.")
            return

        # To make it operate like linspace, we have to know the starting and ending points.
        start_u = self.knotvector_u[self.degree_u]
        stop_u = self.knotvector_u[-(self.degree_u+1)]
        start_v = self.knotvector_v[self.degree_v]
        stop_v = self.knotvector_v[-(self.degree_v+1)]

        # Set delta values
        self.delta_u = (stop_u - start_u) / float(value)
        self.delta_v = (stop_v - start_v) / float(value)

    @property
    def delta_u(self):
        """ Evaluation delta for the u-direction.

        Evaluation delta corresponds to the *step size* while ``evaluate()`` function iterates on the knot vector to
        generate surface points. Decreasing step size results in generation of more surface points.
        Therefore; smaller the delta value, smoother the surface.

        Please note that ``delta_u`` and ``sample_size_u`` properties correspond to the same variable with different
        descriptions. Therefore, setting ``delta_u`` will also set ``sample_size_u``.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets evaluation delta for the u-direction
        :setter: Sets evaluation delta for the u-direction
        :type: float
        """
        return self._delta[0]

    @delta_u.setter
    def delta_u(self, value):
        # Delta value for surface evaluation should be between 0 and 1
        if float(value) <= 0 or float(value) >= 1:
            raise ValueError("Surface evaluation delta (u-direction) must be between 0.0 and 1.0")

        # Clean up the surface points
        self.reset(evalpts=True)

        # Set new delta value
        self._delta[0] = float(value)

    @property
    def delta_v(self):
        """ Evaluation delta for the v-direction.

        Evaluation delta corresponds to the *step size* while ``evaluate()`` function iterates on the knot vector to
        generate surface points. Decreasing step size results in generation of more surface points.
        Therefore; smaller the delta value, smoother the surface.

        Please note that ``delta_v`` and ``sample_size_v`` properties correspond to the same variable with different
        descriptions. Therefore, setting ``delta_v`` will also set ``sample_size_v``.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets evaluation delta for the v-direction
        :setter: Sets evaluation delta for the v-direction
        :type: float
        """
        return self._delta[1]

    @delta_v.setter
    def delta_v(self, value):
        # Delta value for surface evaluation should be between 0 and 1
        if float(value) <= 0 or float(value) >= 1:
            raise ValueError("Surface evaluation delta (v-direction) should be between 0.0 and 1.0")

        # Clean up the surface points
        self.reset(evalpts=True)

        # Set new delta value
        self._delta[1] = float(value)

    @property
    def delta(self):
        """ Evaluation delta for both u- and v-directions.

        Evaluation delta corresponds to the *step size* while ``evaluate()`` function iterates on the knot vector to
        generate surface points. Decreasing step size results in generation of more surface points.
        Therefore; smaller the delta value, smoother the surface.

        Please note that ``delta`` and ``sample_size`` properties correspond to the same variable with different
        descriptions. Therefore, setting ``delta`` will also set ``sample_size``.

        The following figure illustrates the working principles of the delta property:

        .. math::

            \\left[{{u_{0}},{u_{start}} + \\delta ,({u_{start}} + \\delta ) + \\delta , \\ldots ,{u_{end}}} \\right]

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets evaluation delta as a tuple of values corresponding to u- and v-directions
        :setter: Sets evaluation delta for both u- and v-directions
        :type: float
        """
        return self.delta_u, self.delta_v

    @delta.setter
    def delta(self, value):
        if isinstance(value, (int, float)):
            self.delta_u = value
            self.delta_v = value
        elif isinstance(value, (list, tuple)):
            if len(value) == 2:
                self.delta_u = value[0]
                self.delta_v = value[1]
            else:
                raise ValueError("Surface requires 2 delta values")
        else:
            raise ValueError("Cannot set delta. Please input a numeric value or a list or tuple with 2 numeric values")

    @property
    def tessellator(self):
        """ Tessellation component.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the tessellation component
        :setter: Sets the tessellation component
        """
        return self._tsl_component

    @tessellator.setter
    def tessellator(self, value):
        if not isinstance(value, tessellate.AbstractTessellate):
            warnings.warn("Tessellation component must be an instance of AbstractTessellate class")
            return

        self._tsl_component = value

    @property
    def vertices(self):
        """ Vertices generated by the tessellation operation.

        If the tessellation component is set to None, the result will be an empty list.

        :getter: Gets the vertices
        """
        if self.tessellator is None:
            return list()
        if not self.tessellator.is_tessellated():
            self.tessellate()
        return self.tessellator.vertices

    @property
    def faces(self):
        """ Faces (triangles, quads, etc.) generated by the tessellation operation.

        If the tessellation component is set to None, the result will be an empty list.

        :getter: Gets the faces
        """
        if self.tessellator is None:
            return list()
        if not self.tessellator.is_tessellated():
            self.tessellate()
        return self.tessellator.faces

    @property
    def trims(self):
        """ Curves for trimming the surface.

        Surface trims are 2-dimensional curves which are introduced on the parametric space of the surfaces. Trim curves
        can be a spline curve, an analytic curve or a 2-dimensional freeform shape. To visualize the trimmed surfaces,
        you need to use a tessellator that supports trimming. The following code snippet illustrates changing the default
        surface tessellator to the trimmed surface tessellator, :class:`.tessellate.TrimTessellate`.

        .. code-block:: python
            :linenos:

            from geomdl import tessellate

            # Assuming that "surf" variable stores the surface instance
            surf.tessellator = tessellate.TrimTessellate()

        In addition, using `trims` initialization argument of the visualization classes, trim curves can be visualized
        together with their underlying surfaces. Please refer to the visualization configuration class initialization
        arguments for more details.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the array of trim curves
        :setter: Sets the array of trim curves
        """
        return tuple(self._trims)

    @trims.setter
    def trims(self, value):
        # Input type validation
        if not isinstance(value, (list, tuple)):
            raise GeomdlException("'trims' setter only accepts a list or a tuple containing the trimming curves")
        # Trim curve validation
        for i, v in enumerate(value):
            try:
                self.add_trim(v)
            except GeomdlException:
                raise GeomdlException("Invalid geometry at index " + str(i))

    @property
    def data(self):
        """ Returns a dict which contains the geometry data.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.
        """
        return dict(
            type=self.type,
            rational=self.rational,
            dimension=self.dimension,
            pdimension=self.pdimension,
            delta=tuple(self._delta),
            sample_size=self.sample_size,
            precision=self._precision,
            degree=tuple(self._degree),
            knotvector=tuple(self._knot_vector),
            size=tuple(self._control_points_size),
            control_points=tuple(self._control_points),
            trims=tuple([t.data for t in self._trims])
        )

    def add_trim(self, trim):
        """ Adds a trim to the surface.

        A trim is a 2-dimensional curve defined on the parametric domain of the surface. Therefore, x-coordinate
        of the trimming curve corresponds to u parametric direction of the surfaceand y-coordinate of the trimming
        curve corresponds to v parametric direction of the surface.

        :attr:`trims` uses this method to add trims to the surface.

        :param trim: surface trimming curve
        :type trim: abstract.Geometry
        """
        if trim.dimension != 2:
            raise GeomdlException("Input geometry should be 2-dimensional")
        self._trims.append(trim)

    def set_ctrlpts(self, ctrlpts, *args, **kwargs):
        """ Sets the control points and checks if the data is consistent.

        This method is designed to provide a consistent way to set control points whether they are weighted or not.
        It directly sets the control points member of the class, and therefore it doesn't return any values.
        The input will be an array of coordinates. If you are working in the 3-dimensional space, then your coordinates
        will be an array of 3 elements representing *(x, y, z)* coordinates.

        .. note::

            The v index varies first. That is, a row of v control points for the first u value is found first.
            Then, the row of v control points for the next u value.

        :param ctrlpts: input control points as a list of coordinates
        :type ctrlpts: list
        :param args: number of control points corresponding to each parametric dimension
        :type args: tuple[int, int]
        """
        # Validate input
        for arg, degree in zip(args, self._degree):
            if degree <= 0:
                raise GeomdlException("Set the degree first")
            if arg < degree + 1:
                raise GeomdlException("Number of control points should be at least degree + 1")

        if len(ctrlpts[0]) < 2:
            raise GeomdlException("A surface should be at least 2-dimensional")

        if self.rational and len(ctrlpts[0]) < 3:
            raise GeomdlException("Rational surfaces expect weighted control points, e.g. (x * w, y * w, z * w, w)")

        # Clean up the surface and control points
        self.reset(evalpts=True, ctrlpts=True)

        # Call parent function
        super(Surface, self).set_ctrlpts(ctrlpts, *args, **kwargs)

    def render(self, **kwargs):
        """ Renders the surface using the visualization component.

        The visualization component must be set using :py:attr:`~vis` property before calling this method.

        Keyword Arguments:
            * ``cpcolor``: sets the color of the control points grid
            * ``evalcolor``: sets the color of the surface
            * ``trimcolor``: sets the color of the trim curves
            * ``filename``: saves the plot with the input name
            * ``plot``: controls plot window visibility. *Default: True*
            * ``animate``: activates animation (if supported). *Default: False*
            * ``extras``: adds line plots to the figure. *Default: None*
            * ``colormap``: sets the colormap of the surface

        The ``plot`` argument is useful when you would like to work on the command line without any window context.
        If ``plot`` flag is False, this method saves the plot as an image file (.png file where possible) and disables
        plot window popping out. If you don't provide a file name, the name of the image file will be pulled from the
        configuration class.

        ``extras`` argument can be used to add extra line plots to the figure. This argument expects a list of dicts
        in the format described below:

        .. code-block:: python
            :linenos:

            [
                dict(  # line plot 1
                    points=[[1, 2, 3], [4, 5, 6]],  # list of points
                    name="My line Plot 1",  # name displayed on the legend
                    color="red",   # color of the line plot
                    size=6.5  # size of the line plot
                ),
                dict(  # line plot 2
                    points=[[7, 8, 9], [10, 11, 12]],  # list of points
                    name="My line Plot 2",  # name displayed on the legend
                    color="navy",   # color of the line plot
                    size=12.5  # size of the line plot
                )
            ]

        Please note that ``colormap`` argument can only work with visualization classes that support colormaps. As an
        example, please see :py:class:`.VisMPL.VisSurfTriangle()` class documentation. This method expects a single
        colormap input.

        :return: the figure object
        """
        if not self._vis_component:
            warnings.warn("No visualization component has been set")
            return

        cpcolor = kwargs.pop('cpcolor', 'blue')
        evalcolor = kwargs.pop('evalcolor', 'green')
        bboxcolor = kwargs.pop('bboxcolor', 'darkorange')
        trimcolor = kwargs.pop('trimcolor', 'black')
        filename = kwargs.pop('filename', None)
        plot_visible = kwargs.pop('plot', True)
        extra_plots = kwargs.pop('extras', None)
        animate_plot = kwargs.pop('animate', False)
        force_tsl = bool(kwargs.pop('force', False))  # force re-tessellation

        # Get colormap and convert to a list
        surf_cmap = kwargs.get('colormap', None)
        surf_cmap = [surf_cmap] if surf_cmap else []

        # Check all parameters are set
        self._check_variables()

        # Check if the surface has been evaluated
        if self._eval_points is None or len(self._eval_points) == 0:
            self.evaluate()

        # Clear the visualization component
        self._vis_component.clear()

        # Add control points
        if self._vis_component.mconf['ctrlpts'] == 'points':
            self._vis_component.add(ptsarr=self.ctrlpts, name="control points", color=cpcolor, plot_type='ctrlpts')

        # Add control points as quads
        if self._vis_component.mconf['ctrlpts'] == 'quads':
            qtsl = tessellate.QuadTessellate()
            qtsl.tessellate(self.ctrlpts, size_u=self.ctrlpts_size_u, size_v=self.ctrlpts_size_v)
            self._vis_component.add(ptsarr=[qtsl.vertices, qtsl.faces],
                                    name="control points", color=cpcolor, plot_type='ctrlpts')

        # Add surface points
        if self._vis_component.mconf['evalpts'] == 'points':
            self._vis_component.add(ptsarr=self.evalpts, name=self.name, color=evalcolor, plot_type='evalpts')

        # Add surface points as quads
        if self._vis_component.mconf['evalpts'] == 'quads':
            qtsl = tessellate.QuadTessellate()
            qtsl.tessellate(self.evalpts, size_u=self.sample_size_u, size_v=self.sample_size_v)
            self._vis_component.add(ptsarr=[qtsl.vertices, qtsl.faces],
                                    name=self.name, color=evalcolor, plot_type='evalpts')

        # Add surface points as vertices and triangles
        if self._vis_component.mconf['evalpts'] == 'triangles':
            self.tessellate(force=force_tsl)
            self._vis_component.add(ptsarr=[self.tessellator.vertices, self.tessellator.faces],
                                    name=self.name, color=evalcolor, plot_type='evalpts')

        # Visualize the trim curve
        for idx, trim in enumerate(self._trims):
            self._vis_component.add(ptsarr=self.evaluate_list(trim.evalpts),
                                    name="Trim Curve " + str(idx + 1), color=trimcolor, plot_type='trimcurve')

        # Bounding box
        self._vis_component.add(ptsarr=self.bbox, name="Bounding Box", color=bboxcolor, plot_type='bbox')

        # User-defined plots
        if extra_plots is not None:
            for ep in extra_plots:
                self._vis_component.add(ptsarr=ep['points'], name=ep['name'],
                                        color=(ep['color'], ep['size']), plot_type='extras')

        # Data requested by the visualization module
        if self._vis_component.mconf['others']:
            vis_other = self._vis_component.mconf['others'].split(",")
            for vo in vis_other:
                vo_clean = vo.strip()
                # Send center point of the parametric space to the visualization module
                if vo_clean == "midpt":
                    midprm_u = (max(self.knotvector_u) + min(self.knotvector_u)) / 2.0
                    midprm_v = (max(self.knotvector_v) + min(self.knotvector_v)) / 2.0
                    midpt = self.evaluate_single((midprm_u, midprm_v))
                    self._vis_component.add(ptsarr=[midpt], plot_type=vo_clean)

        # Display the figure
        if animate_plot:
            return self._vis_component.animate(fig_save_as=filename, display_plot=plot_visible, colormap=surf_cmap)
        else:
            return self._vis_component.render(fig_save_as=filename, display_plot=plot_visible, colormap=surf_cmap)

    def tessellate(self, **kwargs):
        """ Tessellates the surface.

        Keyword arguments are directly passed to the tessellation component.
        """
        # Keyword arguments
        force_tessellate = kwargs.pop('force', False)  # force re-tessellation

        # No need to re-tessellate if we have already tessellated the surface
        if self._tsl_component.is_tessellated() and not force_tessellate:
            return

        # Remove duplicate elements from the kwargs dictionary
        kwlist = ["size_u", "size_v", "trims"]
        for kw in kwlist:
            if kw in kwargs:
                kwargs.pop(kw)

        # Call tessellation component for vertex and triangle generation
        self._tsl_component.tessellate(self.evalpts, size_u=self.sample_size_u, size_v=self.sample_size_v,
                                       trims=self.trims, **kwargs)

        # Re-evaluate vertex coordinates
        for idx in range(len(self._tsl_component.vertices)):
            uv = self._tsl_component.vertices[idx].uv
            if self._kv_normalize and not utilities.check_params(uv):
                continue
            self._tsl_component.vertices[idx].data = self.evaluate_single(uv)

    def reset(self, **kwargs):
        """ Resets control points and/or evaluated points.

        Keyword Arguments:
            * ``evalpts``: if True, then resets evaluated points
            * ``ctrlpts`` if True, then resets control points

        """
        reset_ctrlpts = kwargs.get('ctrlpts', False)
        reset_evalpts = kwargs.get('evalpts', False)

        if reset_ctrlpts:
            self._control_points = self._init_array()
            self._control_points_size[0] = 0
            self._control_points_size[1] = 0
            self._bounding_box = self._init_array()

        if reset_evalpts:
            self._eval_points = self._init_array()

        # Reset vertices and triangles
        self._tsl_component.reset()

    # Checks whether the surface evaluation is possible or not
    def _check_variables(self):
        works = True
        param_list = []
        if self.degree_u == 0:
            works = False
            param_list.append('degree_u')
        if self.degree_v == 0:
            works = False
            param_list.append('degree_v')
        if len(self._control_points) == 0:
            works = False
            param_list.append('ctrlpts')
        if len(self.knotvector_u) == 0:
            works = False
            param_list.append('knotvector_u')
        if len(self.knotvector_v) == 0:
            works = False
            param_list.append('knotvector_v')
        if not works:
            raise ValueError("Please set the following variables before evaluation: " + ",".join(param_list))

    @abc.abstractmethod
    def evaluate(self, **kwargs):
        """ Evaluates the parametric surface.

        .. note::

            This is an abstract method and it must be implemented in the subclass.

        """
        # Check all parameters are set before the evaluation
        self._check_variables()

    @abc.abstractmethod
    def evaluate_single(self, param):
        """ Evaluates the parametric surface at the given (u, v) parameter.

        .. note::

            This is an abstract method and it must be implemented in the subclass.

        :param param: parameter (u, v)
        """
        # Check all variables are set before the evaluation
        self._check_variables()

        if isinstance(param, (int, float)):
            param = [float(param) for _ in range(self.pdimension)]

        # Check parameters
        if self._kv_normalize:
            if not utilities.check_params(param):
                raise GeomdlException("Parameters should be between 0 and 1")

    @abc.abstractmethod
    def evaluate_list(self, param_list):
        """ Evaluates the parametric surface for an input range of (u, v) parameters.

        .. note::

            This is an abstract method and it must be implemented in the subclass.

        :param param_list: array of parameters (u, v)
        """
        # Check all parameters are set before the evaluation
        self._check_variables()

    @abc.abstractmethod
    def derivatives(self, u, v, order, **kwargs):
        """ Evaluates the derivatives of the parametric surface at parameter (u, v).

        .. note::

            This is an abstract method and it must be implemented in the subclass.

        :param u: parameter on the u-direction
        :type u: float
        :param v: parameter on the v-direction
        :type v: float
        :param order: derivative order
        :type order: int
        """
        # Check all variables are set before the evaluation
        self._check_variables()

        # Check parameters
        if self._kv_normalize:
            if not utilities.check_params([u, v]):
                raise GeomdlException("Parameters should be between 0 and 1")


@utl.add_metaclass(abc.ABCMeta)
class Volume(SplineGeometry):
    """ Abstract base class for defining spline volumes.

    Volume ABC is inherited from abc.ABCMeta class which is included in Python standard library by default. Due to
    differences between Python 2 and 3 on defining a metaclass, the compatibility module ``six`` is employed. Using
    ``six`` to set metaclass allows users to use the abstract classes in a correct way.

    The abstract base classes in this module are implemented using a feature called Python Properties. This feature
    allows users to use some of the functions as if they are class fields. You can also consider properties as a
    pythonic way to set getters and setters. You will see "getter" and "setter" descriptions on the documentation of
    these properties.

    The Volume ABC allows users to set the *FindSpan* function to be used in evaluations with ``find_span_func``
    keyword as an input to the class constructor. NURBS-Python includes a binary and a linear search variation of the
    FindSpan function in the ``helpers`` module.
    You may also implement and use your own *FindSpan* function. Please see the ``helpers`` module for details.

    Code segment below illustrates a possible implementation of Volume abstract base class:

    .. code-block:: python
        :linenos:

        from geomdl import abstract

        class MyVolumeClass(abstract.Volume):
            def __init__(self, **kwargs):
            super(MyVolumeClass, self).__init__(**kwargs)
            # Add your constructor code here

            def evaluate(self, **kwargs):
                # Implement this function
                pass

            def evaluate_single(self, uvw):
                # Implement this function
                pass

            def evaluate_list(self, uvw_list):
                # Implement this function
                pass

    The properties and functions defined in the abstract base class will be automatically available in the subclasses.

    **Keyword Arguments:**

    * ``id``: object ID (as integer)
    * ``precision``: number of decimal places to round to. *Default: 18*
    * ``normalize_kv``: if True, knot vector(s) will be normalized to [0,1] domain. *Default: True*
    * ``find_span_func``: default knot span finding algorithm. *Default:* :func:`.helpers.find_span_linear`
    """

    def __init__(self, **kwargs):
        self._pdim = 3  # number of parametric directions
        self._dinit = 0.1  # evaluation delta init value
        self._name = "volume"  # object name
        super(Volume, self).__init__(**kwargs)
        self._trims = self._init_array()  # trimming surfaces

    @property
    def order_u(self):
        """ Order for the u-direction.

        Defined as ``order = degree + 1``

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the surface order for u-direction
        :setter: Sets the surface order for u-direction
        :type: int
        """
        return self.degree_u + 1

    @order_u.setter
    def order_u(self, value):
        self.degree_u = value - 1

    @property
    def order_v(self):
        """ Order for the v-direction.

        Defined as ``order = degree + 1``

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the surface order for v-direction
        :setter: Sets the surface order for v-direction
        :type: int
        """
        return self.degree_v + 1

    @order_v.setter
    def order_v(self, value):
        self.degree_v = value - 1

    @property
    def order_w(self):
        """ Order for the w-direction.

        Defined as ``order = degree + 1``

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the surface order for v-direction
        :setter: Sets the surface order for v-direction
        :type: int
        """
        return self.degree_w + 1

    @order_w.setter
    def order_w(self, value):
        self.degree_w = value - 1

    @property
    def degree(self):
        """ Degree for u-, v- and w-directions

        :getter: Gets the degree
        :setter: Sets the degree
        :type: list
        """
        return self._degree

    @degree.setter
    def degree(self, value):
        if not isinstance(value, (list, tuple)):
            raise ValueError("Please input a list with a length of " + str(self.pdimension))
        self.degree_u = value[0]
        self.degree_v = value[1]
        self.degree_w = value[2]

    @property
    def degree_u(self):
        """ Degree for the u-direction.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets degree for the u-direction
        :setter: Sets degree for the u-direction
        :type: int
        """
        return self._degree[0]

    @degree_u.setter
    def degree_u(self, value):
        val = int(value)
        if val <= 0:
            raise ValueError("Degree cannot be less than zero")
        # Clean up the surface points
        self.reset(evalpts=True)
        # Set degree u
        self._degree[0] = int(value)

    @property
    def degree_v(self):
        """ Degree for the v-direction.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets degree for the v-direction
        :setter: Sets degree for the v-direction
        :type: int
        """
        return self._degree[1]

    @degree_v.setter
    def degree_v(self, value):
        val = int(value)
        if val <= 0:
            raise ValueError("Degree cannot be less than zero")
        # Clean up the surface points
        self.reset(evalpts=True)
        # Set degree v
        self._degree[1] = val

    @property
    def degree_w(self):
        """ Degree for the w-direction.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets degree for the w-direction
        :setter: Sets degree for the w-direction
        :type: int
        """
        return self._degree[2]

    @degree_w.setter
    def degree_w(self, value):
        val = int(value)
        if val <= 0:
            raise ValueError("Degree cannot be less than zero")
        # Clean up the surface points
        self.reset(evalpts=True)
        # Set degree v
        self._degree[2] = val

    @property
    def knotvector(self):
        """ Knot vector for u-, v- and w-directions

        :getter: Gets the knot vector
        :setter: Sets the knot vector
        :type: list
        """
        return self._knot_vector

    @knotvector.setter
    def knotvector(self, value):
        if not isinstance(value, (list, tuple)):
            raise ValueError("Please input a list with a length of " + str(self.pdimension))
        self.knotvector_u = value[0]
        self.knotvector_v = value[1]
        self.knotvector_w = value[2]

    @property
    def knotvector_u(self):
        """ Knot vector for the u-direction.

        The knot vector will be normalized to [0, 1] domain if the class is initialized with ``normalize_kv=True``
        argument.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets knot vector for the u-direction
        :setter: Sets knot vector for the u-direction
        :type: list
        """
        return self._knot_vector[0]

    @knotvector_u.setter
    def knotvector_u(self, value):
        if self.degree_u == 0 or self.ctrlpts_size_u == 0:
            raise ValueError("Set degree and control points first on the u-direction")

        # Check knot vector validity
        if not knotvector.check(self.degree_u, value, self.ctrlpts_size_u):
            raise ValueError("Input is not a valid knot vector on the u-direction")

        # Clean up the surface points
        self.reset(evalpts=True)

        # Set knot vector
        self._knot_vector[0] = knotvector.normalize(value, decimals=self._precision) if self._kv_normalize else value

    @property
    def knotvector_v(self):
        """ Knot vector for the v-direction.

        The knot vector will be normalized to [0, 1] domain if the class is initialized with ``normalize_kv=True``
        argument.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets knot vector for the v-direction
        :setter: Sets knot vector for the v-direction
        :type: list
        """
        return self._knot_vector[1]

    @knotvector_v.setter
    def knotvector_v(self, value):
        if self.degree_v == 0 or self.ctrlpts_size_v == 0:
            raise ValueError("Set degree and control points first on the v-direction")

        # Check knot vector validity
        if not knotvector.check(self.degree_v, value, self.ctrlpts_size_v):
            raise ValueError("Input is not a valid knot vector on the v-direction")

        # Clean up the surface points
        self.reset(evalpts=True)

        # Set knot vector
        self._knot_vector[1] = knotvector.normalize(value, decimals=self._precision) if self._kv_normalize else value

    @property
    def knotvector_w(self):
        """ Knot vector for the w-direction.

        The knot vector will be normalized to [0, 1] domain if the class is initialized with ``normalize_kv=True``
        argument.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets knot vector for the w-direction
        :setter: Sets knot vector for the w-direction
        :type: list
        """
        return self._knot_vector[2]

    @knotvector_w.setter
    def knotvector_w(self, value):
        if self.degree_w == 0 or self.ctrlpts_size_w == 0:
            raise ValueError("Set degree and control points first for the w-direction")

        # Check knot vector validity
        if not knotvector.check(self.degree_w, value, self.ctrlpts_size_w):
            raise ValueError("Input is not a valid knot vector for the w-direction")

        # Clean up the surface points
        self.reset(evalpts=True)

        # Set knot vector
        self._knot_vector[2] = knotvector.normalize(value, decimals=self._precision) if self._kv_normalize else value

    @property
    def ctrlpts(self):
        """ 1-dimensional array of control points.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the control points
        :setter: Sets the control points
        :type: list
        """
        return self._control_points

    @ctrlpts.setter
    def ctrlpts(self, value):
        if self.ctrlpts_size_u <= 0 or self.ctrlpts_size_v <= 0 or self.ctrlpts_size_w <= 0:
            raise ValueError("Please set the number of control points on the u-, v- and w-directions")
        self.set_ctrlpts(value, self.ctrlpts_size_u, self.ctrlpts_size_v, self.ctrlpts_size_w)

    @property
    def ctrlpts_size_u(self):
        """ Number of control points for the u-direction.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets number of control points for the u-direction
        :setter: Sets number of control points for the u-direction
        """
        return self._control_points_size[0]

    @ctrlpts_size_u.setter
    def ctrlpts_size_u(self, value):
        if not isinstance(value, int):
            raise TypeError("Number of control points for the u-direction must be an integer number")
        if value <= 0:
            raise ValueError("Control points size cannot be less than and equal to zero")

        # Assume that user is doing this right
        self._control_points_size[0] = value

    @property
    def ctrlpts_size_v(self):
        """ Number of control points for the v-direction.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets number of control points for the v-direction
        :setter: Sets number of control points for the v-direction
        """
        return self._control_points_size[1]

    @ctrlpts_size_v.setter
    def ctrlpts_size_v(self, value):
        if not isinstance(value, int):
            raise TypeError("Number of control points for the v-direction must be an integer number")
        if value <= 0:
            raise ValueError("Control points size cannot be less than and equal to zero")

        # Assume that user is doing this right
        self._control_points_size[1] = value

    @property
    def ctrlpts_size_w(self):
        """ Number of control points for the w-direction.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets number of control points for the w-direction
        :setter: Sets number of control points for the w-direction
        """
        return self._control_points_size[2]

    @ctrlpts_size_w.setter
    def ctrlpts_size_w(self, value):
        if not isinstance(value, int):
            raise TypeError("Number of control points for the w-direction must be an integer number")
        if value <= 0:
            raise ValueError("Control points size cannot be less than and equal to zero")

        # Assume that user is doing this right
        self._control_points_size[2] = value

    @property
    def sample_size_u(self):
        """ Sample size for the u-direction.

        Sample size defines the number of evaluated points to generate. It also sets the ``delta_u`` property.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets sample size for the u-direction
        :setter: Sets sample size for the u-direction
        :type: int
        """
        ss = math.floor((1.0 / self.delta_u) + 0.5)
        return int(ss)

    @sample_size_u.setter
    def sample_size_u(self, value):
        if not isinstance(value, int):
            raise ValueError("Sample size must be an integer value")

        if (self.knotvector_u is None or len(self.knotvector_u) == 0) or self.degree_u == 0:
            warnings.warn("Cannot determine 'delta_u' value. Please set knot vectors and degrees before sample size.")
            return

        # To make it operate like linspace, we have to know the starting and ending points.
        start_u = self.knotvector_u[self.degree_u]
        stop_u = self.knotvector_u[-(self.degree_u + 1)]

        # Set delta values
        self.delta_u = (stop_u - start_u) / float(value)

    @property
    def sample_size_v(self):
        """ Sample size for the v-direction.

        Sample size defines the number of evaluated points to generate. It also sets the ``delta_v`` property.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets sample size for the v-direction
        :setter: Sets sample size for the v-direction
        :type: int
        """
        ss = math.floor((1.0 / self.delta_v) + 0.5)
        return int(ss)

    @sample_size_v.setter
    def sample_size_v(self, value):
        if not isinstance(value, int):
            raise ValueError("Sample size must be an integer value")

        if (self.knotvector_v is None or len(self.knotvector_v) == 0) or self.degree_v == 0:
            warnings.warn("Cannot determine 'delta_v' value. Please set knot vectors and degrees before sample size.")
            return

        # To make it operate like linspace, we have to know the starting and ending points.
        start_v = self.knotvector_v[self.degree_v]
        stop_v = self.knotvector_v[-(self.degree_v + 1)]

        # Set delta values
        self.delta_v = (stop_v - start_v) / float(value)

    @property
    def sample_size_w(self):
        """ Sample size for the w-direction.

        Sample size defines the number of evaluated points to generate. It also sets the ``delta_w`` property.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets sample size for the w-direction
        :setter: Sets sample size for the w-direction
        :type: int
        """
        ss = math.floor((1.0 / self.delta_w) + 0.5)
        return int(ss)

    @sample_size_w.setter
    def sample_size_w(self, value):
        if not isinstance(value, int):
            raise ValueError("Sample size must be an integer value")

        if (self.knotvector_w is None or len(self.knotvector_w) == 0) or self.degree_w == 0:
            warnings.warn("Cannot determine 'delta_w' value. Please set knot vectors and degrees before sample size.")
            return

        # To make it operate like linspace, we have to know the starting and ending points.
        start_w = self.knotvector_w[self.degree_w]
        stop_w = self.knotvector_w[-(self.degree_w + 1)]

        # Set delta values
        self.delta_w = (stop_w - start_w) / float(value)

    @property
    def sample_size(self):
        """ Sample size for both u- and v-directions.

        Sample size defines the number of surface points to generate. It also sets the ``delta`` property.

        The following figure illustrates the working principles of sample size property:

        .. math::

            \\underbrace {\\left[ {{u_{start}}, \\ldots ,{u_{end}}} \\right]}_{{n_{sample}}}

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets sample size as a tuple of values corresponding to u-, v- and w-directions
        :setter: Sets sample size value for both u-, v- and w-directions
        :type: int
        """
        sample_size_u = math.floor((1.0 / self.delta_u) + 0.5)
        sample_size_v = math.floor((1.0 / self.delta_v) + 0.5)
        sample_size_w = math.floor((1.0 / self.delta_w) + 0.5)
        return int(sample_size_u), int(sample_size_v), int(sample_size_w)

    @sample_size.setter
    def sample_size(self, value):
        if (self.knotvector_u is None or len(self.knotvector_u) == 0) or self.degree_u == 0 or \
                (self.knotvector_v is None or len(self.knotvector_v) == 0 or self.degree_v == 0) or \
                (self.knotvector_w is None or len(self.knotvector_w) == 0 or self.degree_w == 0):
            warnings.warn("Cannot determine 'delta' value. Please set knot vectors and degrees before sample size.")
            return

        # To make it operate like linspace, we have to know the starting and ending points.
        start_u = self.knotvector_u[self.degree_u]
        stop_u = self.knotvector_u[-(self.degree_u + 1)]
        start_v = self.knotvector_v[self.degree_v]
        stop_v = self.knotvector_v[-(self.degree_v + 1)]
        start_w = self.knotvector_w[self.degree_w]
        stop_w = self.knotvector_w[-(self.degree_w + 1)]

        # Set delta values
        self.delta_u = (stop_u - start_u) / float(value)
        self.delta_v = (stop_v - start_v) / float(value)
        self.delta_w = (stop_w - start_w) / float(value)

    @property
    def delta_u(self):
        """ Evaluation delta for the u-direction.

        Evaluation delta corresponds to the *step size* while ``evaluate()`` function iterates on the knot vector to
        generate surface points. Decreasing step size results in generation of more surface points.
        Therefore; smaller the delta value, smoother the surface.

        Please note that ``delta_u`` and ``sample_size_u`` properties correspond to the same variable with different
        descriptions. Therefore, setting ``delta_u`` will also set ``sample_size_u``.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets evaluation delta for the u-direction
        :setter: Sets evaluation delta for the u-direction
        :type: float
        """
        return self._delta[0]

    @delta_u.setter
    def delta_u(self, value):
        # Delta value should be between 0 and 1
        if float(value) <= 0 or float(value) >= 1:
            raise ValueError("Evaluation delta (u-direction) must be between 0.0 and 1.0")

        # Clean up evaluated points
        self.reset(evalpts=True)

        # Set new delta value
        self._delta[0] = float(value)

    @property
    def delta_v(self):
        """ Evaluation delta for the v-direction.

        Evaluation delta corresponds to the *step size* while ``evaluate()`` function iterates on the knot vector to
        generate surface points. Decreasing step size results in generation of more surface points.
        Therefore; smaller the delta value, smoother the surface.

        Please note that ``delta_v`` and ``sample_size_v`` properties correspond to the same variable with different
        descriptions. Therefore, setting ``delta_v`` will also set ``sample_size_v``.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets evaluation delta for the v-direction
        :setter: Sets evaluation delta for the v-direction
        :type: float
        """
        return self._delta[1]

    @delta_v.setter
    def delta_v(self, value):
        # Delta value should be between 0 and 1
        if float(value) <= 0 or float(value) >= 1:
            raise ValueError("Evaluation delta (v-direction) should be between 0.0 and 1.0")

        # Clean up evaluated points
        self.reset(evalpts=True)

        # Set new delta value
        self._delta[1] = float(value)

    @property
    def delta_w(self):
        """ Evaluation delta for the w-direction.

        Evaluation delta corresponds to the *step size* while ``evaluate()`` function iterates on the knot vector to
        generate surface points. Decreasing step size results in generation of more surface points.
        Therefore; smaller the delta value, smoother the surface.

        Please note that ``delta_w`` and ``sample_size_w`` properties correspond to the same variable with different
        descriptions. Therefore, setting ``delta_w`` will also set ``sample_size_w``.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets evaluation delta for the w-direction
        :setter: Sets evaluation delta for the w-direction
        :type: float
        """
        return self._delta[2]

    @delta_w.setter
    def delta_w(self, value):
        # Delta value should be between 0 and 1
        if float(value) <= 0 or float(value) >= 1:
            raise ValueError("Evaluation delta (w-direction) should be between 0.0 and 1.0")

        # Clean up evaluated points
        self.reset(evalpts=True)

        # Set new delta value
        self._delta[2] = float(value)

    @property
    def delta(self):
        """ Evaluation delta for u-, v- and w-directions.

        Evaluation delta corresponds to the *step size* while ``evaluate()`` function iterates on the knot vector to
        generate surface points. Decreasing step size results in generation of more surface points.
        Therefore; smaller the delta value, smoother the surface.

        Please note that ``delta`` and ``sample_size`` properties correspond to the same variable with different
        descriptions. Therefore, setting ``delta`` will also set ``sample_size``.

        The following figure illustrates the working principles of the delta property:

        .. math::

            \\left[{{u_{0}},{u_{start}} + \\delta ,({u_{start}} + \\delta ) + \\delta , \\ldots ,{u_{end}}} \\right]

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets evaluation delta as a tuple of values corresponding to u-, v- and w-directions
        :setter: Sets evaluation delta for u-, v- and w-directions
        :type: float
        """
        return self.delta_u, self.delta_v, self.delta_w

    @delta.setter
    def delta(self, value):
        if isinstance(value, (int, float)):
            self.delta_u = value
            self.delta_v = value
            self.delta_w = value
        elif isinstance(value, (list, tuple)):
            if len(value) == 3:
                self.delta_u = value[0]
                self.delta_v = value[1]
                self.delta_w = value[2]
            else:
                raise ValueError("Surface requires 3 delta values")
        else:
            raise ValueError("Cannot set delta. Please input a numeric value or a list or tuple with 3 numeric values")

    @property
    def trims(self):
        """ Trimming surfaces.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.

        :getter: Gets the array of trim surfaces
        :setter: Sets the array of trim surfaces
        """
        return tuple(self._trims)

    @trims.setter
    def trims(self, value):
        # Input type validation
        if not isinstance(value, (list, tuple)):
            raise GeomdlException("'trims' setter only accepts a list or a tuple containing the trimming surfaces")
        # Trim curve validation
        for i, v in enumerate(value):
            try:
                self.add_trim(v)
            except GeomdlException:
                raise GeomdlException("Invalid geometry at index " + str(i))

    @property
    def data(self):
        """ Returns a dict which contains the geometry data.

        Please refer to the `wiki <https://github.com/orbingol/NURBS-Python/wiki/Using-Python-Properties>`_ for details
        on using this class member.
        """
        return dict(
            type=self.type,
            rational=self.rational,
            dimension=self.dimension,
            pdimension=self.pdimension,
            delta=tuple(self._delta),
            sample_size=self.sample_size,
            precision=self._precision,
            degree=tuple(self._degree),
            knotvector=tuple(self._knot_vector),
            size=tuple(self._control_points_size),
            control_points=tuple(self._control_points),
            trims=tuple([t.data for t in self._trims])
        )

    def add_trim(self, trim):
        """ Adds a trim to the volume.

        :attr:`trims` uses this method to add trims to the volume.

        :param trim: trimming surface
        :type trim: abstract.Surface
        """
        if trim.dimension != 3:
            raise GeomdlException("Input geometry should be 3-dimensional")
        self._trims.append(trim)

    def reset(self, **kwargs):
        """ Resets control points and/or evaluated points.

        Keyword Arguments:
            * ``evalpts``: if True, then resets evaluated points
            * ``ctrlpts`` if True, then resets control points

        """
        reset_ctrlpts = kwargs.get('ctrlpts', False)
        reset_evalpts = kwargs.get('evalpts', False)

        if reset_ctrlpts:
            self._control_points = self._init_array()
            self._control_points_size = [0, 0, 0]
            self._bounding_box = self._init_array()

        if reset_evalpts:
            self._eval_points = self._init_array()

    def _check_variables(self):
        """ Checks whether the evaluation is possible or not. """
        works = True
        param_list = []
        if self.degree_u == 0:
            works = False
            param_list.append('degree_u')
        if self.degree_v == 0:
            works = False
            param_list.append('degree_v')
        if self.degree_w == 0:
            works = False
            param_list.append('degree_w')
        if self._control_points is None or len(self._control_points) == 0:
            works = False
            param_list.append('ctrlpts')
        if self.knotvector_u is None or len(self.knotvector_u) == 0:
            works = False
            param_list.append('knotvector_u')
        if self.knotvector_v is None or len(self.knotvector_v) == 0:
            works = False
            param_list.append('knotvector_v')
        if self.knotvector_w is None or len(self.knotvector_w) == 0:
            works = False
            param_list.append('knotvector_w')
        if not works:
            raise ValueError("Please set the following variables before evaluation: " + ",".join(param_list))

    def set_ctrlpts(self, ctrlpts, *args, **kwargs):
        """ Sets the control points and checks if the data is consistent.

        This method is designed to provide a consistent way to set control points whether they are weighted or not.
        It directly sets the control points member of the class, and therefore it doesn't return any values.
        The input will be an array of coordinates. If you are working in the 3-dimensional space, then your coordinates
        will be an array of 3 elements representing *(x, y, z)* coordinates.

        :param ctrlpts: input control points as a list of coordinates
        :type ctrlpts: list
        :param args: number of control points corresponding to each parametric dimension
        :type args: tuple[int, int, int]
        """
        # Validate input
        for arg, degree in zip(args, self._degree):
            if degree <= 0:
                raise GeomdlException("Set the degree first")
            if arg < degree + 1:
                raise GeomdlException("Number of control points should be at least degree + 1")

        if len(ctrlpts[0]) < 3:
            raise GeomdlException("A volume should be at least 3-dimensional")

        if self.rational and len(ctrlpts[0]) < 4:
            raise GeomdlException("Rational volumes expect weighted control points, e.g. (x * w, y * w, z * w, w)")

        # Clean up the surface and control points
        self.reset(evalpts=True, ctrlpts=True)

        # Call parent function
        super(Volume, self).set_ctrlpts(ctrlpts, *args, **kwargs)

    def render(self, **kwargs):
        """ Renders the volume using the visualization component.

        The visualization component must be set using :py:attr:`~vis` property before calling this method.

        Keyword Arguments:
            * ``cpcolor``: sets the color of the control points
            * ``evalcolor``: sets the color of the volume
            * ``filename``: saves the plot with the input name
            * ``plot``: controls plot window visibility. *Default: True*
            * ``animate``: activates animation (if supported). *Default: False*
            * ``grid_size``: grid size for voxelization. *Default: (8, 8, 8)*
            * ``use_cubes``: use cube voxels instead of cuboid ones. *Default: False*
            * ``num_procs``: number of concurrent processes for voxelization. *Default: 1*

        The ``plot`` argument is useful when you would like to work on the command line without any window context.
        If ``plot`` flag is False, this method saves the plot as an image file (.png file where possible) and disables
        plot window popping out. If you don't provide a file name, the name of the image file will be pulled from the
        configuration class.

        ``extras`` argument can be used to add extra line plots to the figure. This argument expects a list of dicts
        in the format described below:

        .. code-block:: python
            :linenos:

            [
                dict(  # line plot 1
                    points=[[1, 2, 3], [4, 5, 6]],  # list of points
                    name="My line Plot 1",  # name displayed on the legend
                    color="red",   # color of the line plot
                    size=6.5  # size of the line plot
                ),
                dict(  # line plot 2
                    points=[[7, 8, 9], [10, 11, 12]],  # list of points
                    name="My line Plot 2",  # name displayed on the legend
                    color="navy",   # color of the line plot
                    size=12.5  # size of the line plot
                )
            ]

        :return: the figure object
        """
        if not self._vis_component:
            warnings.warn("No visualization component has been set")
            return

        cpcolor = kwargs.pop('cpcolor', 'blue')
        evalcolor = kwargs.pop('evalcolor', 'green')
        bboxcolor = kwargs.pop('bboxcolor', 'darkorange')
        filename = kwargs.pop('filename', None)
        plot_visible = kwargs.pop('plot', True)
        extra_plots = kwargs.pop('extras', None)
        animate_plot = kwargs.pop('animate', False)

        # Check all parameters are set
        self._check_variables()

        # Check if the volume has been evaluated
        if self._eval_points is None or len(self._eval_points) == 0:
            self.evaluate()

        # Clear the visualization component
        self._vis_component.clear()

        # Add control points
        if self._vis_component.mconf['ctrlpts'] == 'points':
            self._vis_component.add(ptsarr=self.ctrlpts, name="control points", color=cpcolor, plot_type='ctrlpts')

        # Add evaluated points
        if self._vis_component.mconf['evalpts'] == 'points':
            self._vis_component.add(ptsarr=self.evalpts, name=self.name, color=evalcolor, plot_type='evalpts')

        # Add evaluated points as voxels
        if self._vis_component.mconf['evalpts'] == 'voxels':
            grid, filled = voxelize.voxelize(self, **kwargs)
            faces = voxelize.convert_bb_to_faces(grid)
            self._vis_component.add(ptsarr=[grid, faces, filled], name=self.name, color=evalcolor, plot_type='evalpts')

        # Bounding box
        self._vis_component.add(ptsarr=self.bbox, name="Bounding Box", color=bboxcolor, plot_type='bbox')

        # User-defined plots
        if extra_plots is not None:
            for ep in extra_plots:
                self._vis_component.add(ptsarr=ep['points'], name=ep['name'],
                                        color=(ep['color'], ep['size']), plot_type='extras')

        # Data requested by the visualization module
        if self._vis_component.mconf['others']:
            vis_other = self._vis_component.mconf['others'].split(",")
            for vo in vis_other:
                vo_clean = vo.strip()
                # Send center point of the parametric space to the visualization module
                if vo_clean == "midpt":
                    midprm_u = (max(self.knotvector_u) + min(self.knotvector_u)) / 2.0
                    midprm_v = (max(self.knotvector_v) + min(self.knotvector_v)) / 2.0
                    midprm_w = (max(self.knotvector_w) + min(self.knotvector_w)) / 2.0
                    midpt = self.evaluate_single((midprm_u, midprm_v, midprm_w))
                    self._vis_component.add(ptsarr=[midpt], plot_type=vo_clean)

        # Display the figure
        if animate_plot:
            return self._vis_component.animate(fig_save_as=filename, display_plot=plot_visible)
        else:
            return self._vis_component.render(fig_save_as=filename, display_plot=plot_visible)

    @abc.abstractmethod
    def evaluate(self, **kwargs):
        """ Evaluates the parametric volume.

        .. note::

            This is an abstract method and it must be implemented in the subclass.

        """
        # Check all parameters are set before the evaluation
        self._check_variables()

    @abc.abstractmethod
    def evaluate_single(self, param):
        """ Evaluates the parametric surface at the given (u, v, w) parameter.

        .. note::

            This is an abstract method and it must be implemented in the subclass.

        :param param: parameter pair (u, v, w)
        """
        # Check all parameters are set before the evaluation
        self._check_variables()

        if isinstance(param, (int, float)):
            param = [float(param) for _ in range(self.pdimension)]

        # Check parameters
        if self._kv_normalize:
            if not utilities.check_params(param):
                raise GeomdlException("Parameters should be between 0 and 1")

    @abc.abstractmethod
    def evaluate_list(self, param_list):
        """ Evaluates the parametric volume for an input range of (u, v, w) parameter pairs.

        .. note::

            This is an abstract method and it must be implemented in the subclass.

        :param param_list: array of parameter pairs (u, v, w)
        """
        # Check all parameters are set before the evaluation
        self._check_variables()