point.py

from __future__ import print_function, division
from .vector import Vector, _check_vector
from .frame import _check_frame
from warnings import warn

__all__ = ['Point']


class Point(object):
    """This object represents a point in a dynamic system.

    It stores the: position, velocity, and acceleration of a point.
    The position is a vector defined as the vector distance from a parent
    point to this point.

    Parameters
    ==========

    name : string
        The display name of the Point

    Examples
    ========

    >>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
    >>> from sympy.physics.vector import init_vprinting
    >>> init_vprinting(pretty_print=False)
    >>> N = ReferenceFrame('N')
    >>> O = Point('O')
    >>> P = Point('P')
    >>> u1, u2, u3 = dynamicsymbols('u1 u2 u3')
    >>> O.set_vel(N, u1 * N.x + u2 * N.y + u3 * N.z)
    >>> O.acc(N)
    u1'*N.x + u2'*N.y + u3'*N.z

    symbols() can be used to create multiple Points in a single step, for example:

    >>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
    >>> from sympy.physics.vector import init_vprinting
    >>> init_vprinting(pretty_print=False)
    >>> from sympy import symbols
    >>> N = ReferenceFrame('N')
    >>> u1, u2 = dynamicsymbols('u1 u2')
    >>> A, B = symbols('A B', cls=Point)
    >>> type(A)
    <class 'sympy.physics.vector.point.Point'>
    >>> A.set_vel(N, u1 * N.x + u2 * N.y)
    >>> B.set_vel(N, u2 * N.x + u1 * N.y)
    >>> A.acc(N) - B.acc(N)
    (u1' - u2')*N.x + (-u1' + u2')*N.y

    """

    def __init__(self, name):
        """Initialization of a Point object. """
        self.name = name
        self._pos_dict = {}
        self._vel_dict = {}
        self._acc_dict = {}
        self._pdlist = [self._pos_dict, self._vel_dict, self._acc_dict]

    def __str__(self):
        return self.name

    __repr__ = __str__

    def _check_point(self, other):
        if not isinstance(other, Point):
            raise TypeError('A Point must be supplied')

    def _pdict_list(self, other, num):
        """Returns a list of points that gives the shortest path with respect
        to position, velocity, or acceleration from this point to the provided
        point.

        Parameters
        ==========
        other : Point
            A point that may be related to this point by position, velocity, or
            acceleration.
        num : integer
            0 for searching the position tree, 1 for searching the velocity
            tree, and 2 for searching the acceleration tree.

        Returns
        =======
        list of Points
            A sequence of points from self to other.

        Notes
        =====

        It isn't clear if num = 1 or num = 2 actually works because the keys to
        ``_vel_dict`` and ``_acc_dict`` are :class:`ReferenceFrame` objects which
        do not have the ``_pdlist`` attribute.

        """
        outlist = [[self]]
        oldlist = [[]]
        while outlist != oldlist:
            oldlist = outlist[:]
            for i, v in enumerate(outlist):
                templist = v[-1]._pdlist[num].keys()
                for i2, v2 in enumerate(templist):
                    if not v.__contains__(v2):
                        littletemplist = v + [v2]
                        if not outlist.__contains__(littletemplist):
                            outlist.append(littletemplist)
        for i, v in enumerate(oldlist):
            if v[-1] != other:
                outlist.remove(v)
        outlist.sort(key=len)
        if len(outlist) != 0:
            return outlist[0]
        raise ValueError('No Connecting Path found between ' + other.name +
                         ' and ' + self.name)

    def a1pt_theory(self, otherpoint, outframe, interframe):
        """Sets the acceleration of this point with the 1-point theory.

        The 1-point theory for point acceleration looks like this:

        ^N a^P = ^B a^P + ^N a^O + ^N alpha^B x r^OP + ^N omega^B x (^N omega^B
        x r^OP) + 2 ^N omega^B x ^B v^P

        where O is a point fixed in B, P is a point moving in B, and B is
        rotating in frame N.

        Parameters
        ==========

        otherpoint : Point
            The first point of the 1-point theory (O)
        outframe : ReferenceFrame
            The frame we want this point's acceleration defined in (N)
        fixedframe : ReferenceFrame
            The intermediate frame in this calculation (B)

        Examples
        ========

        >>> from sympy.physics.vector import Point, ReferenceFrame
        >>> from sympy.physics.vector import dynamicsymbols
        >>> from sympy.physics.vector import init_vprinting
        >>> init_vprinting(pretty_print=False)
        >>> q = dynamicsymbols('q')
        >>> q2 = dynamicsymbols('q2')
        >>> qd = dynamicsymbols('q', 1)
        >>> q2d = dynamicsymbols('q2', 1)
        >>> N = ReferenceFrame('N')
        >>> B = ReferenceFrame('B')
        >>> B.set_ang_vel(N, 5 * B.y)
        >>> O = Point('O')
        >>> P = O.locatenew('P', q * B.x)
        >>> P.set_vel(B, qd * B.x + q2d * B.y)
        >>> O.set_vel(N, 0)
        >>> P.a1pt_theory(O, N, B)
        (-25*q + q'')*B.x + q2''*B.y - 10*q'*B.z

        """

        _check_frame(outframe)
        _check_frame(interframe)
        self._check_point(otherpoint)
        dist = self.pos_from(otherpoint)
        v = self.vel(interframe)
        a1 = otherpoint.acc(outframe)
        a2 = self.acc(interframe)
        omega = interframe.ang_vel_in(outframe)
        alpha = interframe.ang_acc_in(outframe)
        self.set_acc(outframe, a2 + 2 * (omega ^ v) + a1 + (alpha ^ dist) +
                (omega ^ (omega ^ dist)))
        return self.acc(outframe)

    def a2pt_theory(self, otherpoint, outframe, fixedframe):
        """Sets the acceleration of this point with the 2-point theory.

        The 2-point theory for point acceleration looks like this:

        ^N a^P = ^N a^O + ^N alpha^B x r^OP + ^N omega^B x (^N omega^B x r^OP)

        where O and P are both points fixed in frame B, which is rotating in
        frame N.

        Parameters
        ==========

        otherpoint : Point
            The first point of the 2-point theory (O)
        outframe : ReferenceFrame
            The frame we want this point's acceleration defined in (N)
        fixedframe : ReferenceFrame
            The frame in which both points are fixed (B)

        Examples
        ========

        >>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
        >>> from sympy.physics.vector import init_vprinting
        >>> init_vprinting(pretty_print=False)
        >>> q = dynamicsymbols('q')
        >>> qd = dynamicsymbols('q', 1)
        >>> N = ReferenceFrame('N')
        >>> B = N.orientnew('B', 'Axis', [q, N.z])
        >>> O = Point('O')
        >>> P = O.locatenew('P', 10 * B.x)
        >>> O.set_vel(N, 5 * N.x)
        >>> P.a2pt_theory(O, N, B)
        - 10*q'**2*B.x + 10*q''*B.y

        """

        _check_frame(outframe)
        _check_frame(fixedframe)
        self._check_point(otherpoint)
        dist = self.pos_from(otherpoint)
        a = otherpoint.acc(outframe)
        omega = fixedframe.ang_vel_in(outframe)
        alpha = fixedframe.ang_acc_in(outframe)
        self.set_acc(outframe, a + (alpha ^ dist) + (omega ^ (omega ^ dist)))
        return self.acc(outframe)

    def acc(self, frame):
        """The acceleration Vector of this Point in a ReferenceFrame.

        Parameters
        ==========

        frame : ReferenceFrame
            The frame in which the returned acceleration vector will be defined in

        Examples
        ========

        >>> from sympy.physics.vector import Point, ReferenceFrame
        >>> N = ReferenceFrame('N')
        >>> p1 = Point('p1')
        >>> p1.set_acc(N, 10 * N.x)
        >>> p1.acc(N)
        10*N.x

        """

        _check_frame(frame)
        if not (frame in self._acc_dict):
            if self._vel_dict[frame] != 0:
                return (self._vel_dict[frame]).dt(frame)
            else:
                return Vector(0)
        return self._acc_dict[frame]

    def locatenew(self, name, value):
        """Creates a new point with a position defined from this point.

        Parameters
        ==========

        name : str
            The name for the new point
        value : Vector
            The position of the new point relative to this point

        Examples
        ========

        >>> from sympy.physics.vector import ReferenceFrame, Point
        >>> N = ReferenceFrame('N')
        >>> P1 = Point('P1')
        >>> P2 = P1.locatenew('P2', 10 * N.x)

        """

        if not isinstance(name, str):
            raise TypeError('Must supply a valid name')
        if value == 0:
            value = Vector(0)
        value = _check_vector(value)
        p = Point(name)
        p.set_pos(self, value)
        self.set_pos(p, -value)
        return p

    def pos_from(self, otherpoint):
        """Returns a Vector distance between this Point and the other Point.

        Parameters
        ==========

        otherpoint : Point
            The otherpoint we are locating this one relative to

        Examples
        ========

        >>> from sympy.physics.vector import Point, ReferenceFrame
        >>> N = ReferenceFrame('N')
        >>> p1 = Point('p1')
        >>> p2 = Point('p2')
        >>> p1.set_pos(p2, 10 * N.x)
        >>> p1.pos_from(p2)
        10*N.x

        """

        outvec = Vector(0)
        plist = self._pdict_list(otherpoint, 0)
        for i in range(len(plist) - 1):
            outvec += plist[i]._pos_dict[plist[i + 1]]
        return outvec

    def set_acc(self, frame, value):
        """Used to set the acceleration of this Point in a ReferenceFrame.

        Parameters
        ==========

        frame : ReferenceFrame
            The frame in which this point's acceleration is defined
        value : Vector
            The vector value of this point's acceleration in the frame

        Examples
        ========

        >>> from sympy.physics.vector import Point, ReferenceFrame
        >>> N = ReferenceFrame('N')
        >>> p1 = Point('p1')
        >>> p1.set_acc(N, 10 * N.x)
        >>> p1.acc(N)
        10*N.x

        """

        if value == 0:
            value = Vector(0)
        value = _check_vector(value)
        _check_frame(frame)
        self._acc_dict.update({frame: value})

    def set_pos(self, otherpoint, value):
        """Used to set the position of this point w.r.t. another point.

        Parameters
        ==========

        otherpoint : Point
            The other point which this point's location is defined relative to
        value : Vector
            The vector which defines the location of this point

        Examples
        ========

        >>> from sympy.physics.vector import Point, ReferenceFrame
        >>> N = ReferenceFrame('N')
        >>> p1 = Point('p1')
        >>> p2 = Point('p2')
        >>> p1.set_pos(p2, 10 * N.x)
        >>> p1.pos_from(p2)
        10*N.x

        """

        if value == 0:
            value = Vector(0)
        value = _check_vector(value)
        self._check_point(otherpoint)
        self._pos_dict.update({otherpoint: value})
        otherpoint._pos_dict.update({self: -value})

    def set_vel(self, frame, value):
        """Sets the velocity Vector of this Point in a ReferenceFrame.

        Parameters
        ==========

        frame : ReferenceFrame
            The frame in which this point's velocity is defined
        value : Vector
            The vector value of this point's velocity in the frame

        Examples
        ========

        >>> from sympy.physics.vector import Point, ReferenceFrame
        >>> N = ReferenceFrame('N')
        >>> p1 = Point('p1')
        >>> p1.set_vel(N, 10 * N.x)
        >>> p1.vel(N)
        10*N.x

        """

        if value == 0:
            value = Vector(0)
        value = _check_vector(value)
        _check_frame(frame)
        self._vel_dict.update({frame: value})

    def v1pt_theory(self, otherpoint, outframe, interframe):
        """Sets the velocity of this point with the 1-point theory.

        The 1-point theory for point velocity looks like this:

        ^N v^P = ^B v^P + ^N v^O + ^N omega^B x r^OP

        where O is a point fixed in B, P is a point moving in B, and B is
        rotating in frame N.

        Parameters
        ==========

        otherpoint : Point
            The first point of the 2-point theory (O)
        outframe : ReferenceFrame
            The frame we want this point's velocity defined in (N)
        interframe : ReferenceFrame
            The intermediate frame in this calculation (B)

        Examples
        ========

        >>> from sympy.physics.vector import Point, ReferenceFrame
        >>> from sympy.physics.vector import dynamicsymbols
        >>> from sympy.physics.vector import init_vprinting
        >>> init_vprinting(pretty_print=False)
        >>> q = dynamicsymbols('q')
        >>> q2 = dynamicsymbols('q2')
        >>> qd = dynamicsymbols('q', 1)
        >>> q2d = dynamicsymbols('q2', 1)
        >>> N = ReferenceFrame('N')
        >>> B = ReferenceFrame('B')
        >>> B.set_ang_vel(N, 5 * B.y)
        >>> O = Point('O')
        >>> P = O.locatenew('P', q * B.x)
        >>> P.set_vel(B, qd * B.x + q2d * B.y)
        >>> O.set_vel(N, 0)
        >>> P.v1pt_theory(O, N, B)
        q'*B.x + q2'*B.y - 5*q*B.z

        """

        _check_frame(outframe)
        _check_frame(interframe)
        self._check_point(otherpoint)
        dist = self.pos_from(otherpoint)
        v1 = self.vel(interframe)
        v2 = otherpoint.vel(outframe)
        omega = interframe.ang_vel_in(outframe)
        self.set_vel(outframe, v1 + v2 + (omega ^ dist))
        return self.vel(outframe)

    def v2pt_theory(self, otherpoint, outframe, fixedframe):
        """Sets the velocity of this point with the 2-point theory.

        The 2-point theory for point velocity looks like this:

        ^N v^P = ^N v^O + ^N omega^B x r^OP

        where O and P are both points fixed in frame B, which is rotating in
        frame N.

        Parameters
        ==========

        otherpoint : Point
            The first point of the 2-point theory (O)
        outframe : ReferenceFrame
            The frame we want this point's velocity defined in (N)
        fixedframe : ReferenceFrame
            The frame in which both points are fixed (B)

        Examples
        ========

        >>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
        >>> from sympy.physics.vector import init_vprinting
        >>> init_vprinting(pretty_print=False)
        >>> q = dynamicsymbols('q')
        >>> qd = dynamicsymbols('q', 1)
        >>> N = ReferenceFrame('N')
        >>> B = N.orientnew('B', 'Axis', [q, N.z])
        >>> O = Point('O')
        >>> P = O.locatenew('P', 10 * B.x)
        >>> O.set_vel(N, 5 * N.x)
        >>> P.v2pt_theory(O, N, B)
        5*N.x + 10*q'*B.y

        """

        _check_frame(outframe)
        _check_frame(fixedframe)
        self._check_point(otherpoint)
        dist = self.pos_from(otherpoint)
        v = otherpoint.vel(outframe)
        omega = fixedframe.ang_vel_in(outframe)
        self.set_vel(outframe, v + (omega ^ dist))
        return self.vel(outframe)

    def vel(self, frame):
        """The velocity Vector of this Point in the ReferenceFrame.

        Parameters
        ==========

        frame : ReferenceFrame
            The frame in which the returned velocity vector will be defined in

        Examples
        ========

        >>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
        >>> N = ReferenceFrame('N')
        >>> p1 = Point('p1')
        >>> p1.set_vel(N, 10 * N.x)
        >>> p1.vel(N)
        10*N.x

        Velocities will be automatically calculated if possible, otherwise a ``ValueError`` will be returned. If it is possible to calculate multiple different velocities from the relative points, the points defined most directly relative to this point will be used. In the case of inconsistent relative positions of points, incorrect velocities may be returned. It is up to the user to define prior relative positions and velocities of points in a self-consistent way.

        >>> p = Point('p')
        >>> q = dynamicsymbols('q')
        >>> p.set_vel(N, 10 * N.x)
        >>> p2 = Point('p2')
        >>> p2.set_pos(p, q*N.x)
        >>> p2.vel(N)
        (Derivative(q(t), t) + 10)*N.x

        """

        _check_frame(frame)
        if not (frame in self._vel_dict):
            valid_neighbor_found = False
            is_cyclic = False
            visited = []
            queue = [self]
            candidate_neighbor = []
            while queue: #BFS to find nearest point
                node = queue.pop(0)
                if node not in visited:
                    visited.append(node)
                    for neighbor, neighbor_pos in node._pos_dict.items():
                        if neighbor in visited:
                            continue
                        try:
                            neighbor_pos.express(frame) #Checks if pos vector is valid
                        except ValueError:
                            continue
                        if neighbor in queue:
                            is_cyclic = True
                        try :
                            neighbor_velocity = neighbor._vel_dict[frame] #Checks if point has its vel defined in req frame
                        except KeyError:
                            queue.append(neighbor)
                            continue
                        candidate_neighbor.append(neighbor)
                        if not valid_neighbor_found:
                            self.set_vel(frame, self.pos_from(neighbor).dt(frame) + neighbor_velocity)
                            valid_neighbor_found = True
            if is_cyclic:
                warn('Kinematic loops are defined among the positions of points. This is likely not desired and may cause errors in your calculations.')
            if len(candidate_neighbor) > 1:
                warn('Velocity automatically calculated based on point ' +
                    candidate_neighbor[0].name + ' but it is also possible from points(s):' +
                    str(candidate_neighbor[1:]) + '. Velocities from these points are not necessarily the same. This may cause errors in your calculations.')
            if valid_neighbor_found:
                return self._vel_dict[frame]
            else:
                raise ValueError('Velocity of point ' + self.name + ' has not been'
                             ' defined in ReferenceFrame ' + frame.name)

        return self._vel_dict[frame]

    def partial_velocity(self, frame, *gen_speeds):
        """Returns the partial velocities of the linear velocity vector of this
        point in the given frame with respect to one or more provided
        generalized speeds.

        Parameters
        ==========
        frame : ReferenceFrame
            The frame with which the velocity is defined in.
        gen_speeds : functions of time
            The generalized speeds.

        Returns
        =======
        partial_velocities : tuple of Vector
            The partial velocity vectors corresponding to the provided
            generalized speeds.

        Examples
        ========

        >>> from sympy.physics.vector import ReferenceFrame, Point
        >>> from sympy.physics.vector import dynamicsymbols
        >>> N = ReferenceFrame('N')
        >>> A = ReferenceFrame('A')
        >>> p = Point('p')
        >>> u1, u2 = dynamicsymbols('u1, u2')
        >>> p.set_vel(N, u1 * N.x + u2 * A.y)
        >>> p.partial_velocity(N, u1)
        N.x
        >>> p.partial_velocity(N, u1, u2)
        (N.x, A.y)

        """
        partials = [self.vel(frame).diff(speed, frame, var_in_dcm=False) for
                    speed in gen_speeds]

        if len(partials) == 1:
            return partials[0]
        else:
            return tuple(partials)