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curvature.py
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curvature.py
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r"""
.. role:: raw-math(raw) :format: latex html
--------------------
Curvature
--------------------
In MembraneCurvature, we calculate Gaussian and mean curvature from a cloud of points.
Gaussian curvature is defined by
.. math:: K = \frac{\partial_{xx}\partial_{yy}-\partial_{xy}^2}
{(1+\partial_x^2+\partial_y^2)^2}.
Mean curvature is defined by
.. math:: H =
\frac{(1+\partial_x^2)\partial_{yy}+(1+\partial_y^2)\partial_{xx}-2\partial_x\partial_y\partial_{xy}}
{2(1+\partial_x^2+\partial_y^2)^{3/2}}.
Notes
---------
Numpy cannot calculate the gradient for arrays with inner array of
`length==1` unless `axis=0` is specified. Therefore in the functions here included
for mean and Gaussian curvature, shape of arrays must be at least (2,2).
In general, to calculate a numerical gradients shape of arrays must be >=(`edge_order` +
1).
Functions
---------
"""
import numpy as np
def gaussian_curvature(Z):
"""
Calculate gaussian curvature from Z cloud points.
Parameters
----------
Z: np.ndarray.
Multidimensional array of shape (n,n).
Returns
-------
K : np.ndarray.
The result of gaussian curvature of Z. Returns multidimensional
array object with values of gaussian curvature of shape `(n, n)`.
"""
Zy, Zx = np.gradient(Z)
Zxy, Zxx = np.gradient(Zx)
Zyy, _ = np.gradient(Zy)
K = (Zxx * Zyy - (Zxy ** 2)) / (1 + (Zx ** 2) + (Zy ** 2)) ** 2
return K
def mean_curvature(Z):
"""
Calculates mean curvature from Z cloud points.
Parameters
----------
Z: np.ndarray.
Multidimensional array of shape (n,n).
Returns
-------
H : np.ndarray.
The result of gaussian curvature of Z. Returns multidimensional
array object with values of gaussian curvature of shape `(n, n)`.
"""
Zy, Zx = np.gradient(Z)
Zxy, Zxx = np.gradient(Zx)
Zyy, _ = np.gradient(Zy)
H = (Zx**2 + 1) * Zyy - 2 * Zx * Zy * Zxy + (Zy**2 + 1) * Zxx
H = -H / (2 * (Zx**2 + Zy**2 + 1)**(1.5))
return H