microstructpy.geometry.Sphere¶
-
class
microstructpy.geometry.Sphere(**kwargs)[source]¶ Bases:
microstructpy.geometry.n_sphere.NSphereA 3D sphere.
This class represents a three-dimensional circle. It is defined by a center point and size parameter, which can be either radius or diameter.
Without input parameters, this defaults to a unit sphere centered at the origin.
Parameters: - r (float) – (optional) The radius of the sphere. Defaults to 1.
- radius (float) – (optional) Same as
r. - d (float) – (optional) Alias for
2*r. - diameter (float) – (optional) Alias for
2*r. - size (float) – (optional) Alias for
2*r. - center (list, float, numpy.ndarray) – (optional) The coordinates of
the center. Defaults to
[0, 0, 0]. - position (list, float, numpy.ndarray) – (optional)
Alias for
center.
-
approximate()¶ Approximate the n-sphere with itself
Other geometries can be approximated by a set of circles or spheres. For the n-sphere, this approximation is exact.
Returns: A list containing [(x, y, z, …, r)] Return type: list
-
classmethod
best_fit(points)¶ Find n-sphere of best fit for set of points.
This function takes a list of points and computes an n-sphere of best fit, in an algebraic sense. This method was developed using the a published writeup, which was extended from 2D to ND. [1]
Parameters: points (list, numpy.ndarray) – List of points to fit. Returns: An instance of the class that fits the points. Return type: NSphere [1] Circle fitting writup by Randy Bullock, https://dtcenter.org/met/users/docs/write_ups/circle_fit.pdf
-
plot(**kwargs)[source]¶ Plot the sphere.
This function uses the
mpl_toolkits.mplot3d.Axes3D.plot_surface()method to add the sphere to the current axes. The keyword arguments are passed through to plot_surface.Parameters: **kwargs (dict) – Keyword arguments for plot_surface.
-
reflect(points)¶ Reflect points across surface.
This function reflects a point or set of points across the surface of the n-sphere. Points at the center of the n-sphere are not reflected.
Parameters: points (list or numpy.ndarray) – Points to reflect. Returns: Reflected points. Return type: numpy.ndarray
-
classmethod
volume_expectation(**kwargs)[source]¶ Expected value of volume.
This function computes the expected value for the volume of a sphere. The keyword arguments are identical to the
Spherefunction. The values for these keywords can be either constants orscipy.statsdistributions.The expected value is computed by the following formula:
\[\begin{split}\mathbb{E}[V] &= \mathbb{E}[\frac{4}{3}\pi R^3] \\ &= \frac{4}{3}\pi \mathbb{E}[R^3] \\ &= \frac{4}{3}\pi (\mu_R^3 + 3 \mu_R \sigma_R^2 + \gamma_{1, R} \sigma_R^3)\end{split}\]Parameters: **kwargs – Keyword arguments, see microstructpy.geometry.Sphere.Returns: Expected value of the volume of the sphere. Return type: float
-
within(points)¶ Test if points are within n-sphere.
This function tests whether a point or set of points are within the n-sphere. For the set of points, a list of booleans is returned to indicate which points are within the n-sphere.
Parameters: points (list or numpy.ndarray) – Point or list of points. Returns: Set to True for points in geometry. Return type: bool or numpy.ndarray
