microstructpy.geometry.Sphere¶
- class microstructpy.geometry.Sphere(**kwargs)[source]¶
Bases:
microstructpy.geometry.n_sphere.NSphere
A 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
- 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
- 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
- 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
Sphere
function. The values for these keywords can be either constants orscipy.stats
distributions.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
- 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