microstructpy.geometry.ellipsoid module

class microstructpy.geometry.ellipsoid.Ellipsoid(**kwargs)

Bases: object

A 3D Ellipsoid

This class contains the data and functions for a 3D ellispoid. It is defined by its center, axes, and orientation.

If multiple keywords are given for the shape of the ellipsoid, there is no guarantee for which keywords are used.

Parameters:

• center (list, optional) – The ellipsoid center. Defaults to (0, 0, 0).
• axes (list, optional) – List of 3 semi-axes. Defaults to (1, 1, 1).
• size (float, optional) – The diameter of a sphere with equal volume. Defaults to 2.
• ratio_ab (float, optional) – The ratio of a to b.
• ratio_ac (float, optional) – The ratio of a to c.
• ratio_bc (float, optional) – The ratio of b to c.
• ratio_ba (float, optional) – The ratio of b to a.
• ratio_ca (float, optional) – The ratio of c to a.
• ratio_cb (float, optional) – The ratio of c to b.
• rot_seq (list, optional) –

  List of rotations (deg). Each element of the list should be an (axis, angle) tuple. The options for the axis are: ‘x’, ‘y’, ‘z’, 1, 2, or 3. For example:

  rot_seq = [('x', 10), (2, -20), ('z', 85), ('x', 21)]
  

• rot_seq_deg (list, optional) – Alias for rot_seq, with degrees stated explicitly.
• rot_seq_rad (list, optional) – Same format as rot_seq, except the angles are expressed in radians.
• matrix (3x3 numpy.ndarray, optional) – A 3x3 rotation matrix expressing the orientation of the ellipsoid. Defaults to the identity.
• position – Alias for center.
• a – Alias for axes[0].
• b – Alias for axes[1].
• c – Alias for axes[2].
• orientation – Alias for matrix.

approximate(x1=None)

Approximate Ellipsoid with Spheres

This function approximates the ellipsoid by a set of spheres. It does so by approximating the x-z and y-z elliptical cross sections with circles, then scaling those circles and promoting them to spheres.

See the documentation for microstructpy.geometry.Ellipse.approximate() for more details.

Parameters:x1 (float) – Center position of the first sphere. Returns:An Nx4 list of the (x, y, z, r) data of the spheres that approximate the ellipsoid. Return type:numpy.ndarray

best_fit(points)

Find ellipsoid of best fit.

This function takes a list of 3D points and computes the ellipsoid of best fit for the points. It uses a published algorithm to fit the ellipsoid, then attempts to define the axes in such a way that they most align with this ellipsoid’s axes. [1]

Parameters:points (list) – Points to fit ellipsoid Returns:The ellipsoid that best fits the points. Return type:Ellipsoid

[1]	Turner, D. A., Anderson, I. J., Mason, J. C., and Cox, M. G., “An Algorithm for Fitting an Ellipsoid to Data,” National Physical Laboratory, 1999, The United Kingdom. (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.36.2773&rep=rep1&type=pdf)

plot(**kwargs)

Plot the ellipsoid.

This function uses the Axes3D.plot_surface method to add an ellipsoid to the current axes. The keyword arguments are passed through to the plot_surface function.

Parameters:**kwargs (dict) – Keyword arguments for matplotlib.

reflect(points)

Reflect points across surface.

This function reflects a point or set of points across the surface of the ellipsoid. Points at the center of the ellipsoid are not reflected.

Parameters:points (list) – Points to reflect. Returns:Reflected points. Return type:numpy.ndarray

classmethod volume_expectation(**kwargs)

Expected value of volume.

This function computes the expected value for the volume of an ellipsoid. The keyword arguments are the same as the input parameters for the class, microstructpy.geometry.Ellipsoid. The values for these keywords can be either constants or scipy.stats distributions.

The expected value is computed by the following formula:

\[\begin{split}\begin{align} \mathbb{E}[V] &= \mathbb{E}[\frac{4}{3}\pi A B C] \\ &= \frac{4}{3}\pi \mathbb{E}[A] \mathbb{E}[B] \mathbb{E}[C] \\ &= \frac{4}{3}\pi \mu_A \mu_B \mu_C \end{align}\end{split}\]

If the ellisoid is specified by size and aspect ratios, then the expected volume is computed by:

\[\begin{split}\begin{align} \mathbb{E}[V] &= \mathbb{E}[\frac{\pi}{6} S^3] \\ &= \frac{\pi}{6} (\mu_S^3 + 3 \mu_S \sigma_S^2 + \gamma_{1, S} \sigma_S^3) \end{align}\end{split}\]

If the ellipsoid is specified using a combination of semi-axes and aspect ratios, then the expected volume is the mean of 1000 random samples:

\[\mathbb{E}[V] \approx \frac{1}{n} \sum_{i=1}^n V_i\]

where \(n=1000\).

Parameters:**kwargs – Keyword arguments, see microstructpy.geometry.ellipsoid.Ellipsoid. Returns:Expected value of the volume of the sphere. Return type:float

within(points)

Test if points are within ellipsoid.

This function tests whether a point or set of points are within the ellipsoid. For the set of points, a list of booleans is returned to indicate which points are within the ellipsoid.

Parameters:points (list) – Point or list of points. Returns:Set to True for points in geometry. Return type:bool or numpy.ndarray

axes

the 3 semi-axes of the ellipsoid

Type:tuple

bound_max

maximum bounding sphere, (x, y, z, r)

Type:tuple

bound_min

minimum interior sphere, (x, y, z, r)

Type:tuple

coefficients

coeffificients of equation, \((A, B, C, D, E, F, G, H, K, L)\) in \(Ax^2 + Bxy + Cxz + Dy^2 + Eyz + Fz^2 + Gx + Hy + Kz + L = 0\)

Type:tuple

limits

list of (lower, upper) bounds for the bounding box

Type:list

matrix

A 3x3 rotation matrix

Type:numpy.ndarray

matrix_quadeq

Matrix of the quadratic equation

Type:numpy.ndarray

matrix_quadform

Matrix of the quadratic form

Type:numpy.ndarray

n_dim

number of dimensions, 3

Type:int

orientation

A 3x3 rotation matrix

Type:numpy.ndarray

ratio_ab

ratio of x-axis length to y-axis length

Type:float

ratio_ac

ratio of x-axis length to z-axis length

Type:float

ratio_ba

ratio of y-axis length to x-axis length

Type:float

ratio_bc

ratio of y-axis length to z-axis length

Type:float

ratio_ca

ratio of z-axis length to x-axis length

Type:float

ratio_cb

ratio of z-axis length to y-axis length

Type:float

rot_seq_deg

rotation sequence, with angles in degrees

Type:list

rot_seq_rad

rotation sequence, with angles in radiands

Type:list

sample_limits

list of (lower, upper) bounds for the sampling region

Type:list

size

diameter of equivalent volume sphere

Type:float

volume

volume of ellipsoid, \(V = \frac{4}{3}\pi a b c\)

Type:float