Source code for microstructpy.geometry.ellipse

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from __future__ import division

import matplotlib.pyplot as plt
import numpy as np
from matplotlib import patches

from microstructpy.geometry._ellipse_best_fit import _best_fit

__author__ = 'Kenneth (Kip) Hart'


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# Ellipse Class                                                               #
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[docs]class Ellipse(object): r"""A 2-D ellipse geometry. This class contains a 2-D ellipse. It is defined by a center point, axes and an orientation. Without any parameters, the ellipse defaults to a unit circle. Args: a (float): *(optional)* Semi-major axis of ellipse. Defaults to 1. b (float): *(optional)* Semi-minor axis of ellipse. Defaults to 1. center (list): *(optional)* The ellipse center. Defaults to (0, 0). axes (list): *(optional)* A 2-element list of semi-axes, equivalent to ``[a, b]``. Defaults to [1, 1]. size (float): *(optional)* The diameter of a circle with equivalent area. Defaults to 1. aspect_ratio (float): *(optional)* The ratio of x-axis to y-axis length. Defaults to 1. angle (float): *(optional)* The counterclockwise rotation angle, in degrees, measured from the +x axis. angle_deg (float): *(optional)* The rotation angle, in degrees. angle_rad (float): *(optional)* The rotation angle, in radians. matrix (numpy.ndarray): *(optional)* The 2x2 rotation matrix. orientation (numpy.ndarray): *(optional)* Alias for ``matrix``. """ # ----------------------------------------------------------------------- # # Constructor # # ----------------------------------------------------------------------- # def __init__(self, **kwargs): # Check Values for kw in ('a', 'b', 'size', 'aspect_ratio'): if kw in kwargs and kwargs[kw] <= 0: raise ValueError(kw + ' should be positive.') if 'axes' in kwargs: for i, ax in kwargs['axes']: if ax <= 0: raise ValueError('axes[{}] should be positive'.format(i)) for kw in ['matrix', 'orientation']: if kw in kwargs: m = np.array(kwargs[kw]) if m.shape != (2, 2): raise ValueError(kw + ' should be 2x2.') if not np.all(np.isclose(m * m.T, np.eye(2))): raise ValueError(kw + ' should be orthonormal.') # position if 'center' in kwargs: self.center = kwargs['center'] elif 'position' in kwargs: self.center = kwargs['position'] else: self.center = (0, 0) # axes if ('a' in kwargs) and ('b' in kwargs): assert kwargs['a'] > 0 assert kwargs['b'] > 0 self.a = kwargs['a'] self.b = kwargs['b'] elif 'axes' in kwargs: self.a, self.b = kwargs['axes'] elif ('size' in kwargs) and ('aspect_ratio' in kwargs): assert kwargs['size'] > 0 assert kwargs['aspect_ratio'] > 0 # Derivation for converting (r, k) into (a, b) # Area formula: pi*a*b = pi*r^2 # Aspect ratio: k = a/b # # Plug AR into area: k*b^2 = r^2 # Solve for b: b = r/sqrt(k) # Solve for a: a = k * b r = 0.5 * kwargs['size'] k = kwargs['aspect_ratio'] self.b = r / np.sqrt(k) self.a = k * self.b elif ('a' in kwargs) and ('size' in kwargs): assert kwargs['a'] > 0 assert kwargs['size'] > 0 r = 0.5 * kwargs['size'] self.a = kwargs['a'] self.b = (r * r) / self.a elif ('b' in kwargs) and ('size' in kwargs): assert kwargs['b'] > 0 assert kwargs['size'] > 0 r = 0.5 * kwargs['size'] self.b = kwargs['b'] self.a = (r * r) / self.b elif ('a' in kwargs) and ('aspect_ratio' in kwargs): assert kwargs['a'] > 0 assert kwargs['aspect_ratio'] > 0 self.a = kwargs['a'] self.b = self.a / kwargs['aspect_ratio'] elif ('b' in kwargs) and ('aspect_ratio' in kwargs): assert kwargs['b'] > 0 assert kwargs['aspect_ratio'] > 0 self.b = kwargs['b'] self.a = self.b * kwargs['aspect_ratio'] else: self.a = 1 self.b = 1 # orientation if 'angle_deg' in kwargs: self.angle = kwargs['angle_deg'] elif 'angle_rad' in kwargs: self.angle = 180 * kwargs['angle_rad'] / np.pi elif 'angle' in kwargs: self.angle = kwargs['angle'] elif 'matrix' in kwargs: ct = kwargs['matrix'][0][0] st = kwargs['matrix'][1][0] self.angle = 180 * np.arctan2(st, ct) / np.pi elif 'orientation' in kwargs: ct = kwargs['orientation'][0][0] st = kwargs['orientation'][1][0] self.angle = 180 * np.arctan2(st, ct) / np.pi else: self.angle = 0 # ----------------------------------------------------------------------- # # Best Fit Function # # ----------------------------------------------------------------------- #
[docs] def best_fit(self, points): r"""Find ellipse of best fit for points This function computes the ellipse of best fit for a set of points. It solves for the quadratic form of the ellipse using linear least squares regression, then computes the center, axes, and orientation of the ellipse from the quadratic coefficients. The current instance of the class is used as an initial guess for the ellipse of best fit. Since an ellipse can be expressed multiple ways (e.g. rotate 90 degrees and flip the axes), this initial guess is used to choose from the multiple parameter sets. Args: points (list or numpy.ndarray): An Nx2 list of points to fit. Returns: .Ellipse: An instance of the class that best fits the points. """ # NOQA: E501 pts = np.array(points) pt_cen = pts.mean(axis=0) pts -= pt_cen.reshape(1, -1) width, height, phi, xc, yc = _best_fit(pts, self) xc += pt_cen[0] yc += pt_cen[1] # Find pair closest to self s = np.sin(phi) c = np.cos(phi) rot = np.array([[c, -s], [s, c]]) x_ax_seed = np.array(self.matrix)[:, 0] x_dot, y_dot = rot.T.dot(x_ax_seed) if np.abs(x_dot) > np.abs(y_dot): if x_dot > 0: x_ax_fit = rot[:, 0] else: x_ax_fit = - rot[:, 0] a = width b = height else: if y_dot > 0: x_ax_fit = rot[:, 1] else: x_ax_fit = - rot[:, 1] a = height b = width ang_diff = np.arcsin(np.cross(x_ax_seed, x_ax_fit)) ang_rad = self.angle_rad + ang_diff return type(self)(center=(xc, yc), a=a, b=b, angle_rad=ang_rad)
# ----------------------------------------------------------------------- # # String and Representation Functions # # ----------------------------------------------------------------------- # def __str__(self): str_str = 'center: ' + str(tuple(self.center)) + '\n' str_str += 'a: ' + str(self.a) + '\n' str_str += 'b: ' + str(self.b) + '\n' str_str += 'angle: ' + str(self.angle) return str_str def __repr__(self): repr_str = 'Ellipse(' repr_str += 'center=' + repr(tuple(self.center)) repr_str += ', a=' + repr(self.a) + ', b=' + repr(self.b) repr_str += ', angle=' + repr(self.angle) repr_str += ')' return repr_str # ----------------------------------------------------------------------- # # Size and Orientation Getters # # ----------------------------------------------------------------------- # @property def size(self): """float: Diameter of equivalent area circle""" return 2 * np.sqrt(self.a * self.b) @property def aspect_ratio(self): """float: Ratio of x-axis length to y-axis length""" return self.a / self.b @property def axes(self): """ tuple: List of semi-axes.""" return self.a, self.b @property def angle_deg(self): """float: Rotation angle, in degrees""" return self.angle @property def angle_rad(self): """float: Rotation angle, in radians""" return self.angle * np.pi / 180 @property def matrix(self): """numpy.ndarray: Rotation matrix""" ct = np.cos(self.angle_rad) st = np.sin(self.angle_rad) return np.array([[ct, -st], [st, ct]]) @property def orientation(self): """numpy.ndarray: Rotation matrix""" return self.matrix # ----------------------------------------------------------------------- # # Number of Dimensions # # ----------------------------------------------------------------------- # @property def n_dim(self): """int: Number of dimensions, 2""" return 2 # ----------------------------------------------------------------------- # # Area Property # # ----------------------------------------------------------------------- # @property def area(self): r""" float: Area of ellipse, :math:`A = \pi a b`""" return np.pi * self.a * self.b @property def volume(self): r""" float: Same as :any:`microstructpy.geometry.Ellipse.area`""" return self.area # ----------------------------------------------------------------------- # # Expected Area # # ----------------------------------------------------------------------- #
[docs] @classmethod def area_expectation(cls, **kwargs): r"""Expected value of area. This function computes the expected value for the area of an ellipse. The keyword arguments are the same as the input parameters of the class. The keyword values can be either constants (ints or floats) or distributions from the SciPy :mod:`scipy.stats` module. If an ellipse is specified by size, the expected value is computed as follows. .. math:: \mathbb{E}[A] &= \frac{\pi}{4} \mathbb[S^2] \\ &= \frac{\pi}{4} (\mu_S^2 + \sigma_S^2) If the ellipse is specified by independent distributions for each semi-axis, the expected value is computed by: .. math:: \mathbb{E}[A] = \pi\, \mathbb{E}[A B] = \pi \mu_A \mu_B If the ellipse is specified by the second semi-axis and the aspect ratio, the expected value is computed by: .. math:: \mathbb{E}[A] &= \pi\, \mathbb{E}[K B^2] \\ &= \pi \mu_K (\mu_B^2 + \sigma_B^2) Finally, if the ellipse is specified by the first semi-axis and the aspect ratio, the expected value is computed by Monte Carlo: .. math:: \mathbb{E}[A] &= \pi\, \mathbb{E}\left[\frac{A^2}{K}\right] \\ &\approx \frac{\pi}{n} \sum_{i=1}^n \frac{A_i}{K_i} where :math:`n=1000`. Args: **kwargs: Keyword arguments, see :class:`microstructpy.geometry.Ellipse`. Returns: float: Expected value of the area of the ellipse. """ # NOQA: E501 if 'size' in kwargs: s_dist = kwargs['size'] if type(s_dist) in (float, int): return 0.25 * np.pi * s_dist * s_dist else: return 0.25 * np.pi * s_dist.moment(2) elif ('a' in kwargs) and ('b' in kwargs): exp = np.pi for kw in ('a', 'b'): dist = kwargs[kw] if type(dist) in (float, int): mu = dist else: mu = dist.moment(1) exp *= mu return exp elif ('b' in kwargs) and ('aspect_ratio' in kwargs): exp = np.pi try: exp *= kwargs['b'].moment(2) except AttributeError: exp *= kwargs['b'] * kwargs['b'] try: exp *= kwargs['aspect_ratio'].moment(1) except AttributeError: exp *= kwargs['aspect_ratio'] return exp elif ('a' in kwargs) and ('aspect_ratio' in kwargs): n = 1000 try: a = kwargs['a'].rvs(size=n) except AttributeError: a = np.full(n, kwargs['a']) try: k = kwargs['aspect_ratio'].rvs(size=n) except AttributeError: k = np.full(n, kwargs['aspect_ratio']) return np.pi * np.mean((a * a) / k) else: e_str = 'Could not calculate expected area from keywords ' e_str += str(kwargs.keys()) + '.' raise KeyError(e_str)
# ----------------------------------------------------------------------- # # Bounding Circles # # ----------------------------------------------------------------------- # @property def bound_max(self): """tuple: Maximum bounding circle of ellipse, (x, y, r)""" r = max(self.a, self.b) return tuple(list(self.center) + [r]) @property def bound_min(self): """tuple: Minimum interior circle of ellipse, (x, y, r)""" r = min(self.a, self.b) return tuple(list(self.center) + [r]) # ----------------------------------------------------------------------- # # Circle Approximation of Ellipse # # ----------------------------------------------------------------------- #
[docs] def approximate(self, x1=None): """Approximate ellipse with a set of circles. This function converts an ellipse into a set of circles. It implements a published algorithm by Ilin and Bernacki. [#ilin]_ Example: >>> import matplotlib.pyplot as plt >>> import microstructpy as msp >>> import numpy as np >>> ellipse = msp.geometry.Ellipse(a=3, b=1) >>> approx = ellipse.approximate(0.7) >>> approx array([[ 0. , 0. , 1. ], [ 0.7 , 0. , 0.96889112], [ 1.38067777, 0. , 0.87276349], [ 2.00213905, 0. , 0.7063497 ], [ 2.5234414 , 0. , 0.45169729], [ 2.66666667, 0. , 0.33333333], [-0.7 , 0. , 0.96889112], [-1.38067777, 0. , 0.87276349], [-2.00213905, 0. , 0.7063497 ], [-2.5234414 , 0. , 0.45169729], [-2.66666667, 0. , 0.33333333]]) >>> ellipse.plot(edgecolor='k', facecolor='none', lw=3) >>> t = np.linspace(0, 2 * np.pi) >>> for x, y, r in approx: ... plt.plot(x + r * np.cos(t), y + r * np.sin(t), 'b') >>> plt.xticks(np.unique(np.concatenate((approx[:, 0], (-3, 3))))) >>> plt.yticks(np.unique(np.concatenate((approx[:, 1], (-1, 1))))) >>> plt.axis('scaled') >>> plt.grid(True, linestyle=':') >>> plt.show() Executing the code above produces :numref:`f_api_ellipse_approx`. .. _f_api_ellipse_approx: .. figure:: ../../auto_examples/geometry/images/sphx_glr_plot_ellipse_001.png Circular approximation of ellipse, after Ilin and Bernacki. Args: x1 (float or None): *(optional)* Position of the first circle along the +x axis. Defaults to 0.5x the shortest semi-axis. Returns: numpy.ndarray: An Nx3 list of the (x, y, r) data of each circle approximating the ellipse. Raises: AssertionError: Thrown if max(a, b) < x1. .. [#ilin] Ilin, D.N., and Bernacki, M., "Advancing Layer Algorithm of Dense Ellipse Packing for Generating Statistically Equivalent Polygonal Structures," Granular Matter, vol. 18(3), pp. 43, 2016. """ # NOQA: E501 if x1 is None: x1 = 0.5 * min(self.a, self.b) flip = self.a < self.b if flip: a = self.b b = self.a else: a = self.a b = self.b if a == b: return np.append(self.center, a).reshape(1, -1) a_str = 'Center of first circle, x1=' + str(x1) + ', must be less than' a_str += ' semi-major axis, a=' + str(a) assert x1 < a, a_str R_N = b * b / a # Eq. 8 x_N = a - R_N # Eq 9 def R_i(x): return b * np.sqrt(1 - x * x / (a * a - b * b)) # Eq. 6 def y_i(x): ratio = x / a return b * np.sqrt(1 - ratio * ratio) # Eq. 1 circles = [(0, 0, b)] y_vals = [b] adding_circles = x1 < x_N if adding_circles: circles.append((x1, 0, R_i(x1))) y_vals.append(y_i(x1)) while adding_circles: y_ratio = y_vals[-1] / y_vals[-2] x_diff = circles[-1][0] - circles[-2][0] x_ip1 = y_ratio * x_diff + circles[-1][0] # Eq. 7 adding_circles = x_ip1 < x_N if adding_circles: circle = (x_ip1, 0, R_i(x_ip1)) circles.append(circle) y_vals.append(y_i(x_ip1)) circles.append((x_N, 0, R_N)) reflect = [(-x, y, r) for x, y, r in circles[1:]] all_circles = np.array(circles + reflect) if flip: all_circles[:, [0, 1]] = all_circles[:, [1, 0]] rot_cens = all_circles[:, :-1].dot(self.matrix.T) all_circles[:, :-1] = rot_cens + np.array(self.center) return all_circles
# ----------------------------------------------------------------------- # # Plot Function # # ----------------------------------------------------------------------- #
[docs] def plot(self, **kwargs): """Plot the ellipse. This function adds a :class:`matplotlib.patches.Ellipse` patch to the current axes using matplotlib. The keyword arguments are passed to the patch. Args: **kwargs (dict): Keyword arguments for matplotlib. """ # NOQA: E501 p = patches.Ellipse(self.center, 2 * self.a, 2 * self.b, self.angle_deg, **kwargs) plt.gca().add_patch(p)
# ----------------------------------------------------------------------- # # Limits # # ----------------------------------------------------------------------- # @property def limits(self): """list: List of (lower, upper) bounds for the bounding box""" theta = self.angle_rad tan_t = np.tan(theta) tan_tx = - self.b / self.a * tan_t tx_max = np.arctan(tan_tx) tx_min = np.pi + tx_max if np.isclose(tan_t, 0) and np.cos(theta) > 0: ty_max = 0.5 * np.pi ty_min = - 0.5 * np.pi elif np.isclose(tan_t, 0) and np.cos(theta) < 0: ty_max = -0.5 * np.pi ty_min = 0.5 * np.pi else: tan_ty = self.b / (self.a * np.tan(theta)) ty_max = (np.arctan(tan_ty) + np.pi) % np.pi ty_min = np.pi + ty_max def xp(t): return self.a * np.cos(t) def yp(t): return self.b * np.sin(t) def xr(t): return np.cos(theta) * xp(t) - np.sin(theta) * yp(t) def yr(t): return np.sin(theta) * xp(t) + np.cos(theta) * yp(t) x_max = xr(tx_max) + self.center[0] x_min = xr(tx_min) + self.center[0] y_max = yr(ty_max) + self.center[1] y_min = yr(ty_min) + self.center[1] return [sorted((x_min, x_max)), sorted((y_min, y_max))] @property def sample_limits(self): """list: List of (lower, upper) bounds for the sampling region""" return self.limits # ----------------------------------------------------------------------- # # Within Test # # ----------------------------------------------------------------------- #
[docs] def within(self, points): """Test if points are within ellipse. This function tests whether a point or set of points are within the ellipse. For the set of points, a list of booleans is returned to indicate which points are within the ellipse. Args: points (list or numpy.ndarray): Point or list of points. Returns: bool or numpy.ndarray: Set to True for points in ellipse. """ pts = np.array(points) single_pt = pts.ndim == 1 if single_pt: pts = pts.reshape(1, -1) rel_pos = pts - np.array(self.center) rot_pos = rel_pos.dot(self.orientation) scl_pos = rot_pos / np.array(self.axes).reshape(1, -1) sq_dist = np.sum(scl_pos * scl_pos, axis=-1) mask = sq_dist <= 1 if single_pt: return mask[0] else: return mask
# ----------------------------------------------------------------------- # # Reflect # # ----------------------------------------------------------------------- #
[docs] def reflect(self, points): """Reflect points across surface. This function reflects a point or set of points across the surface of the ellipse. Points at the center of the ellipse are not reflected. Args: points (list or numpy.ndarray): Nx2 list of points to reflect. Returns: numpy.ndarray: Reflected points. """ pts = np.array(points) single_pt = pts.ndim == 1 if single_pt: pts = pts.reshape(1, -1) rel_pos = pts - np.array(self.center) rot_pos = rel_pos.dot(self.orientation) scl_pos = rot_pos / np.array(self.axes).reshape(1, -1) dist = np.sqrt(np.sum(scl_pos * scl_pos, axis=-1)) mask = dist > 0 if not np.any(mask): return np.array([]) new_dist = 2 - dist[mask] scl = new_dist / dist[mask] new_scl_pos = scl_pos[mask] * scl new_rel_pos = new_scl_pos.dot(self.orientation.T) new_pos = new_rel_pos + np.array(self.center) if single_pt: return new_pos[0] else: return new_pos