"""
Ellipse
=======
Defines the objects related to ellipse computations:
- :func:`colour.algebra.ellipse_coefficients_general_form`
- :func:`colour.algebra.ellipse_coefficients_canonical_form`
- :func:`colour.algebra.point_at_angle_on_ellipse`
- :func:`colour.algebra.ellipse_fitting_Halir1998`
References
----------
- :cite:`Halir1998` : Halir, R., & Flusser, J. (1998). Numerically Stable
Direct Least Squares Fitting Of Ellipses (pp. 1-8).
http://citeseerx.ist.psu.edu/viewdoc/download;\
jsessionid=BEEAFC85DE53308286D626302F4A3E3C?doi=10.1.1.1.7559&rep=rep1&type=pdf
- :cite:`Wikipedia` : Wikipedia. (n.d.). Ellipse. Retrieved November 24,
2018, from https://en.wikipedia.org/wiki/Ellipse
"""
from __future__ import annotations
import numpy as np
from colour.hints import ArrayLike, Literal, NDArrayFloat, cast
from colour.utilities import (
CanonicalMapping,
ones,
tsplit,
tstack,
validate_method,
)
__author__ = "Colour Developers"
__copyright__ = "Copyright 2013 Colour Developers"
__license__ = "BSD-3-Clause - https://opensource.org/licenses/BSD-3-Clause"
__maintainer__ = "Colour Developers"
__email__ = "colour-developers@colour-science.org"
__status__ = "Production"
__all__ = [
"ellipse_coefficients_general_form",
"ellipse_coefficients_canonical_form",
"point_at_angle_on_ellipse",
"ellipse_fitting_Halir1998",
"ELLIPSE_FITTING_METHODS",
"ellipse_fitting",
]
[docs]
def point_at_angle_on_ellipse(phi: ArrayLike, coefficients: ArrayLike) -> NDArrayFloat:
"""
Return the coordinates of the point at angle :math:`\\phi` in degrees on
the ellipse with given canonical form coefficients.
Parameters
----------
phi
Point at angle :math:`\\phi` in degrees to retrieve the coordinates
of.
coefficients
General form ellipse coefficients as follows: the center coordinates
:math:`x_c` and :math:`y_c`, semi-major axis length :math:`a_a`,
semi-minor axis length :math:`a_b` and rotation angle :math:`\\theta`
in degrees of its semi-major axis :math:`a_a`.
Returns
-------
:class:`numpy.ndarray`
Coordinates of the point at angle :math:`\\phi`
Examples
--------
>>> coefficients = np.array([0.5, 0.5, 2, 1, 45])
>>> point_at_angle_on_ellipse(45, coefficients) # doctest: +ELLIPSIS
array([ 1., 2.])
"""
phi = np.radians(phi)
x_c, y_c, a_a, a_b, theta = tsplit(coefficients)
theta = np.radians(theta)
cos_phi = np.cos(phi)
sin_phi = np.sin(phi)
cos_theta = np.cos(theta)
sin_theta = np.sin(theta)
x = x_c + a_a * cos_theta * cos_phi - a_b * sin_theta * sin_phi
y = y_c + a_a * sin_theta * cos_phi + a_b * cos_theta * sin_phi
return tstack([x, y])
[docs]
def ellipse_fitting_Halir1998(a: ArrayLike) -> NDArrayFloat:
"""
Return the coefficients of the implicit second-order polynomial/quadratic
curve that fits given point array :math:`a` using
*Halir and Flusser (1998)* method.
The implicit second-order polynomial is expressed as follows:
:math:`F\\left(x, y\\right)` = ax^2 + bxy + cy^2 + dx + ey + f = 0`
with an ellipse-specific constraint such as :math:`b^2 -4ac < 0` and where
:math:`a, b, c, d, e, f` are coefficients of the ellipse and
:math:`F\\left(x, y\\right)` are coordinates of points lying on it.
Parameters
----------
a
Point array :math:`a` to be fitted.
Returns
-------
:class:`numpy.ndarray`
Coefficients of the implicit second-order polynomial/quadratic
curve that fits given point array :math:`a`.
References
----------
:cite:`Halir1998`
Examples
--------
>>> a = np.array([[2, 0], [0, 1], [-2, 0], [0, -1]])
>>> ellipse_fitting_Halir1998(a) # doctest: +ELLIPSIS
array([ 0.2425356..., 0. , 0.9701425..., 0. , 0. ,
-0.9701425...])
>>> ellipse_coefficients_canonical_form(ellipse_fitting_Halir1998(a))
array([-0., -0., 2., 1., 0.])
"""
x, y = tsplit(a)
# Quadratic part of the design matrix.
D1 = tstack([x**2, x * y, y**2])
# Linear part of the design matrix.
D2 = tstack([x, y, ones(x.shape)])
D1_T = np.transpose(D1)
D2_T = np.transpose(D2)
# Quadratic part of the scatter matrix.
S1 = np.dot(D1_T, D1)
# Combined part of the scatter matrix.
S2 = np.dot(D1_T, D2)
# Linear part of the scatter matrix.
S3 = np.dot(D2_T, D2)
T = -np.dot(np.linalg.inv(S3), np.transpose(S2))
# Reduced scatter matrix.
M = S1 + np.dot(S2, T)
M = np.array([M[2, :] / 2, -M[1, :], M[0, :] / 2])
_w, v = np.linalg.eig(M)
A1 = v[:, np.nonzero(4 * v[0, :] * v[2, :] - v[1, :] ** 2 > 0)[0]]
A2 = np.dot(T, A1)
A = cast(NDArrayFloat, np.ravel([A1, A2]))
return A
ELLIPSE_FITTING_METHODS: CanonicalMapping = CanonicalMapping(
{"Halir 1998": ellipse_fitting_Halir1998}
)
ELLIPSE_FITTING_METHODS.__doc__ = """
Supported ellipse fitting methods.
References
----------
:cite:`Halir1998`
"""
[docs]
def ellipse_fitting(
a: ArrayLike, method: Literal["Halir 1998"] | str = "Halir 1998"
) -> NDArrayFloat:
"""
Return the coefficients of the implicit second-order polynomial/quadratic
curve that fits given point array :math:`a` using
given method.
The implicit second-order polynomial is expressed as follows:
:math:`F\\left(x, y\\right)` = ax^2 + bxy + cy^2 + dx + ey + f = 0`
with an ellipse-specific constraint such as :math:`b^2 -4ac < 0` and where
:math:`a, b, c, d, e, f` are coefficients of the ellipse and
:math:`F\\left(x, y\\right)` are coordinates of points lying on it.
Parameters
----------
a
Point array :math:`a` to be fitted.
method
Computation method.
Returns
-------
:class:`numpy.ndarray`
Coefficients of the implicit second-order polynomial/quadratic
curve that fits given point array :math:`a`.
References
----------
:cite:`Halir1998`
Examples
--------
>>> a = np.array([[2, 0], [0, 1], [-2, 0], [0, -1]])
>>> ellipse_fitting(a) # doctest: +ELLIPSIS
array([ 0.2425356..., 0. , 0.9701425..., 0. , 0. ,
-0.9701425...])
>>> ellipse_coefficients_canonical_form(ellipse_fitting(a))
array([-0., -0., 2., 1., 0.])
"""
method = validate_method(method, tuple(ELLIPSE_FITTING_METHODS))
function = ELLIPSE_FITTING_METHODS[method]
return function(a)