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####################################################################################################
#
# Patro - A Python library to make patterns for fashion design
# Copyright (C) 2017 Fabrice Salvaire
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
#
####################################################################################################
"""Module to implement conic geometry like circle and ellipse.
* circle with angular domain
* circle with start angle and arc length
* curvilinear distance on circle
* line-circle intersection
* circle-circle intersection
* point constructed from a virtual circle and a point on a tangent : right triangle
* point from tangent circle and segment ???
* ellipse with angular domain and rotation
"""
# Fixme:
#
# Ellipse passing by two points
# https://www.w3.org/TR/SVG/implnote.html#ArcConversionEndpointToCenter
#
####################################################################################################
__all__ = [
'Circle2D',
'Ellipse2D',
]
####################################################################################################
import math
from math import fabs, sqrt, radians, pi, cos, sin # , degrees
import numpy as np
from Patro.Common.Math.Functions import sign # , epsilon_float
from .BoundingBox import bounding_box_from_points
from .Mixin import AngularDomainMixin, CenterMixin, AngularDomain
from .Primitive import Primitive, Primitive2DMixin
from .Segment import Segment2D
from .Transformation import Transformation2D
####################################################################################################
_module_logger = logging.getLogger(__name__)
####################################################################################################
class PointNotOnCircleError(ValueError):
pass
####################################################################################################
class Circle2D(Primitive2DMixin, CenterMixin, AngularDomainMixin, Primitive):
##############################################
@classmethod
def from_two_points(cls, center, point):
"""Construct a circle from a center point and passing by another point"""
return cls(center, (point - center).magnitude)
##############################################
@classmethod
def from_triangle_circumcenter(cls, triangle):
"""Construct a circle passing by three point"""
return cls.from_two_points(triangle.circumcenter, triangle.p0)
##############################################
@classmethod
def from_triangle_in_circle(cls, triangle):
"""Construct the in circle of a triangle"""
return triangle.in_circle
##############################################
# @classmethod
# def from_start_angle_distance(cls, center, radius, start_angle, distance):
# """Construct a circle from a center point, a starting angle and a distance point"""
# if distance > 2*pi*radius:
# domain = None
# else:
# stop_angle = start_angle + math.degrees(distance / radius)
# domain = AngularDomain(start_angle, stop_angle)
# return cls(center, radius, domain)
##############################################
# Fixme: tangent constructs ...
##############################################
def __init__(self, center, radius,
domain=None,
diameter=False,
start_angle=None,
distance=None,
):
"""Construct a 2D circle from a center point and a radius.
If the circle is not closed, *domain* is a an :class:`AngularDomain` instance in degrees.
If *start_angle and *distance* is given then the stop angle is computed from them.
self._radius = radius
if start_angle is not None and distance is not None:
if distance > 2*pi*radius:
self._domain = None
else:
stop_angle = start_angle + math.degrees(distance / radius)
self._domain = AngularDomain(start_angle, stop_angle)
else:
self.domain = domain # Fixme: name ???
##############################################
Fabrice Salvaire
committed
def clone(self):
return self.__class__(self._center, self._radius, self._domain)
##############################################
def apply_transformation(self, transformation):
Fabrice Salvaire
committed
self._center = transformation * self._center
# Fixme: shear -> ellipse
if self._radius is not None:
self._radius = transformation * self._radius
##############################################
return '{0}({1._center}, {1._radius}, {1._domain})'.format(self.__class__.__name__, self)
@property
def radius(self):
return self._radius
@radius.setter
def radius(self, value):
self._radius = value
@property
def diameter(self):
return self._radius * 2
##############################################
@property
def eccentricity(self):
return 1
if self._domain is not None:
return 2*pi * self._radius
else:
return self._radius * self._domain.length
##############################################
return self.__vector_cls__.from_polar(self._radius, angle) + self._center
##############################################
def point_in_circle_frame(self, point):
return point - self._center
##############################################
def angle_for_point(self, point):
offset = self.point_in_circle_frame(point)
# distance = offset.magnitude_square
# if not epsilon_float(distance, self._radius**2):
# raise PointNotOnCircleError # ValueError('Point is not on circle')
# Fixme:
orientation = offset.orientation
if orientation < 0:
orientation = 360 + orientation
return orientation
##############################################
def point_at_distance(self, distance):
angle = math.degrees(distance / self._radius)
return self.point_at_angle(angle)
##############################################
point = self.__vector_cls__.from_polar(self._radius, angle) + self._center
tangent = (point - self._center).normal
return Line2D(point, tangent)
##############################################
@property
def bounding_box(self):
# Fixme: wrong for arc
return self._center.bounding_box.enlarge(self._radius)
##############################################
def signed_distance_to_point(self, point):
# d = |P - C| - R
# < 0 if inside
# = 0 on circle
# > 0 if outside
return (point - self._center).magnitude - self._radius
##############################################
def _circle_distance_to_point(self, point):
return abs(self.signed_distance_to_point(point))
##############################################
def distance_to_point(self, point):
if self._domain is not None:
# Fixme: check !!!
# try:
angle = self.angle_for_point(point)
# print('distance_to_circle', point, angle)
if self._domain.is_inside(angle):
# print('point is inside')
return self._circle_distance_to_point(point)
# except PointNotOnCircleError:
# pass
# print('point is outside')
return min([(point - vertex).magnitude for vertex in (self.start_point, self.stop_point)])
else:
return self._circle_distance_to_point(point)
##############################################
def is_point_inside(self, point):
return (point - self._center).magnitude_square <= self._radius**2
##############################################
def intersect_segment(self, segment):
r"""Compute the intersection of a circle and a segment.
* http://mathworld.wolfram.com/Circle-LineIntersection.html
* Rhoad et al. 1984, p. 429
* Rhoad, R.; Milauskas, G.; and Whipple, R. Geometry for Enjoyment and Challenge,
* rev. ed. Evanston, IL: McDougal, Littell & Company, 1984.
System of equations
.. math::
\begin{split}
x^2 + y^2 = r^2 \\
dx \times y = dy \times x - D
\end{split}
where
.. math::
\begin{align}
dx &= x1 - x0 \\
dy &= y1 - y0 \\
D &= x0 \times y1 - x1 \times y0
\end{align}
"""
# Fixme: check domain !!!
dx = segment.vector.x
dy = segment.vector.y
dr2 = dx**2 + dy**2
p0 = segment.p0 - self.center
p1 = segment.p1 - self.center
D = p0.cross_product(p1)
# from sympy import *
# x, y, dx, dy, D, r = symbols('x y dx dy D r')
# system = [x**2 + y**2 - r**2, dx*y - dy*x + D]
# vars = [x, y]
# solution = nonlinsolve(system, vars)
# solution.subs(dx**2 + dy**2, dr**2)
Vector2D = self.__vector_cls__
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discriminant = self.radius**2 * dr2 - D**2
if discriminant < 0:
return None
elif discriminant == 0: # tangent line
x = ( D * dy ) / dr2
y = (- D * dx ) / dr2
return Vector2D(x, y) + self.center
else: # intersection
x_a = D * dy
y_a = -D * dx
x_b = sign(dy) * dx * sqrt(discriminant)
y_b = fabs(dy) * sqrt(discriminant)
x0 = (x_a - x_b) / dr2
y0 = (y_a - y_b) / dr2
x1 = (x_a + x_b) / dr2
y1 = (y_a + y_b) / dr2
p0 = Vector2D(x0, y0) + self.center
p1 = Vector2D(x1, y1) + self.center
return p0, p1
##############################################
def intersect_circle(self, circle):
# Fixme: check domain !!!
# http://mathworld.wolfram.com/Circle-CircleIntersection.html
v = circle.center - self.center
d = sign(v.x) * v.magnitude
# Equations
# x**2 + y**2 = R**2
# (x-d)**2 + y**2 = r**2
x = (d**2 - circle.radius**2 + self.radius**2) / (2*d)
y2 = self.radius**2 - x**2
if y2 < 0:
return None
else:
p = self.center + v.normalise() * x
if y2 == 0:
return p
else:
n = v.normal() * sqrt(y2)
return p - n, p - n
##############################################
def bezier_approximation(self):
# http://spencermortensen.com/articles/bezier-circle/
# > First approximation:
#
# 1) The endpoints of the cubic Bézier curve must coincide with the endpoints of the
# circular arc, and their first derivatives must agree there.
#
# 2) The midpoint of the cubic Bézier curve must lie on the circle.
#
# B(t) = (1-t)**3 * P0 + 3*(1-t)**2*t * P1 + 3*(1-t)*t**2 * P2 + t**3 * P3
#
# For an unitary circle : P0 = (0,1) P1 = (c,1) P2 = (1,c) P3 = (1, 0)
#
# The second constraint provides the value of c = 4/3 * (sqrt(2) - 1)
#
# The maximum radial drift is 0.027253 % with this approximation.
# In this approximation, the Bézier curve always falls outside the circle, except
# momentarily when it dips in to touch the circle at the midpoint and endpoints.
#
# >Better approximation:
#
# 2) The maximum radial distance from the circle to the Bézier curve must be as small as
# possible.
#
# The first constraint yields the parametric form of the Bézier curve:
# B(t) = (x,y), where:
# x(t) = 3*c*(1-t)**2*t + 3*(1-t)*t**2 + t**3
# y(t) = 3*c*t**2*(1-t) + 3*t*(1-t)**2 + (1-t)**3
#
# The radial distance from the arc to the Bézier curve is: d(t) = sqrt(x**2 + y**2) - 1
#
# The Bézier curve touches the right circular arc at its initial endpoint, then drifts
# outside the arc, inside, outside again, and finally returns to touch the arc at its
# endpoint.
#
# roots of d : 0, (3*c +- sqrt(-9*c**2 - 24*c + 16) - 2)/(6*c - 4), 1
#
# This radial distance function, d(t), has minima at t = 0, 1/2, 1,
# and maxima at t = 1/2 +- sqrt(12 - 20*c - 3*c**22)/(4 - 6*c)
#
# Because the Bézier curve is symmetric about t = 1/2 , the two maxima have the same
# value. The radial deviation is minimized when the magnitude of this maximum is equal to
# the magnitude of the minimum at t = 1/2.
#
# This gives the ideal value for c = 0.551915024494
# The maximum radial drift is 0.019608 % with this approximation.
# P0 = (0,1) P1 = (c,1) P2 = (1,c) P3 = (1,0)
# P0 = (1,0) P1 = (1,-c) P2 = (c,-1) P3 = (0,-1)
# P0 = (0,-1) P1 = (-c,-1) P2 = (-1,-c) P3 = (-1,0)
# P0 = (-1,0) P1 = (-1,c) P2 = (-c,1) P3 = (0,1)
raise NotImplementedError
####################################################################################################
class Ellipse2D(Primitive2DMixin, CenterMixin, AngularDomainMixin, Primitive):
A general ellipse in 2D is represented by a center point `C`, an orthonormal set of
axis-direction vectors :math:`{U_0 , U_1 }`, and associated extents :math:`e_i` with :math:`e_0
\ge e_1 > 0`. The ellipse points are
.. math::
\begin{equation}
P = C + x_0 U_0 + x_1 U_1
\end{equation}
where
.. math::
\left(\frac{x_0}{e_0}\right)^2 + \left(\frac{x_1}{e_1}\right)^2 = 1
\end{equation}
If :math:`e_0 = e_1`, then the ellipse is a circle with center `C` and radius :math:`e_0`.
The orthonormality of the axis directions and Equation (1) imply :math:`x_i = U_i \dot (P −
C)`. Substituting this into Equation (2) we obtain
.. math::
(P − C)^T M (P − C) = 1
where :math:`M = R D R^T`, `R` is an orthogonal matrix whose columns are :math:`U_0` and
:math:`U_1` , and `D` is a diagonal matrix whose diagonal entries are :math:`1/e_0^2` and
:math:`1/e_1^2`.
An ellipse can also be parameterised by an angle :math:`\theta`
.. math::
\begin{pmatrix} x \\ y \end{pmatrix} =
\begin{bmatrix}
\cos\phi & \sin\phi \\
-\sin\phi & \cos\phi
\end{bmatrix}
\begin{pmatrix} r_x \cos\theta \\ r_y \sin\theta \end{pmatrix}
+ \begin{pmatrix} C_x \\ C_y \end{pmatrix}
where :math:`\phi` is the angle from the x-axis, :math:`r_x` is the semi-major and :math:`r_y`
semi-minor axes.
##############################################
@classmethod
def svg_arc(cls, point1, point2, radius_x, radius_y, angle, large_arc, sweep):
"""Implement SVG Arc.
Parameters
* *point1* is the start point and *point2* is the end point.
* *radius_x* and *radius_y* are the radii of the ellipse, also known as its semi-major and
semi-minor axes.
* *angle* is the angle from the x-axis of the current coordinate system to the x-axis of the ellipse.
* if the *large arc* flag is unset then arc spanning less than or equal to 180 degrees is
chosen, else an arc spanning greater than 180 degrees is chosen.
* if the *sweep* flag is unset then the line joining centre to arc sweeps through decreasing
angles, else if it sweeps through increasing angles.
References
* https://www.w3.org/TR/SVG/implnote.html#ArcConversionEndpointToCenter
* https://www.w3.org/TR/SVG/implnote.html#ArcCorrectionOutOfRangeRadii
"""
# Ensure radii are non-zero
return Segment2D(point1, point2)
# Ensure radii are positive
radius_x = abs(radius_x)
radius_y = abs(radius_y)
radius_x2 = radius_x**2
radius_y2 = radius_y**2
# We define a new referential with the origin is set to the middle of P1 — P2
origin_prime = (point1 + point2)/2
# P1 is exprimed in this referential where the ellipse major axis line up with the x axis
point1_prime = Transformation2D.Rotation(-angle) * (point1 - point2)/2
radii_scale = point1_prime.x**2/radius_x2 + point1_prime.y**2/radius_y2
radius_x = radii_scale * radius_x
radius_y = radii_scale * radius_y
radius_x2 = radius_x**2
radius_y2 = radius_y**2
den = radius_x2 * point1_prime.y**2 + radius_y2 * point1_prime.x**2
num = radius_x2*radius_y2 - den
sign = 1 if large_arc != sweep else -1
# print(point1_prime)
# print(point1_prime.anti_normal)
# print(ratio)
# print(point1_prime.anti_normal.scale(ratio, 1/ratio))
sign *= -1 # Fixme: solve mirroring artefacts for y-axis pointing to the top
center_prime = sign * math.sqrt(num / den) * point1_prime.anti_normal.scale(ratio, 1/ratio)
center = Transformation2D.Rotation(angle) * center_prime + origin_prime
vector1 = (point1_prime - center_prime).divide(radius_x, radius_y)
vector2 = - (point1_prime + center_prime).divide(radius_x, radius_y)
theta = cls.__vector_cls__(1, 0).angle_with(vector1)
delta_theta = vector1.angle_with(vector2)
# if theta < 0:
# theta = 180 + theta
# if delta_theta < 0:
# delta_theta = 180 + delta_theta
delta_theta = delta_theta % 360
# print('theta', theta, delta_theta)
if not sweep and delta_theta > 0:
delta_theta -= 360
elif sweep and delta_theta < 0:
delta_theta += 360
# print('theta', theta, delta_theta, theta + delta_theta)
domain = domain = AngularDomain(theta, theta + delta_theta)
return cls(center, radius_x, radius_y, angle, domain)
#######################################
def __init__(self, center, radius_x, radius_y, angle, domain=None):
self.radius_x = radius_x
self.radius_y = radius_y
self.angle = angle
self.domain = domain
self._bounding_box = None
##############################################
Fabrice Salvaire
committed
def clone(self):
return self.__class__(
self._center,
Fabrice Salvaire
committed
self._angle,
self._domain,
)
##############################################
def apply_transformation(self, transformation):
Fabrice Salvaire
committed
self._center = transformation * self._center
self._radius_x = transformation * self._radius_x
self._radius_y = transformation * self._radius_y
Fabrice Salvaire
committed
self._bounding_box = None
##############################################
return '{0}({1._center}, {1._radius_x}, {1._radius_x}, {1._angle})'.format(self.__class__.__name__, self)
@property
@radius_x.setter
def radius_x(self, value):
self._radius_x = float(value)
@property
@radius_y.setter
def radius_y(self, value):
self._radius_y = float(value)
@property
def angle(self):
return self._angle
@angle.setter
def angle(self, value):
self._angle = float(value)
@property
def major_vector(self):
# Fixme: x < y
return self.__vector_cls__.from_polar(self._angle, self._radius_x)
@property
def minor_vector(self):
# Fixme: x < y
return self.__vector_cls__.from_polar(self._angle + 90, self._radius_y)
##############################################
@property
def eccentricity(self):
# focal distance
# c = sqrt(self._radius_x**2 - self._radius_y**2)
# e = c / a
return sqrt(1 - (self._radius_y/self._radius_x)**2)
##############################################
def matrix(self):
# unit circle -> scale(a, b) -> rotation -> translation(xc, yc)
angle = radians(self._angle)
c = cos(angle)
s = sin(angle)
c2 = c**2
s2 = s**2
a2 = a**2
b2 = b**2
xc = self._center.x
yc = self._center.y
xc2 = xc**2
yc2 = yc**2
A = a2*s + b2*c2
B = 2*(b2 - a2)*c*s
C = a2*c2 * b2*s2
D = -2*A*xc - B*yc
E = -B*xc - 2*C*yc
F = A*xc2 + B*xc*yc + C*yc2 - a2*b2
return np.array((
( A, B/2, D/2),
(B/2, C, E/2),
(D/2, E/2, F),
##############################################
def point_in_ellipse_frame(self, point):
return (point - self._center).rotate(-self._angle)
def point_from_ellipse_frame(self, point):
return self._center + point.rotate(self._angle)
##############################################
# point = self.__vector_cls__.from_ellipse(self._radius_x, self._radius_y, angle)
# return self.point_from_ellipse_frame(point)
point = self.__vector_cls__.from_ellipse(self._radius_x, self._radius_y, self._angle + angle)
##############################################
@property
def bounding_box(self):
if self._bounding_box is None:
radius_x, radius_y = self._radius_x, self._radius_y
if self._angle == 0:
bounding_box = self._center.bounding_box
bounding_box.x.enlarge(radius_x)
bounding_box.y.enlarge(radius_y)
self._bounding_box = bounding_box
else:
angle_x = self._angle
angle_y = angle_x + 90
Vector2D = self.__vector_cls__
points = [self._center + offset for offset in (
Vector2D.from_polar(angle_x, radius_x),
Vector2D.from_polar(angle_x, -radius_x),
Vector2D.from_polar(angle_y, radius_y),
Vector2D.from_polar(angle_y, -radius_y),
)]
self._bounding_box = bounding_box_from_points(points)
return self._bounding_box
##############################################
@staticmethod
def _robust_length(x, y):
if x < y:
x, y = y, x
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##############################################
def _distance_point_bisection(self, r0, z0, z1, g):
n0 = r0 * z0
s0 = z1 - 1
if g < 0:
s1 = 0
else:
s1 = self._robust_length(n0 , z1) - 1
s = 0
MAX_ITERATION = 1074 # for double
for i in range(MAX_ITERATION):
s = (s0 + s1) / 2
if s == s0 or s == s1:
break
ratio0 = n0 / (s + r0)
ratio1 = z1 / (s + 1)
g = ratio0**2 + ratio1**2 -1
if g > 0:
s0 = s
elif g < 0:
s1 = s
else:
break
return s
##############################################
def _eberly_distance(self, point):
"""Compute distance to point using the algorithm described in
Distance from a Point to an Ellipse, an Ellipsoid, or a Hyperellipsoid
David Eberly, Geometric Tools, Redmond WA 98052
September 28, 2018
https://www.geometrictools.com/Documentation/Documentation.html
https://www.geometrictools.com/Documentation/DistancePointEllipseEllipsoid.pdf
The point is expressed in the ellipse coordinate system.
The preconditions are e0 ≥ e1 > 0, y0 ≥ 0, and y1 ≥ 0.
# Fixme: make a 3D plot to check the algorithm on a 2D grid and rotated ellipse
y0, y1 = point
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if y1 > 0:
if y0 > 0:
z0 = y0 / e0
z1 = y1 / e1
g = z0**2 + z1**2 - 1
if g != 0:
r0 = (e0 / e1)**2
sbar = self._distance_point_bisection(r0, z0, z1, g)
x0 = r0 * y0 / (sbar + r0)
x1 = y1 / (sbar + 1)
distance = math.sqrt((x0 - y0)**2 + (x1 - y1)**2)
else:
x0 = y0
x1 = y1
distance = 0
else:
# y0 == 0
x0 = 0
x1 = e1
distance = abs(y1 - e1)
else:
# y1 == 0
numer0 = e0 * y0
denom0 = e0**2 - e1**2
if numer0 < denom0:
xde0 = numer0 / denom0
x0 = e0 * xde0
x1 = e1 * math.sqrt(1 - xde0**2)
distance = math.sqrt((x0 - y0)**2 + x1**2)
else:
x0 = e0
x1 = 0
distance = abs(y0 - e0)
return distance, self.__vector_cls__(x0, x1)
##############################################
def distance_to_point(self, point, return_point=False, is_inside=False):
# Fixme: can be transform the problem to a circle using transformation ???
point_in_frame = self.point_in_ellipse_frame(point)
point_in_frame_abs = self.__vector_cls__(abs(point_in_frame.x), abs(point_in_frame.y))
distance, point_in_ellipse = self._eberly_distance(point_in_frame_abs)
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if is_inside:
# Fixme: right ???
return (
(point_in_frame_abs - self._center).magnitude_square
<=
(point_in_ellipse - self._center).magnitude_square
)
elif return_point:
point_in_ellipse = self.__vector_cls__(
sign(point_in_frame.x)*(point_in_ellipse.x),
sign(point_in_frame.y)*(point_in_ellipse.y),
)
point_in_ellipse = self.point_from_ellipse_frame(point_in_ellipse)
return distance, point_in_ellipse
else:
return distance
##############################################
def is_point_inside(self, point):
return self.distance_to_point(point, is_inside=True)
##############################################
def intersect_segment(self, segment):
# Fixme: to be checked
# Map segment in ellipse frame and scale y axis so as to transform the ellipse to a circle
points = [self.point_in_ellipse_frame(point) for point in segment.points]
points = [self.__vector_cls__(point.x, point.y * y_scale) for point in points]
segment_in_frame = Segment2D(*points)
circle = Circle2D(self.__vector_cls__(0, 0), self._radius_x)
points = circle.intersect_segment(segment_in_frame)
points = [self.__vector_cls__(point.x, point.y / y_scale) for point in points]
points = [self.point_from_ellipse_frame(point) for point in points]
return points
##############################################
def intersect_conic(self, conic):
"""
Reference
* Intersection of Ellipses
* David Eberly, Geometric Tools, Redmond WA 98052
* June 23, 2015
* https://www.geometrictools.com/
* https://www.geometrictools.com/Documentation/IntersectionOfEllipses.pdf
"""
raise NotImplementedError