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#
# Anti-Grain Geometry - Version 2.4
# Copyright (C) 2002-2005 Maxim Shemanarev (McSeem)
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions
# are met:
#
# 1. Redistributions of source code must retain the above copyright
# notice, this list of conditions and the following disclaimer.
#
# 2. Redistributions in binary form must reproduce the above copyright
# notice, this list of conditions and the following disclaimer in
# the documentation and/or other materials provided with the
# distribution.
#
# 3. The name of the author may not be used to endorse or promote
# products derived from this software without specific prior
# written permission.
#
# THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
# IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
# ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT,
# INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
# (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
# SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
# HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
# STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING
# IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
# POSSIBILITY OF SUCH DAMAGE.
# ----------------------------------------------------------------------------
#
# Python translation by Nicolas P. Rougier
# Copyright (C) 2013 Nicolas P. Rougier. All rights reserved.
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions are met:
#
# 1. Redistributions of source code must retain the above copyright notice,
# this list of conditions and the following disclaimer.
#
# 2. Redistributions in binary form must reproduce the above copyright
# notice, this list of conditions and the following disclaimer in the
# documentation and/or other materials provided with the distribution.
#
# THIS SOFTWARE IS PROVIDED BY NICOLAS P. ROUGIER ''AS IS'' AND ANY EXPRESS OR
# IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
# MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO
# EVENT SHALL NICOLAS P. ROUGIER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT,
# INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
# (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
# LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
# ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
# (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
# THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
#
# The views and conclusions contained in the software and documentation are
# those of the authors and should not be interpreted as representing official
# policies, either expressed or implied, of Nicolas P. Rougier.
#
import math
import numpy as np
curve_distance_epsilon = 1e-30
curve_collinearity_epsilon = 1e-30
curve_angle_tolerance_epsilon = 0.01
curve_recursion_limit = 32
m_cusp_limit = 0.0
m_angle_tolerance = 10 * math.pi / 180.0
m_approximation_scale = 1.0
m_distance_tolerance_square = (0.5 / m_approximation_scale)**2
def calc_sq_distance(x1, y1, x2, y2):
dx = x2 - x1
dy = y2 - y1
return dx * dx + dy * dy
def _curve3_recursive_bezier(points, x1, y1, x2, y2, x3, y3, level=0):
if level > curve_recursion_limit:
return
# Calculate all the mid-points of the line segments
x12 = (x1 + x2) / 2.
y12 = (y1 + y2) / 2.
x23 = (x2 + x3) / 2.
y23 = (y2 + y3) / 2.
x123 = (x12 + x23) / 2.
y123 = (y12 + y23) / 2.
dx = x3 - x1
dy = y3 - y1
d = math.fabs((x2 - x3) * dy - (y2 - y3) * dx)
if d > curve_collinearity_epsilon:
# Regular case
if d * d <= m_distance_tolerance_square * (dx * dx + dy * dy):
# If the curvature doesn't exceed the distance_tolerance value
# we tend to finish subdivisions.
if m_angle_tolerance < curve_angle_tolerance_epsilon:
points.append((x123, y123))
return
# Angle & Cusp Condition
da = math.fabs(
math.atan2(y3 - y2, x3 - x2) - math.atan2(y2 - y1, x2 - x1))
if da >= math.pi:
da = 2 * math.pi - da
if da < m_angle_tolerance:
# Finally we can stop the recursion
points.append((x123, y123))
return
else:
# Collinear case
da = dx * dx + dy * dy
if da == 0:
d = calc_sq_distance(x1, y1, x2, y2)
else:
d = ((x2 - x1) * dx + (y2 - y1) * dy) / da
if d > 0 and d < 1:
# Simple collinear case, 1---2---3, we can leave just two
# endpoints
return
if(d <= 0):
d = calc_sq_distance(x2, y2, x1, y1)
elif d >= 1:
d = calc_sq_distance(x2, y2, x3, y3)
else:
d = calc_sq_distance(x2, y2, x1 + d * dx, y1 + d * dy)
if d < m_distance_tolerance_square:
points.append((x2, y2))
return
# Continue subdivision
_curve3_recursive_bezier(points, x1, y1, x12, y12, x123, y123, level + 1)
_curve3_recursive_bezier(points, x123, y123, x23, y23, x3, y3, level + 1)
def _curve4_recursive_bezier(points, x1, y1, x2, y2, x3, y3, x4, y4, level=0):
if level > curve_recursion_limit:
return
# Calculate all the mid-points of the line segments
x12 = (x1 + x2) / 2.
y12 = (y1 + y2) / 2.
x23 = (x2 + x3) / 2.
y23 = (y2 + y3) / 2.
x34 = (x3 + x4) / 2.
y34 = (y3 + y4) / 2.
x123 = (x12 + x23) / 2.
y123 = (y12 + y23) / 2.
x234 = (x23 + x34) / 2.
y234 = (y23 + y34) / 2.
x1234 = (x123 + x234) / 2.
y1234 = (y123 + y234) / 2.
# Try to approximate the full cubic curve by a single straight line
dx = x4 - x1
dy = y4 - y1
d2 = math.fabs(((x2 - x4) * dy - (y2 - y4) * dx))
d3 = math.fabs(((x3 - x4) * dy - (y3 - y4) * dx))
s = int((d2 > curve_collinearity_epsilon) << 1) + \
int(d3 > curve_collinearity_epsilon)
if s == 0:
# All collinear OR p1==p4
k = dx * dx + dy * dy
if k == 0:
d2 = calc_sq_distance(x1, y1, x2, y2)
d3 = calc_sq_distance(x4, y4, x3, y3)
else:
k = 1. / k
da1 = x2 - x1
da2 = y2 - y1
d2 = k * (da1 * dx + da2 * dy)
da1 = x3 - x1
da2 = y3 - y1
d3 = k * (da1 * dx + da2 * dy)
if d2 > 0 and d2 < 1 and d3 > 0 and d3 < 1:
# Simple collinear case, 1---2---3---4
# We can leave just two endpoints
return
if d2 <= 0:
d2 = calc_sq_distance(x2, y2, x1, y1)
elif d2 >= 1:
d2 = calc_sq_distance(x2, y2, x4, y4)
else:
d2 = calc_sq_distance(x2, y2, x1 + d2 * dx, y1 + d2 * dy)
if d3 <= 0:
d3 = calc_sq_distance(x3, y3, x1, y1)
elif d3 >= 1:
d3 = calc_sq_distance(x3, y3, x4, y4)
else:
d3 = calc_sq_distance(x3, y3, x1 + d3 * dx, y1 + d3 * dy)
if d2 > d3:
if d2 < m_distance_tolerance_square:
points.append((x2, y2))
return
else:
if d3 < m_distance_tolerance_square:
points.append((x3, y3))
return
elif s == 1:
# p1,p2,p4 are collinear, p3 is significant
if d3 * d3 <= m_distance_tolerance_square * (dx * dx + dy * dy):
if m_angle_tolerance < curve_angle_tolerance_epsilon:
points.append((x23, y23))
return
# Angle Condition
da1 = math.fabs(
math.atan2(y4 - y3, x4 - x3) - math.atan2(y3 - y2, x3 - x2))
if da1 >= math.pi:
da1 = 2 * math.pi - da1
if da1 < m_angle_tolerance:
points.extend([(x2, y2), (x3, y3)])
return
if m_cusp_limit != 0.0:
if da1 > m_cusp_limit:
points.append((x3, y3))
return
elif s == 2:
# p1,p3,p4 are collinear, p2 is significant
if d2 * d2 <= m_distance_tolerance_square * (dx * dx + dy * dy):
if m_angle_tolerance < curve_angle_tolerance_epsilon:
points.append((x23, y23))
return
# Angle Condition
# ---------------
da1 = math.fabs(
math.atan2(y3 - y2, x3 - x2) - math.atan2(y2 - y1, x2 - x1))
if da1 >= math.pi:
da1 = 2 * math.pi - da1
if da1 < m_angle_tolerance:
points.extend([(x2, y2), (x3, y3)])
return
if m_cusp_limit != 0.0:
if da1 > m_cusp_limit:
points.append((x2, y2))
return
elif s == 3:
# Regular case
if (d2 + d3) * (d2 + d3) <= m_distance_tolerance_square * (
dx * dx + dy * dy):
# If the curvature doesn't exceed the distance_tolerance value
# we tend to finish subdivisions.
if m_angle_tolerance < curve_angle_tolerance_epsilon:
points.append((x23, y23))
return
# Angle & Cusp Condition
k = math.atan2(y3 - y2, x3 - x2)
da1 = math.fabs(k - math.atan2(y2 - y1, x2 - x1))
da2 = math.fabs(math.atan2(y4 - y3, x4 - x3) - k)
if da1 >= math.pi:
da1 = 2 * math.pi - da1
if da2 >= math.pi:
da2 = 2 * math.pi - da2
if da1 + da2 < m_angle_tolerance:
# Finally we can stop the recursion
points.append((x23, y23))
return
if m_cusp_limit != 0.0:
if da1 > m_cusp_limit:
points.append((x2, y2))
return
if da2 > m_cusp_limit:
points.append((x3, y3))
return
# Continue subdivision
_curve4_recursive_bezier(
points, x1, y1, x12, y12, x123, y123, x1234, y1234, level + 1)
_curve4_recursive_bezier(
points, x1234, y1234, x234, y234, x34, y34, x4, y4, level + 1)
def curve3_bezier(p1, p2, p3):
"""
Generate the vertices for a quadratic Bezier curve.
The vertices returned by this function can be passed to a LineVisual or
ArrowVisual.
Parameters
----------
p1 : array
2D coordinates of the start point
p2 : array
2D coordinates of the first curve point
p3 : array
2D coordinates of the end point
Returns
-------
coords : list
Vertices for the Bezier curve.
See Also
--------
curve4_bezier
Notes
-----
For more information about Bezier curves please refer to the `Wikipedia`_
page.
.. _Wikipedia: https://en.wikipedia.org/wiki/B%C3%A9zier_curve
"""
x1, y1 = p1
x2, y2 = p2
x3, y3 = p3
points = []
_curve3_recursive_bezier(points, x1, y1, x2, y2, x3, y3)
dx, dy = points[0][0] - x1, points[0][1] - y1
if (dx * dx + dy * dy) > 1e-10:
points.insert(0, (x1, y1))
dx, dy = points[-1][0] - x3, points[-1][1] - y3
if (dx * dx + dy * dy) > 1e-10:
points.append((x3, y3))
return np.array(points).reshape(len(points), 2)
def curve4_bezier(p1, p2, p3, p4):
"""
Generate the vertices for a third order Bezier curve.
The vertices returned by this function can be passed to a LineVisual or
ArrowVisual.
Parameters
----------
p1 : array
2D coordinates of the start point
p2 : array
2D coordinates of the first curve point
p3 : array
2D coordinates of the second curve point
p4 : array
2D coordinates of the end point
Returns
-------
coords : list
Vertices for the Bezier curve.
See Also
--------
curve3_bezier
Notes
-----
For more information about Bezier curves please refer to the `Wikipedia`_
page.
.. _Wikipedia: https://en.wikipedia.org/wiki/B%C3%A9zier_curve
"""
x1, y1 = p1
x2, y2 = p2
x3, y3 = p3
x4, y4 = p4
points = []
_curve4_recursive_bezier(points, x1, y1, x2, y2, x3, y3, x4, y4)
dx, dy = points[0][0] - x1, points[0][1] - y1
if (dx * dx + dy * dy) > 1e-10:
points.insert(0, (x1, y1))
dx, dy = points[-1][0] - x4, points[-1][1] - y4
if (dx * dx + dy * dy) > 1e-10:
points.append((x4, y4))
return np.array(points).reshape(len(points), 2)
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