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603 lines
19 KiB
Python
603 lines
19 KiB
Python
"""
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A module providing some utility functions regarding Bézier path manipulation.
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"""
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from functools import lru_cache
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import math
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import warnings
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import numpy as np
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from matplotlib import _api
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# same algorithm as 3.8's math.comb
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@np.vectorize
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@lru_cache(maxsize=128)
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def _comb(n, k):
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if k > n:
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return 0
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k = min(k, n - k)
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i = np.arange(1, k + 1)
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return np.prod((n + 1 - i)/i).astype(int)
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class NonIntersectingPathException(ValueError):
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pass
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# some functions
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def get_intersection(cx1, cy1, cos_t1, sin_t1,
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cx2, cy2, cos_t2, sin_t2):
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"""
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Return the intersection between the line through (*cx1*, *cy1*) at angle
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*t1* and the line through (*cx2*, *cy2*) at angle *t2*.
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"""
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# line1 => sin_t1 * (x - cx1) - cos_t1 * (y - cy1) = 0.
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# line1 => sin_t1 * x + cos_t1 * y = sin_t1*cx1 - cos_t1*cy1
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line1_rhs = sin_t1 * cx1 - cos_t1 * cy1
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line2_rhs = sin_t2 * cx2 - cos_t2 * cy2
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# rhs matrix
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a, b = sin_t1, -cos_t1
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c, d = sin_t2, -cos_t2
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ad_bc = a * d - b * c
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if abs(ad_bc) < 1e-12:
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raise ValueError("Given lines do not intersect. Please verify that "
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"the angles are not equal or differ by 180 degrees.")
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# rhs_inverse
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a_, b_ = d, -b
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c_, d_ = -c, a
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a_, b_, c_, d_ = (k / ad_bc for k in [a_, b_, c_, d_])
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x = a_ * line1_rhs + b_ * line2_rhs
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y = c_ * line1_rhs + d_ * line2_rhs
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return x, y
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def get_normal_points(cx, cy, cos_t, sin_t, length):
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"""
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For a line passing through (*cx*, *cy*) and having an angle *t*, return
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locations of the two points located along its perpendicular line at the
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distance of *length*.
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"""
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if length == 0.:
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return cx, cy, cx, cy
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cos_t1, sin_t1 = sin_t, -cos_t
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cos_t2, sin_t2 = -sin_t, cos_t
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x1, y1 = length * cos_t1 + cx, length * sin_t1 + cy
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x2, y2 = length * cos_t2 + cx, length * sin_t2 + cy
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return x1, y1, x2, y2
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# BEZIER routines
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# subdividing bezier curve
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# http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/Bezier/bezier-sub.html
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def _de_casteljau1(beta, t):
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next_beta = beta[:-1] * (1 - t) + beta[1:] * t
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return next_beta
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def split_de_casteljau(beta, t):
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"""
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Split a Bézier segment defined by its control points *beta* into two
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separate segments divided at *t* and return their control points.
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"""
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beta = np.asarray(beta)
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beta_list = [beta]
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while True:
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beta = _de_casteljau1(beta, t)
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beta_list.append(beta)
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if len(beta) == 1:
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break
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left_beta = [beta[0] for beta in beta_list]
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right_beta = [beta[-1] for beta in reversed(beta_list)]
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return left_beta, right_beta
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def find_bezier_t_intersecting_with_closedpath(
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bezier_point_at_t, inside_closedpath, t0=0., t1=1., tolerance=0.01):
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"""
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Find the intersection of the Bézier curve with a closed path.
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The intersection point *t* is approximated by two parameters *t0*, *t1*
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such that *t0* <= *t* <= *t1*.
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Search starts from *t0* and *t1* and uses a simple bisecting algorithm
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therefore one of the end points must be inside the path while the other
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doesn't. The search stops when the distance of the points parametrized by
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*t0* and *t1* gets smaller than the given *tolerance*.
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Parameters
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----------
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bezier_point_at_t : callable
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A function returning x, y coordinates of the Bézier at parameter *t*.
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It must have the signature::
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bezier_point_at_t(t: float) -> tuple[float, float]
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inside_closedpath : callable
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A function returning True if a given point (x, y) is inside the
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closed path. It must have the signature::
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inside_closedpath(point: tuple[float, float]) -> bool
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t0, t1 : float
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Start parameters for the search.
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tolerance : float
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Maximal allowed distance between the final points.
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Returns
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-------
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t0, t1 : float
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The Bézier path parameters.
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"""
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start = bezier_point_at_t(t0)
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end = bezier_point_at_t(t1)
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start_inside = inside_closedpath(start)
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end_inside = inside_closedpath(end)
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if start_inside == end_inside and start != end:
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raise NonIntersectingPathException(
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"Both points are on the same side of the closed path")
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while True:
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# return if the distance is smaller than the tolerance
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if np.hypot(start[0] - end[0], start[1] - end[1]) < tolerance:
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return t0, t1
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# calculate the middle point
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middle_t = 0.5 * (t0 + t1)
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middle = bezier_point_at_t(middle_t)
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middle_inside = inside_closedpath(middle)
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if start_inside ^ middle_inside:
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t1 = middle_t
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if end == middle:
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# Edge case where infinite loop is possible
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# Caused by large numbers relative to tolerance
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return t0, t1
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end = middle
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else:
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t0 = middle_t
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if start == middle:
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# Edge case where infinite loop is possible
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# Caused by large numbers relative to tolerance
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return t0, t1
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start = middle
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start_inside = middle_inside
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class BezierSegment:
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"""
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A d-dimensional Bézier segment.
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Parameters
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----------
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control_points : (N, d) array
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Location of the *N* control points.
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"""
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def __init__(self, control_points):
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self._cpoints = np.asarray(control_points)
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self._N, self._d = self._cpoints.shape
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self._orders = np.arange(self._N)
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coeff = [math.factorial(self._N - 1)
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// (math.factorial(i) * math.factorial(self._N - 1 - i))
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for i in range(self._N)]
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self._px = (self._cpoints.T * coeff).T
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def __call__(self, t):
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"""
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Evaluate the Bézier curve at point(s) *t* in [0, 1].
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Parameters
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----------
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t : (k,) array-like
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Points at which to evaluate the curve.
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Returns
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-------
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(k, d) array
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Value of the curve for each point in *t*.
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"""
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t = np.asarray(t)
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return (np.power.outer(1 - t, self._orders[::-1])
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* np.power.outer(t, self._orders)) @ self._px
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def point_at_t(self, t):
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"""
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Evaluate the curve at a single point, returning a tuple of *d* floats.
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"""
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return tuple(self(t))
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@property
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def control_points(self):
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"""The control points of the curve."""
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return self._cpoints
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@property
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def dimension(self):
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"""The dimension of the curve."""
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return self._d
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@property
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def degree(self):
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"""Degree of the polynomial. One less the number of control points."""
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return self._N - 1
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@property
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def polynomial_coefficients(self):
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r"""
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The polynomial coefficients of the Bézier curve.
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.. warning:: Follows opposite convention from `numpy.polyval`.
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Returns
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-------
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(n+1, d) array
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Coefficients after expanding in polynomial basis, where :math:`n`
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is the degree of the Bézier curve and :math:`d` its dimension.
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These are the numbers (:math:`C_j`) such that the curve can be
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written :math:`\sum_{j=0}^n C_j t^j`.
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Notes
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-----
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The coefficients are calculated as
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.. math::
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{n \choose j} \sum_{i=0}^j (-1)^{i+j} {j \choose i} P_i
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where :math:`P_i` are the control points of the curve.
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"""
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n = self.degree
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# matplotlib uses n <= 4. overflow plausible starting around n = 15.
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if n > 10:
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warnings.warn("Polynomial coefficients formula unstable for high "
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"order Bezier curves!", RuntimeWarning)
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P = self.control_points
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j = np.arange(n+1)[:, None]
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i = np.arange(n+1)[None, :] # _comb is non-zero for i <= j
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prefactor = (-1)**(i + j) * _comb(j, i) # j on axis 0, i on axis 1
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return _comb(n, j) * prefactor @ P # j on axis 0, self.dimension on 1
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def axis_aligned_extrema(self):
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"""
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Return the dimension and location of the curve's interior extrema.
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The extrema are the points along the curve where one of its partial
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derivatives is zero.
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Returns
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-------
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dims : array of int
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Index :math:`i` of the partial derivative which is zero at each
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interior extrema.
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dzeros : array of float
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Of same size as dims. The :math:`t` such that :math:`d/dx_i B(t) =
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0`
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"""
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n = self.degree
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if n <= 1:
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return np.array([]), np.array([])
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Cj = self.polynomial_coefficients
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dCj = np.arange(1, n+1)[:, None] * Cj[1:]
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dims = []
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roots = []
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for i, pi in enumerate(dCj.T):
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r = np.roots(pi[::-1])
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roots.append(r)
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dims.append(np.full_like(r, i))
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roots = np.concatenate(roots)
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dims = np.concatenate(dims)
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in_range = np.isreal(roots) & (roots >= 0) & (roots <= 1)
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return dims[in_range], np.real(roots)[in_range]
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def split_bezier_intersecting_with_closedpath(
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bezier, inside_closedpath, tolerance=0.01):
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"""
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Split a Bézier curve into two at the intersection with a closed path.
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Parameters
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----------
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bezier : (N, 2) array-like
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Control points of the Bézier segment. See `.BezierSegment`.
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inside_closedpath : callable
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A function returning True if a given point (x, y) is inside the
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closed path. See also `.find_bezier_t_intersecting_with_closedpath`.
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tolerance : float
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The tolerance for the intersection. See also
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`.find_bezier_t_intersecting_with_closedpath`.
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Returns
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-------
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left, right
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Lists of control points for the two Bézier segments.
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"""
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bz = BezierSegment(bezier)
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bezier_point_at_t = bz.point_at_t
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t0, t1 = find_bezier_t_intersecting_with_closedpath(
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bezier_point_at_t, inside_closedpath, tolerance=tolerance)
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_left, _right = split_de_casteljau(bezier, (t0 + t1) / 2.)
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return _left, _right
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# matplotlib specific
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def split_path_inout(path, inside, tolerance=0.01, reorder_inout=False):
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"""
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Divide a path into two segments at the point where ``inside(x, y)`` becomes
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False.
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"""
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from .path import Path
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path_iter = path.iter_segments()
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ctl_points, command = next(path_iter)
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begin_inside = inside(ctl_points[-2:]) # true if begin point is inside
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ctl_points_old = ctl_points
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iold = 0
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i = 1
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for ctl_points, command in path_iter:
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iold = i
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i += len(ctl_points) // 2
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if inside(ctl_points[-2:]) != begin_inside:
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bezier_path = np.concatenate([ctl_points_old[-2:], ctl_points])
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break
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ctl_points_old = ctl_points
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else:
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raise ValueError("The path does not intersect with the patch")
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bp = bezier_path.reshape((-1, 2))
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left, right = split_bezier_intersecting_with_closedpath(
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bp, inside, tolerance)
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if len(left) == 2:
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codes_left = [Path.LINETO]
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codes_right = [Path.MOVETO, Path.LINETO]
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elif len(left) == 3:
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codes_left = [Path.CURVE3, Path.CURVE3]
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codes_right = [Path.MOVETO, Path.CURVE3, Path.CURVE3]
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elif len(left) == 4:
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codes_left = [Path.CURVE4, Path.CURVE4, Path.CURVE4]
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codes_right = [Path.MOVETO, Path.CURVE4, Path.CURVE4, Path.CURVE4]
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else:
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raise AssertionError("This should never be reached")
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verts_left = left[1:]
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verts_right = right[:]
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if path.codes is None:
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path_in = Path(np.concatenate([path.vertices[:i], verts_left]))
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path_out = Path(np.concatenate([verts_right, path.vertices[i:]]))
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else:
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path_in = Path(np.concatenate([path.vertices[:iold], verts_left]),
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np.concatenate([path.codes[:iold], codes_left]))
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path_out = Path(np.concatenate([verts_right, path.vertices[i:]]),
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np.concatenate([codes_right, path.codes[i:]]))
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if reorder_inout and not begin_inside:
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path_in, path_out = path_out, path_in
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return path_in, path_out
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def inside_circle(cx, cy, r):
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"""
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Return a function that checks whether a point is in a circle with center
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(*cx*, *cy*) and radius *r*.
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The returned function has the signature::
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f(xy: tuple[float, float]) -> bool
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"""
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r2 = r ** 2
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def _f(xy):
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x, y = xy
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return (x - cx) ** 2 + (y - cy) ** 2 < r2
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return _f
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# quadratic Bezier lines
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def get_cos_sin(x0, y0, x1, y1):
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dx, dy = x1 - x0, y1 - y0
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d = (dx * dx + dy * dy) ** .5
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# Account for divide by zero
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if d == 0:
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return 0.0, 0.0
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return dx / d, dy / d
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def check_if_parallel(dx1, dy1, dx2, dy2, tolerance=1.e-5):
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"""
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Check if two lines are parallel.
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Parameters
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----------
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dx1, dy1, dx2, dy2 : float
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The gradients *dy*/*dx* of the two lines.
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tolerance : float
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The angular tolerance in radians up to which the lines are considered
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parallel.
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Returns
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-------
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is_parallel
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- 1 if two lines are parallel in same direction.
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- -1 if two lines are parallel in opposite direction.
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- False otherwise.
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"""
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theta1 = np.arctan2(dx1, dy1)
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theta2 = np.arctan2(dx2, dy2)
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dtheta = abs(theta1 - theta2)
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if dtheta < tolerance:
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return 1
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elif abs(dtheta - np.pi) < tolerance:
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return -1
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else:
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return False
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def get_parallels(bezier2, width):
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"""
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Given the quadratic Bézier control points *bezier2*, returns
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control points of quadratic Bézier lines roughly parallel to given
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one separated by *width*.
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"""
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# The parallel Bezier lines are constructed by following ways.
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# c1 and c2 are control points representing the start and end of the
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# Bezier line.
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# cm is the middle point
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c1x, c1y = bezier2[0]
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cmx, cmy = bezier2[1]
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c2x, c2y = bezier2[2]
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parallel_test = check_if_parallel(c1x - cmx, c1y - cmy,
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cmx - c2x, cmy - c2y)
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if parallel_test == -1:
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_api.warn_external(
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"Lines do not intersect. A straight line is used instead.")
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cos_t1, sin_t1 = get_cos_sin(c1x, c1y, c2x, c2y)
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cos_t2, sin_t2 = cos_t1, sin_t1
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else:
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# t1 and t2 is the angle between c1 and cm, cm, c2. They are
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# also an angle of the tangential line of the path at c1 and c2
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cos_t1, sin_t1 = get_cos_sin(c1x, c1y, cmx, cmy)
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cos_t2, sin_t2 = get_cos_sin(cmx, cmy, c2x, c2y)
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# find c1_left, c1_right which are located along the lines
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# through c1 and perpendicular to the tangential lines of the
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# Bezier path at a distance of width. Same thing for c2_left and
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# c2_right with respect to c2.
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c1x_left, c1y_left, c1x_right, c1y_right = (
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get_normal_points(c1x, c1y, cos_t1, sin_t1, width)
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)
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c2x_left, c2y_left, c2x_right, c2y_right = (
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get_normal_points(c2x, c2y, cos_t2, sin_t2, width)
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)
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# find cm_left which is the intersecting point of a line through
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# c1_left with angle t1 and a line through c2_left with angle
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# t2. Same with cm_right.
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try:
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cmx_left, cmy_left = get_intersection(c1x_left, c1y_left, cos_t1,
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sin_t1, c2x_left, c2y_left,
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|
cos_t2, sin_t2)
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|
cmx_right, cmy_right = get_intersection(c1x_right, c1y_right, cos_t1,
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|
sin_t1, c2x_right, c2y_right,
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|
cos_t2, sin_t2)
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except ValueError:
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|
# Special case straight lines, i.e., angle between two lines is
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|
# less than the threshold used by get_intersection (we don't use
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|
# check_if_parallel as the threshold is not the same).
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|
cmx_left, cmy_left = (
|
|
0.5 * (c1x_left + c2x_left), 0.5 * (c1y_left + c2y_left)
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|
)
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|
cmx_right, cmy_right = (
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|
0.5 * (c1x_right + c2x_right), 0.5 * (c1y_right + c2y_right)
|
|
)
|
|
|
|
# the parallel Bezier lines are created with control points of
|
|
# [c1_left, cm_left, c2_left] and [c1_right, cm_right, c2_right]
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|
path_left = [(c1x_left, c1y_left),
|
|
(cmx_left, cmy_left),
|
|
(c2x_left, c2y_left)]
|
|
path_right = [(c1x_right, c1y_right),
|
|
(cmx_right, cmy_right),
|
|
(c2x_right, c2y_right)]
|
|
|
|
return path_left, path_right
|
|
|
|
|
|
def find_control_points(c1x, c1y, mmx, mmy, c2x, c2y):
|
|
"""
|
|
Find control points of the Bézier curve passing through (*c1x*, *c1y*),
|
|
(*mmx*, *mmy*), and (*c2x*, *c2y*), at parametric values 0, 0.5, and 1.
|
|
"""
|
|
cmx = .5 * (4 * mmx - (c1x + c2x))
|
|
cmy = .5 * (4 * mmy - (c1y + c2y))
|
|
return [(c1x, c1y), (cmx, cmy), (c2x, c2y)]
|
|
|
|
|
|
def make_wedged_bezier2(bezier2, width, w1=1., wm=0.5, w2=0.):
|
|
"""
|
|
Being similar to `get_parallels`, returns control points of two quadratic
|
|
Bézier lines having a width roughly parallel to given one separated by
|
|
*width*.
|
|
"""
|
|
|
|
# c1, cm, c2
|
|
c1x, c1y = bezier2[0]
|
|
cmx, cmy = bezier2[1]
|
|
c3x, c3y = bezier2[2]
|
|
|
|
# t1 and t2 is the angle between c1 and cm, cm, c3.
|
|
# They are also an angle of the tangential line of the path at c1 and c3
|
|
cos_t1, sin_t1 = get_cos_sin(c1x, c1y, cmx, cmy)
|
|
cos_t2, sin_t2 = get_cos_sin(cmx, cmy, c3x, c3y)
|
|
|
|
# find c1_left, c1_right which are located along the lines
|
|
# through c1 and perpendicular to the tangential lines of the
|
|
# Bezier path at a distance of width. Same thing for c3_left and
|
|
# c3_right with respect to c3.
|
|
c1x_left, c1y_left, c1x_right, c1y_right = (
|
|
get_normal_points(c1x, c1y, cos_t1, sin_t1, width * w1)
|
|
)
|
|
c3x_left, c3y_left, c3x_right, c3y_right = (
|
|
get_normal_points(c3x, c3y, cos_t2, sin_t2, width * w2)
|
|
)
|
|
|
|
# find c12, c23 and c123 which are middle points of c1-cm, cm-c3 and
|
|
# c12-c23
|
|
c12x, c12y = (c1x + cmx) * .5, (c1y + cmy) * .5
|
|
c23x, c23y = (cmx + c3x) * .5, (cmy + c3y) * .5
|
|
c123x, c123y = (c12x + c23x) * .5, (c12y + c23y) * .5
|
|
|
|
# tangential angle of c123 (angle between c12 and c23)
|
|
cos_t123, sin_t123 = get_cos_sin(c12x, c12y, c23x, c23y)
|
|
|
|
c123x_left, c123y_left, c123x_right, c123y_right = (
|
|
get_normal_points(c123x, c123y, cos_t123, sin_t123, width * wm)
|
|
)
|
|
|
|
path_left = find_control_points(c1x_left, c1y_left,
|
|
c123x_left, c123y_left,
|
|
c3x_left, c3y_left)
|
|
path_right = find_control_points(c1x_right, c1y_right,
|
|
c123x_right, c123y_right,
|
|
c3x_right, c3y_right)
|
|
|
|
return path_left, path_right
|