"""
A module providing some utility functions regarding Bezier path manipulation.
"""
from functools import lru_cache
import math
import warnings
import numpy as np
import matplotlib.cbook as cbook
# same algorithm as 3.8's math.comb
@np.vectorize
@lru_cache(maxsize=128)
def _comb(n, k):
if k > n:
return 0
k = min(k, n - k)
i = np.arange(1, k + 1)
return np.prod((n + 1 - i)/i).astype(int)
[docs]class NonIntersectingPathException(ValueError):
pass
# some functions
[docs]def get_intersection(cx1, cy1, cos_t1, sin_t1,
cx2, cy2, cos_t2, sin_t2):
"""
Return the intersection between the line through (*cx1*, *cy1*) at angle
*t1* and the line through (*cx2*, *cy2*) at angle *t2*.
"""
# line1 => sin_t1 * (x - cx1) - cos_t1 * (y - cy1) = 0.
# line1 => sin_t1 * x + cos_t1 * y = sin_t1*cx1 - cos_t1*cy1
line1_rhs = sin_t1 * cx1 - cos_t1 * cy1
line2_rhs = sin_t2 * cx2 - cos_t2 * cy2
# rhs matrix
a, b = sin_t1, -cos_t1
c, d = sin_t2, -cos_t2
ad_bc = a * d - b * c
if abs(ad_bc) < 1e-12:
raise ValueError("Given lines do not intersect. Please verify that "
"the angles are not equal or differ by 180 degrees.")
# rhs_inverse
a_, b_ = d, -b
c_, d_ = -c, a
a_, b_, c_, d_ = [k / ad_bc for k in [a_, b_, c_, d_]]
x = a_ * line1_rhs + b_ * line2_rhs
y = c_ * line1_rhs + d_ * line2_rhs
return x, y
[docs]def get_normal_points(cx, cy, cos_t, sin_t, length):
"""
For a line passing through (*cx*, *cy*) and having an angle *t*, return
locations of the two points located along its perpendicular line at the
distance of *length*.
"""
if length == 0.:
return cx, cy, cx, cy
cos_t1, sin_t1 = sin_t, -cos_t
cos_t2, sin_t2 = -sin_t, cos_t
x1, y1 = length * cos_t1 + cx, length * sin_t1 + cy
x2, y2 = length * cos_t2 + cx, length * sin_t2 + cy
return x1, y1, x2, y2
# BEZIER routines
# subdividing bezier curve
# http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/Bezier/bezier-sub.html
def _de_casteljau1(beta, t):
next_beta = beta[:-1] * (1 - t) + beta[1:] * t
return next_beta
[docs]def split_de_casteljau(beta, t):
"""
Split a Bezier segment defined by its control points *beta* into two
separate segments divided at *t* and return their control points.
"""
beta = np.asarray(beta)
beta_list = [beta]
while True:
beta = _de_casteljau1(beta, t)
beta_list.append(beta)
if len(beta) == 1:
break
left_beta = [beta[0] for beta in beta_list]
right_beta = [beta[-1] for beta in reversed(beta_list)]
return left_beta, right_beta
[docs]def find_bezier_t_intersecting_with_closedpath(
bezier_point_at_t, inside_closedpath, t0=0., t1=1., tolerance=0.01):
"""
Find the intersection of the Bezier curve with a closed path.
The intersection point *t* is approximated by two parameters *t0*, *t1*
such that *t0* <= *t* <= *t1*.
Search starts from *t0* and *t1* and uses a simple bisecting algorithm
therefore one of the end points must be inside the path while the other
doesn't. The search stops when the distance of the points parametrized by
*t0* and *t1* gets smaller than the given *tolerance*.
Parameters
----------
bezier_point_at_t : callable
A function returning x, y coordinates of the Bezier at parameter *t*.
It must have the signature::
bezier_point_at_t(t: float) -> Tuple[float, float]
inside_closedpath : callable
A function returning True if a given point (x, y) is inside the
closed path. It must have the signature::
inside_closedpath(point: Tuple[float, float]) -> bool
t0, t1 : float
Start parameters for the search.
tolerance : float
Maximal allowed distance between the final points.
Returns
-------
t0, t1 : float
The Bezier path parameters.
"""
start = bezier_point_at_t(t0)
end = bezier_point_at_t(t1)
start_inside = inside_closedpath(start)
end_inside = inside_closedpath(end)
if start_inside == end_inside and start != end:
raise NonIntersectingPathException(
"Both points are on the same side of the closed path")
while True:
# return if the distance is smaller than the tolerance
if np.hypot(start[0] - end[0], start[1] - end[1]) < tolerance:
return t0, t1
# calculate the middle point
middle_t = 0.5 * (t0 + t1)
middle = bezier_point_at_t(middle_t)
middle_inside = inside_closedpath(middle)
if start_inside ^ middle_inside:
t1 = middle_t
end = middle
end_inside = middle_inside
else:
t0 = middle_t
start = middle
start_inside = middle_inside
[docs]class BezierSegment:
"""
A d-dimensional Bezier segment.
Parameters
----------
control_points : (N, d) array
Location of the *N* control points.
"""
def __init__(self, control_points):
self._cpoints = np.asarray(control_points)
self._N, self._d = self._cpoints.shape
self._orders = np.arange(self._N)
coeff = [math.factorial(self._N - 1)
// (math.factorial(i) * math.factorial(self._N - 1 - i))
for i in range(self._N)]
self._px = (self._cpoints.T * coeff).T
def __call__(self, t):
"""
Evaluate the Bezier curve at point(s) t in [0, 1].
Parameters
----------
t : float (k,), array_like
Points at which to evaluate the curve.
Returns
-------
float (k, d), array_like
Value of the curve for each point in *t*.
"""
t = np.asarray(t)
return (np.power.outer(1 - t, self._orders[::-1])
* np.power.outer(t, self._orders)) @ self._px
[docs] def point_at_t(self, t):
"""Evaluate curve at a single point *t*. Returns a Tuple[float*d]."""
return tuple(self(t))
@property
def control_points(self):
"""The control points of the curve."""
return self._cpoints
@property
def dimension(self):
"""The dimension of the curve."""
return self._d
@property
def degree(self):
"""Degree of the polynomial. One less the number of control points."""
return self._N - 1
@property
def polynomial_coefficients(self):
r"""
The polynomial coefficients of the Bezier curve.
.. warning:: Follows opposite convention from `numpy.polyval`.
Returns
-------
float, (n+1, d) array_like
Coefficients after expanding in polynomial basis, where :math:`n`
is the degree of the bezier curve and :math:`d` its dimension.
These are the numbers (:math:`C_j`) such that the curve can be
written :math:`\sum_{j=0}^n C_j t^j`.
Notes
-----
The coefficients are calculated as
.. math::
{n \choose j} \sum_{i=0}^j (-1)^{i+j} {j \choose i} P_i
where :math:`P_i` are the control points of the curve.
"""
n = self.degree
# matplotlib uses n <= 4. overflow plausible starting around n = 15.
if n > 10:
warnings.warn("Polynomial coefficients formula unstable for high "
"order Bezier curves!", RuntimeWarning)
P = self.control_points
j = np.arange(n+1)[:, None]
i = np.arange(n+1)[None, :] # _comb is non-zero for i <= j
prefactor = (-1)**(i + j) * _comb(j, i) # j on axis 0, i on axis 1
return _comb(n, j) * prefactor @ P # j on axis 0, self.dimension on 1
[docs] def axis_aligned_extrema(self):
"""
Return the dimension and location of the curve's interior extrema.
The extrema are the points along the curve where one of its partial
derivatives is zero.
Returns
-------
dims : int, array_like
Index :math:`i` of the partial derivative which is zero at each
interior extrema.
dzeros : float, array_like
Of same size as dims. The :math:`t` such that :math:`d/dx_i B(t) =
0`
"""
n = self.degree
if n <= 1:
return np.array([]), np.array([])
Cj = self.polynomial_coefficients
dCj = np.arange(1, n+1)[:, None] * Cj[1:]
dims = []
roots = []
for i, pi in enumerate(dCj.T):
r = np.roots(pi[::-1])
roots.append(r)
dims.append(np.full_like(r, i))
roots = np.concatenate(roots)
dims = np.concatenate(dims)
in_range = np.isreal(roots) & (roots >= 0) & (roots <= 1)
return dims[in_range], np.real(roots)[in_range]
[docs]def split_bezier_intersecting_with_closedpath(
bezier, inside_closedpath, tolerance=0.01):
"""
Split a Bezier curve into two at the intersection with a closed path.
Parameters
----------
bezier : array-like(N, 2)
Control points of the Bezier segment. See `.BezierSegment`.
inside_closedpath : callable
A function returning True if a given point (x, y) is inside the
closed path. See also `.find_bezier_t_intersecting_with_closedpath`.
tolerance : float
The tolerance for the intersection. See also
`.find_bezier_t_intersecting_with_closedpath`.
Returns
-------
left, right
Lists of control points for the two Bezier segments.
"""
bz = BezierSegment(bezier)
bezier_point_at_t = bz.point_at_t
t0, t1 = find_bezier_t_intersecting_with_closedpath(
bezier_point_at_t, inside_closedpath, tolerance=tolerance)
_left, _right = split_de_casteljau(bezier, (t0 + t1) / 2.)
return _left, _right
# matplotlib specific
[docs]def split_path_inout(path, inside, tolerance=0.01, reorder_inout=False):
"""
Divide a path into two segments at the point where ``inside(x, y)`` becomes
False.
"""
from .path import Path
path_iter = path.iter_segments()
ctl_points, command = next(path_iter)
begin_inside = inside(ctl_points[-2:]) # true if begin point is inside
ctl_points_old = ctl_points
iold = 0
i = 1
for ctl_points, command in path_iter:
iold = i
i += len(ctl_points) // 2
if inside(ctl_points[-2:]) != begin_inside:
bezier_path = np.concatenate([ctl_points_old[-2:], ctl_points])
break
ctl_points_old = ctl_points
else:
raise ValueError("The path does not intersect with the patch")
bp = bezier_path.reshape((-1, 2))
left, right = split_bezier_intersecting_with_closedpath(
bp, inside, tolerance)
if len(left) == 2:
codes_left = [Path.LINETO]
codes_right = [Path.MOVETO, Path.LINETO]
elif len(left) == 3:
codes_left = [Path.CURVE3, Path.CURVE3]
codes_right = [Path.MOVETO, Path.CURVE3, Path.CURVE3]
elif len(left) == 4:
codes_left = [Path.CURVE4, Path.CURVE4, Path.CURVE4]
codes_right = [Path.MOVETO, Path.CURVE4, Path.CURVE4, Path.CURVE4]
else:
raise AssertionError("This should never be reached")
verts_left = left[1:]
verts_right = right[:]
if path.codes is None:
path_in = Path(np.concatenate([path.vertices[:i], verts_left]))
path_out = Path(np.concatenate([verts_right, path.vertices[i:]]))
else:
path_in = Path(np.concatenate([path.vertices[:iold], verts_left]),
np.concatenate([path.codes[:iold], codes_left]))
path_out = Path(np.concatenate([verts_right, path.vertices[i:]]),
np.concatenate([codes_right, path.codes[i:]]))
if reorder_inout and not begin_inside:
path_in, path_out = path_out, path_in
return path_in, path_out
[docs]def inside_circle(cx, cy, r):
"""
Return a function that checks whether a point is in a circle with center
(*cx*, *cy*) and radius *r*.
The returned function has the signature::
f(xy: Tuple[float, float]) -> bool
"""
r2 = r ** 2
def _f(xy):
x, y = xy
return (x - cx) ** 2 + (y - cy) ** 2 < r2
return _f
# quadratic Bezier lines
[docs]def get_cos_sin(x0, y0, x1, y1):
dx, dy = x1 - x0, y1 - y0
d = (dx * dx + dy * dy) ** .5
# Account for divide by zero
if d == 0:
return 0.0, 0.0
return dx / d, dy / d
[docs]def check_if_parallel(dx1, dy1, dx2, dy2, tolerance=1.e-5):
"""
Check if two lines are parallel.
Parameters
----------
dx1, dy1, dx2, dy2 : float
The gradients *dy*/*dx* of the two lines.
tolerance : float
The angular tolerance in radians up to which the lines are considered
parallel.
Returns
-------
is_parallel
- 1 if two lines are parallel in same direction.
- -1 if two lines are parallel in opposite direction.
- False otherwise.
"""
theta1 = np.arctan2(dx1, dy1)
theta2 = np.arctan2(dx2, dy2)
dtheta = abs(theta1 - theta2)
if dtheta < tolerance:
return 1
elif abs(dtheta - np.pi) < tolerance:
return -1
else:
return False
[docs]def get_parallels(bezier2, width):
"""
Given the quadratic Bezier control points *bezier2*, returns
control points of quadratic Bezier lines roughly parallel to given
one separated by *width*.
"""
# The parallel Bezier lines are constructed by following ways.
# c1 and c2 are control points representing the begin and end of the
# Bezier line.
# cm is the middle point
c1x, c1y = bezier2[0]
cmx, cmy = bezier2[1]
c2x, c2y = bezier2[2]
parallel_test = check_if_parallel(c1x - cmx, c1y - cmy,
cmx - c2x, cmy - c2y)
if parallel_test == -1:
cbook._warn_external(
"Lines do not intersect. A straight line is used instead.")
cos_t1, sin_t1 = get_cos_sin(c1x, c1y, c2x, c2y)
cos_t2, sin_t2 = cos_t1, sin_t1
else:
# t1 and t2 is the angle between c1 and cm, cm, c2. They are
# also a angle of the tangential line of the path at c1 and c2
cos_t1, sin_t1 = get_cos_sin(c1x, c1y, cmx, cmy)
cos_t2, sin_t2 = get_cos_sin(cmx, cmy, c2x, c2y)
# 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 c2_left and
# c2_right with respect to c2.
c1x_left, c1y_left, c1x_right, c1y_right = (
get_normal_points(c1x, c1y, cos_t1, sin_t1, width)
)
c2x_left, c2y_left, c2x_right, c2y_right = (
get_normal_points(c2x, c2y, cos_t2, sin_t2, width)
)
# find cm_left which is the intersecting point of a line through
# c1_left with angle t1 and a line through c2_left with angle
# t2. Same with cm_right.
try:
cmx_left, cmy_left = get_intersection(c1x_left, c1y_left, cos_t1,
sin_t1, c2x_left, c2y_left,
cos_t2, sin_t2)
cmx_right, cmy_right = get_intersection(c1x_right, c1y_right, cos_t1,
sin_t1, c2x_right, c2y_right,
cos_t2, sin_t2)
except ValueError:
# Special case straight lines, i.e., angle between two lines is
# less than the threshold used by get_intersection (we don't use
# check_if_parallel as the threshold is not the same).
cmx_left, cmy_left = (
0.5 * (c1x_left + c2x_left), 0.5 * (c1y_left + c2y_left)
)
cmx_right, cmy_right = (
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]
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
[docs]def find_control_points(c1x, c1y, mmx, mmy, c2x, c2y):
"""
Find control points of the Bezier 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)]
[docs]def make_wedged_bezier2(bezier2, width, w1=1., wm=0.5, w2=0.):
"""
Being similar to get_parallels, returns control points of two quadratic
Bezier 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 a 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
[docs]@cbook.deprecated(
"3.3", alternative="Path.cleaned() and remove the final STOP if needed")
def make_path_regular(p):
"""
If the ``codes`` attribute of `.Path` *p* is None, return a copy of *p*
with ``codes`` set to (MOVETO, LINETO, LINETO, ..., LINETO); otherwise
return *p* itself.
"""
from .path import Path
c = p.codes
if c is None:
c = np.full(len(p.vertices), Path.LINETO, dtype=Path.code_type)
c[0] = Path.MOVETO
return Path(p.vertices, c)
else:
return p
[docs]@cbook.deprecated("3.3", alternative="Path.make_compound_path()")
def concatenate_paths(paths):
"""Concatenate a list of paths into a single path."""
from .path import Path
return Path.make_compound_path(*paths)
