matplotlib/lib/matplotlib/bezier.py at main · spha-code/matplotlib · GitHub

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"""

A module providing some utility functions regarding Bézier path manipulation.

"""

from functools import lru_cache

import math

import warnings

import numpy as np

from matplotlib import _api

# 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)

class NonIntersectingPathException(ValueError):

pass

# some functions

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

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

def split_de_casteljau(beta, t):

"""

Split a Bézier 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

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 Bézier 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 Bézier 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 Bézier 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

if end == middle:

# Edge case where infinite loop is possible

# Caused by large numbers relative to tolerance

return t0, t1

end = middle

else:

t0 = middle_t

if start == middle:

# Edge case where infinite loop is possible

# Caused by large numbers relative to tolerance

return t0, t1

start = middle

start_inside = middle_inside

class BezierSegment:

"""

A d-dimensional Bézier segment.

A BezierSegment can be called with an argument, either a scalar or an array-like

object, to evaluate the curve at that/those location(s).

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 Bézier curve at point(s) *t* in [0, 1].

Parameters

----------

t : (k,) array-like

Points at which to evaluate the curve.

Returns

-------

(k, d) array

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

@_api.deprecated(

"3.11", alternative="Call the BezierSegment object with an argument.")

def point_at_t(self, t):

"""

Evaluate the curve at a single point, returning a tuple of *d* floats.

"""

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 Bézier curve.

.. warning:: Follows opposite convention from `numpy.polyval`.

Returns

-------

(n+1, d) array

Coefficients after expanding in polynomial basis, where :math:`n`

is the degree of the Bézier 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

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 : array of int

Index :math:`i` of the partial derivative which is zero at each

interior extrema.

dzeros : array of float

Of same size as dims. The :math:`t` such that :math:`d/dx_i B(t) =

"""

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]

def split_bezier_intersecting_with_closedpath(

bezier, inside_closedpath, tolerance=0.01):

"""

Split a Bézier curve into two at the intersection with a closed path.

Parameters

----------

bezier : (N, 2) array-like

Control points of the Bézier 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 Bézier segments.

"""

bz = BezierSegment(bezier)

t0, t1 = find_bezier_t_intersecting_with_closedpath(

lambda t: tuple(bz(t)), inside_closedpath, tolerance=tolerance)

_left, _right = split_de_casteljau(bezier, (t0 + t1) / 2.)

return _left, _right

# matplotlib specific

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

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

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

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

def get_parallels(bezier2, width):

"""

Given the quadratic Bézier control points *bezier2*, returns

control points of quadratic Bézier 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 start 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:

_api.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 an 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

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

matplotlib/lib/matplotlib/bezier.py at main · spha-code/matplotlib · GitHub

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