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Jondale Stratton
2026-01-17 23:40:16 -05:00
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# coding=utf-8
#
# Copyright (C) 2010 Nick Drobchenko, nick@cnc-club.ru
# Copyright (C) 2005 Aaron Spike, aaron@ekips.org
#
# 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 2 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, write to the Free Software
# Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
#
# pylint: disable=invalid-name,too-many-locals
#
"""
Bezier calculations
"""
import cmath
import math
import numpy
from .transforms import DirectedLineSegment
from .localization import inkex_gettext as _
# bez = ((bx0,by0),(bx1,by1),(bx2,by2),(bx3,by3))
def pointdistance(point_a, point_b):
"""The straight line distance between two points"""
return math.sqrt(
((point_b[0] - point_a[0]) ** 2) + ((point_b[1] - point_a[1]) ** 2)
)
def between_point(point_a, point_b, time=0.5):
"""Returns the point between point a and point b"""
return point_a[0] + time * (point_b[0] - point_a[0]), point_a[1] + time * (
point_b[1] - point_a[1]
)
def percent_point(point_a, point_b, percent=50.0):
"""Returns between_point but takes percent instead of 0.0-1.0"""
return between_point(point_a, point_b, percent / 100.0)
def root_wrapper(root_a, root_b, root_c, root_d):
"""Get the Cubic function, moic formular of roots, simple root"""
if root_a:
# Monics formula, see
# http://en.wikipedia.org/wiki/Cubic_function#Monic_formula_of_roots
mono_a, mono_b, mono_c = (root_b / root_a, root_c / root_a, root_d / root_a)
m = 2.0 * mono_a**3 - 9.0 * mono_a * mono_b + 27.0 * mono_c
k = mono_a**2 - 3.0 * mono_b
n = m**2 - 4.0 * k**3
w1 = -0.5 + 0.5 * cmath.sqrt(-3.0)
w2 = -0.5 - 0.5 * cmath.sqrt(-3.0)
if n < 0:
m1 = pow(complex((m + cmath.sqrt(n)) / 2), 1.0 / 3)
n1 = pow(complex((m - cmath.sqrt(n)) / 2), 1.0 / 3)
else:
if m + math.sqrt(n) < 0:
m1 = -pow(-(m + math.sqrt(n)) / 2, 1.0 / 3)
else:
m1 = pow((m + math.sqrt(n)) / 2, 1.0 / 3)
if m - math.sqrt(n) < 0:
n1 = -pow(-(m - math.sqrt(n)) / 2, 1.0 / 3)
else:
n1 = pow((m - math.sqrt(n)) / 2, 1.0 / 3)
return (
-1.0 / 3 * (mono_a + m1 + n1),
-1.0 / 3 * (mono_a + w1 * m1 + w2 * n1),
-1.0 / 3 * (mono_a + w2 * m1 + w1 * n1),
)
if root_b:
det = root_c**2.0 - 4.0 * root_b * root_d
if det:
return (
(-root_c + cmath.sqrt(det)) / (2.0 * root_b),
(-root_c - cmath.sqrt(det)) / (2.0 * root_b),
)
return (-root_c / (2.0 * root_b),)
if root_c:
return (1.0 * (-root_d / root_c),)
return ()
def bezlenapprx(sp1, sp2):
"""Return the aproximate length between two beziers"""
return (
pointdistance(sp1[1], sp1[2])
+ pointdistance(sp1[2], sp2[0])
+ pointdistance(sp2[0], sp2[1])
)
def cspbezsplit(sp1, sp2, time=0.5):
"""Split a cubic bezier at the time period"""
m1 = tpoint(sp1[1], sp1[2], time)
m2 = tpoint(sp1[2], sp2[0], time)
m3 = tpoint(sp2[0], sp2[1], time)
m4 = tpoint(m1, m2, time)
m5 = tpoint(m2, m3, time)
m = tpoint(m4, m5, time)
return [[sp1[0][:], sp1[1][:], m1], [m4, m, m5], [m3, sp2[1][:], sp2[2][:]]]
def cspbezsplitatlength(sp1, sp2, length=0.5, tolerance=0.001):
"""Split a cubic bezier at length"""
bez = (sp1[1][:], sp1[2][:], sp2[0][:], sp2[1][:])
time = beziertatlength(bez, length, tolerance)
return cspbezsplit(sp1, sp2, time)
def cspseglength(sp1, sp2, tolerance=0.001):
"""Get cubic bezier segment length"""
bez = (sp1[1][:], sp1[2][:], sp2[0][:], sp2[1][:])
return bezierlength(bez, tolerance)
def csplength(csp):
"""Get cubic bezier length"""
total = 0
lengths = []
for sp in csp:
lengths.append([])
for i in range(1, len(sp)):
l = cspseglength(sp[i - 1], sp[i])
lengths[-1].append(l)
total += l
return lengths, total
def bezierparameterize(bez):
"""Return the bezier parameter size
Converts the bezier parametrisation from the default form
P(t) = (1-t)³ P_1 + 3(1-t)²t P_2 + 3(1-t)t² P_3 + t³ x_4
to the a form which can be differentiated more easily
P(t) = a t³ + b t² + c t + P0
Args:
bez (List[Tuple[float, float]]): the Bezier curve. The elements of the list the
coordinates of the points (in this order): Start point, Start control point,
End control point, End point.
Returns:
Tuple[float, float, float, float, float, float, float, float]:
the values ax, ay, bx, by, cx, cy, x0, y0
"""
((bx0, by0), (bx1, by1), (bx2, by2), (bx3, by3)) = bez
# parametric bezier
x0 = bx0
y0 = by0
cx = 3 * (bx1 - x0)
bx = 3 * (bx2 - bx1) - cx
ax = bx3 - x0 - cx - bx
cy = 3 * (by1 - y0)
by = 3 * (by2 - by1) - cy
ay = by3 - y0 - cy - by
return ax, ay, bx, by, cx, cy, x0, y0
def linebezierintersect(arg_a, bez):
"""Where a line and bezier intersect"""
((lx1, ly1), (lx2, ly2)) = arg_a
# parametric line
dd = lx1
cc = lx2 - lx1
bb = ly1
aa = ly2 - ly1
if aa:
coef1 = cc / aa
coef2 = 1
else:
coef1 = 1
coef2 = aa / cc
ax, ay, bx, by, cx, cy, x0, y0 = bezierparameterize(bez)
# cubic intersection coefficients
a = coef1 * ay - coef2 * ax
b = coef1 * by - coef2 * bx
c = coef1 * cy - coef2 * cx
d = coef1 * (y0 - bb) - coef2 * (x0 - dd)
roots = root_wrapper(a, b, c, d)
retval = []
for i in roots:
if isinstance(i, complex) and i.imag == 0:
i = i.real
if not isinstance(i, complex) and 0 <= i <= 1:
retval.append(bezierpointatt(bez, i))
return retval
def bezierpointatt(bez, t):
"""Get coords at the given time point along a bezier curve"""
ax, ay, bx, by, cx, cy, x0, y0 = bezierparameterize(bez)
x = ax * (t**3) + bx * (t**2) + cx * t + x0
y = ay * (t**3) + by * (t**2) + cy * t + y0
return x, y
def bezierslopeatt(bez, t):
"""Get slope at the given time point along a bezier curve
The slope is computed as (dx, dy) where dx = df_x(t)/dt and dy = df_y(t)/dt.
Note that for lines P1=P2 and P3=P4, so the slope at the end points is dx=dy=0
(slope not defined).
Args:
bez (List[Tuple[float, float]]): the Bezier curve. The elements of the list the
coordinates of the points (in this order): Start point, Start control point,
End control point, End point.
t (float): time in the interval [0, 1]
Returns:
Tuple[float, float]: x and y increment
"""
ax, ay, bx, by, cx, cy, _, _ = bezierparameterize(bez)
dx = 3 * ax * (t**2) + 2 * bx * t + cx
dy = 3 * ay * (t**2) + 2 * by * t + cy
return dx, dy
def beziertatslope(bez, d):
"""Reverse; get time from slope along a bezier curve"""
ax, ay, bx, by, cx, cy, _, _ = bezierparameterize(bez)
(dy, dx) = d
# quadratic coefficients of slope formula
if dx:
slope = 1.0 * (dy / dx)
a = 3 * ay - 3 * ax * slope
b = 2 * by - 2 * bx * slope
c = cy - cx * slope
elif dy:
slope = 1.0 * (dx / dy)
a = 3 * ax - 3 * ay * slope
b = 2 * bx - 2 * by * slope
c = cx - cy * slope
else:
return []
roots = root_wrapper(0, a, b, c)
retval = []
for i in roots:
if isinstance(i, complex) and i.imag == 0:
i = i.real
if not isinstance(i, complex) and 0 <= i <= 1:
retval.append(i)
return retval
def tpoint(p1, p2, t):
"""Linearly interpolate between p1 and p2.
t = 0.0 returns p1, t = 1.0 returns p2.
:return: Interpolated point
:rtype: tuple
:param p1: First point as sequence of two floats
:param p2: Second point as sequence of two floats
:param t: Number between 0.0 and 1.0
:type t: float
"""
x1, y1 = p1
x2, y2 = p2
return x1 + t * (x2 - x1), y1 + t * (y2 - y1)
def beziersplitatt(bez, t):
"""Split bezier at given time"""
((bx0, by0), (bx1, by1), (bx2, by2), (bx3, by3)) = bez
m1 = tpoint((bx0, by0), (bx1, by1), t)
m2 = tpoint((bx1, by1), (bx2, by2), t)
m3 = tpoint((bx2, by2), (bx3, by3), t)
m4 = tpoint(m1, m2, t)
m5 = tpoint(m2, m3, t)
m = tpoint(m4, m5, t)
return ((bx0, by0), m1, m4, m), (m, m5, m3, (bx3, by3))
def addifclose(bez, l, error=0.001):
"""Gravesen, Add if the line is closed, in-place addition to array l"""
box = 0
for i in range(1, 4):
box += pointdistance(bez[i - 1], bez[i])
chord = pointdistance(bez[0], bez[3])
if (box - chord) > error:
first, second = beziersplitatt(bez, 0.5)
addifclose(first, l, error)
addifclose(second, l, error)
else:
l[0] += (box / 2.0) + (chord / 2.0)
# balfax, balfbx, balfcx, balfay, balfby, balfcy = 0, 0, 0, 0, 0, 0
def balf(t, args):
"""Bezier Arc Length Function"""
ax, bx, cx, ay, by, cy = args
retval = (ax * (t**2) + bx * t + cx) ** 2 + (ay * (t**2) + by * t + cy) ** 2
return math.sqrt(retval)
def simpson(start, end, maxiter, tolerance, bezier_args):
"""Calculate the length of a bezier curve using Simpson's algorithm:
http://steve.hollasch.net/cgindex/curves/cbezarclen.html
Args:
start (int): Start time (between 0 and 1)
end (int): End time (between start time and 1)
maxiter (int): Maximum number of iterations. If not a power of 2, the algorithm
will behave like the value is set to the next power of 2.
tolerance (float): maximum error ratio
bezier_args (list): arguments as computed by bezierparametrize()
Returns:
float: the appoximate length of the bezier curve
"""
n = 2
multiplier = (end - start) / 6.0
endsum = balf(start, bezier_args) + balf(end, bezier_args)
interval = (end - start) / 2.0
asum = 0.0
bsum = balf(start + interval, bezier_args)
est1 = multiplier * (endsum + (2.0 * asum) + (4.0 * bsum))
est0 = 2.0 * est1
# print(multiplier, endsum, interval, asum, bsum, est1, est0)
while n < maxiter and abs(est1 - est0) > tolerance:
n *= 2
multiplier /= 2.0
interval /= 2.0
asum += bsum
bsum = 0.0
est0 = est1
for i in range(1, n, 2):
bsum += balf(start + (i * interval), bezier_args)
est1 = multiplier * (endsum + (2.0 * asum) + (4.0 * bsum))
# print(multiplier, endsum, interval, asum, bsum, est1, est0)
return est1
def bezierlength(bez, tolerance=0.001, time=1.0):
"""Get length of bezier curve"""
ax, ay, bx, by, cx, cy, _, _ = bezierparameterize(bez)
return simpson(0.0, time, 4096, tolerance, [3 * ax, 2 * bx, cx, 3 * ay, 2 * by, cy])
def beziertatlength(bez, l=0.5, tolerance=0.001):
"""Get bezier curve time at the length specified"""
curlen = bezierlength(bez, tolerance, 1.0)
time = 1.0
tdiv = time
targetlen = l * curlen
diff = curlen - targetlen
while abs(diff) > tolerance:
tdiv /= 2.0
if diff < 0:
time += tdiv
else:
time -= tdiv
curlen = bezierlength(bez, tolerance, time)
diff = curlen - targetlen
return time
def maxdist(bez):
"""Get maximum distance within bezier curve"""
seg = DirectedLineSegment(bez[0], bez[3])
return max(seg.distance_to_point(*bez[1]), seg.distance_to_point(*bez[2]))
def cspsubdiv(csp, flat):
"""Sub-divide cubic sub-paths"""
for sp in csp:
subdiv(sp, flat)
def subdiv(sp, flat, i=1):
"""sub divide bezier curve"""
while i < len(sp):
p0 = sp[i - 1][1]
p1 = sp[i - 1][2]
p2 = sp[i][0]
p3 = sp[i][1]
bez = (p0, p1, p2, p3)
mdist = maxdist(bez)
if mdist <= flat:
i += 1
else:
one, two = beziersplitatt(bez, 0.5)
sp[i - 1][2] = one[1]
sp[i][0] = two[2]
p = [one[2], one[3], two[1]]
sp[i:1] = [p]
def csparea(csp):
r"""Get total area of cubic superpath.
.. hint::
The results may be slightly inaccurate for paths containing arcs due
to the loss of accuracy during arc -> cubic bezier conversion.
The function works as follows: For each subpath,
#. compute the area of the polygon created by the path's vertices:
For a line with coordinates :math:`(x_0, y_0)` and :math:`(x_1, y_1)`, the area
of the trapezoid of its projection on the x axis is given by
.. math::
\frac{1}{2} (y_1 + y_0) (x_1 - x_0)
Summing the contribution of all lines of the polygon yields the polygon's area
(lines from left to right have a positive contribution, while those right-to
left have a negative area contribution, canceling out the computed area not
inside the polygon), so we find (setting :math:`x_{0} = x_N` etc.):
.. math::
A = \frac{1}{2} * \sum_{i=1}^N (x_i y_i - x_{i-1} y_{i-1} + x_i y_{i-1}
- x_{i-1} y_{i})
The first two terms cancel out in the summation over all points, and the second
two terms can be regrouped as
.. math::
A = \frac{1}{2} * \sum_{i=1}^N x_i (y_{i+1} -y_{i-1})
#. The contribution by the bezier curve is considered: We compute
the integral :math:`\int_{x(t=0)}^{x(t=1)} y dx`, i.e. the area between the x
axis and the curve, where :math:`y = y(t)` (the Bezier curve). By substitution
:math:`dx = x'(t) dt`, performing the integration and
subtracting the trapezoid we already considered above, we find (with control
points :math:`(x_{c1}, y_{c1})` and :math:`(x_{c2}, y_{c2})`)
.. math::
\Delta A &= \int_0^1 y(t) x'(t) dt - \frac{1}{2} (y_1 + y_0) (x_1 - x_0) \\
&= \frac{3}{20} \cdot \begin{pmatrix}
& y_0(& & 2x_{c1} & + x_{c2} & -3x_1&) \\
+ & y_{c1}(& -2x_0 & & + x_{c2} &+ x_1&) \\
+ & y_{c2}(& -x_0 & -x_{c1} & & + 2x_1&) \\
+ & y_1(& 3x_0 & - x_{c1} & -2 x_{c2} &&)
\end{pmatrix}
This is computed for every bezier and added to the area. Again, this is a signed
area: convex beziers have a positive area and concave ones a negative area
contribution.
"""
MAT_AREA = numpy.array(
[[0, 2, 1, -3], [-2, 0, 1, 1], [-1, -1, 0, 2], [3, -1, -2, 0]]
)
area = 0.0
for sp in csp:
if len(sp) < 2:
continue
for x, coord in enumerate(sp): # calculate polygon area
area += 0.5 * sp[x - 1][1][0] * (coord[1][1] - sp[x - 2][1][1])
for i in range(1, len(sp)): # add contribution from cubic Bezier
# EXPLANATION: https://github.com/Pomax/BezierInfo-2/issues/238#issue-554619801
vec_x = numpy.array(
[sp[i - 1][1][0], sp[i - 1][2][0], sp[i][0][0], sp[i][1][0]]
)
vec_y = numpy.array(
[sp[i - 1][1][1], sp[i - 1][2][1], sp[i][0][1], sp[i][1][1]]
)
vex = numpy.matmul(vec_x, MAT_AREA)
area += 0.15 * numpy.matmul(vex, vec_y.T)
return -area
def cspcofm(csp):
r"""Get center of area / gravity for a cubic superpath.
.. hint::
The results may be slightly inaccurate for paths containing arcs due
to the loss of accuracy during arc -> cubic bezier conversion.
The function works similar to :func:`csparea`, only the computations are a bit more
difficult. Again all subpaths are considered. The total center of mass is given by
.. math::
C_y = \frac{1}{A} \int_A y dA
The integral can be expressed as a weighted sum; first, the contributions
of the polygon created by the path's nodes is computed. Second, we compute the
contribution of the Bezier curve; this is again done by an integral from which
the weighted CofM of the trapezoid between end points and horizontal axis is
removed. For the integrals, we have
.. math::
A * C_{y,bez} &= \int_A y dA = \int_{x(t=0)}^{y(t=1)} \int_{0}^{y(x)} y dy dx \\
&= \int_{x(t=0)}^{y(t=1)} \frac 12 y(x)^2 dx
= \int_0^1 \frac 12 y(t)^2 x'(t) dt \\
A * C_{x,bez} &= \int_A x dA = \int_{x(t=0)}^{y(t=1)} x \int_{0}^{y(x)} dy dx \\
&= \int_{x(t=0)}^{y(t=1)} x y(x) dx = \int_0^1 x(t) y(t) x'(t) dt
from which the trapezoids are removed, in case of the y-CofM this amounts to
.. math::
\frac{y_0}{2} (x_1-x_0)y_0 + \left(y_0 + \frac 13 (y_1 - y_0)\right)
\cdot \frac 12 (y_1 - y_0) (x_1 - x_0)
"""
MAT_COFM_0 = numpy.array(
[[0, 35, 10, -45], [-35, 0, 12, 23], [-10, -12, 0, 22], [45, -23, -22, 0]]
)
MAT_COFM_1 = numpy.array(
[[0, 15, 3, -18], [-15, 0, 9, 6], [-3, -9, 0, 12], [18, -6, -12, 0]]
)
MAT_COFM_2 = numpy.array(
[[0, 12, 6, -18], [-12, 0, 9, 3], [-6, -9, 0, 15], [18, -3, -15, 0]]
)
MAT_COFM_3 = numpy.array(
[[0, 22, 23, -45], [-22, 0, 12, 10], [-23, -12, 0, 35], [45, -10, -35, 0]]
)
area = csparea(csp)
xc = 0.0
yc = 0.0
if abs(area) < 1.0e-8:
raise ValueError(_("Area is zero, cannot calculate Center of Mass"))
for sp in csp:
for x, coord in enumerate(sp): # calculate polygon moment
xc += (
sp[x - 1][1][1]
* (sp[x - 2][1][0] - coord[1][0])
* (sp[x - 2][1][0] + sp[x - 1][1][0] + coord[1][0])
/ 6
)
yc += (
sp[x - 1][1][0]
* (coord[1][1] - sp[x - 2][1][1])
* (sp[x - 2][1][1] + sp[x - 1][1][1] + coord[1][1])
/ 6
)
for i in range(1, len(sp)): # add contribution from cubic Bezier
vec_x = numpy.array(
[sp[i - 1][1][0], sp[i - 1][2][0], sp[i][0][0], sp[i][1][0]]
)
vec_y = numpy.array(
[sp[i - 1][1][1], sp[i - 1][2][1], sp[i][0][1], sp[i][1][1]]
)
def _mul(MAT, vec_x=vec_x, vec_y=vec_y):
return numpy.matmul(numpy.matmul(vec_x, MAT), vec_y.T)
vec_t = numpy.array(
[_mul(MAT_COFM_0), _mul(MAT_COFM_1), _mul(MAT_COFM_2), _mul(MAT_COFM_3)]
)
xc += numpy.matmul(vec_x, vec_t.T) / 280
yc += numpy.matmul(vec_y, vec_t.T) / 280
return -xc / area, -yc / area