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import numpy as np
import matplotlib.pyplot as plt
from scipy.special import binom
from numpy.linalg import norm
def num_bezier(n_ctrl, degree=3):
if type(n_ctrl) == np.ndarray:
n_ctrl = len(n_ctrl)
return int((n_ctrl - 1) / degree)
def bernstein(n, i):
bi = binom(n, i)
return lambda t, bi=bi, n=n, i=i: bi * t**i * (1 - t)**(n - i)
def bezier(P, t, d=0):
'''Bezier curve of degree len(P)-1. d is the derivative order (0 gives positions)'''
n = P.shape[0] - 1
if d > 0:
Q = np.diff(P, axis=0)*n
return bezier(Q, t, d-1)
B = np.vstack([bernstein(n, i)(t) for i, p in enumerate(P)])
return (P.T @ B).T
def cubic_bezier(P, t):
return (1.0-t)**3*P[0] + 3*(1.0-t)**2*t*P[1] + 3*(1.0-t)*t**2*P[2] + t**3*P[3]
def bezier_piecewise(Cp, subd=100, degree=3, d=0):
''' sample a piecewise Bezier curve given a sequence of control points'''
num = num_bezier(Cp.shape[0], degree)
X = []
for i in range(num):
P = Cp[i*degree:i*degree+degree+1, :]
t = np.linspace(0, 1., subd)[:-1]
Y = bezier(P, t, d)
X += [Y]
X.append(Cp[-1])
X = np.vstack(X)
return X
def compute_beziers(beziers, subd=100, degree=3):
chain = beziers_to_chain(beziers)
return bezier_piecewise(chain, subd, degree)
def plot_control_polygon(Cp, degree=3, lw=0.5, linecolor=np.ones(3)*0.1):
n_bezier = num_bezier(len(Cp), degree)
for i in range(n_bezier):
cp = Cp[i*degree:i*degree+degree+1, :]
if degree==3:
plt.plot(cp[0:2,0], cp[0:2, 1], ':', color=linecolor, linewidth=lw)
plt.plot(cp[2:,0], cp[2:,1], ':', color=linecolor, linewidth=lw)
plt.plot(cp[:,0], cp[:,1], 'o', color=[0, 0.5, 1.], markersize=4)
else:
plt.plot(cp[:,0], cp[:,1], ':', color=linecolor, linewidth=lw)
plt.plot(cp[:,0], cp[:,1], 'o', color=[0, 0.5, 1.])
def chain_to_beziers(chain, degree=3):
''' Convert Bezier chain to list of curve segments (4 control points each)'''
num = num_bezier(chain.shape[0], degree)
beziers = []
for i in range(num):
beziers.append(chain[i*degree:i*degree+degree+1,:])
return beziers
def beziers_to_chain(beziers):
''' Convert list of Bezier curve segments to a piecewise bezier chain (shares vertices)'''
n = len(beziers)
chain = []
for i in range(n):
chain.append(list(beziers[i][:-1]))
chain.append([beziers[-1][-1]])
return np.array(sum(chain, []))
def split_cubic(bez, t):
p1, p2, p3, p4 = bez
p12 = (p2 - p1) * t + p1
p23 = (p3 - p2) * t + p2
p34 = (p4 - p3) * t + p3
p123 = (p23 - p12) * t + p12
p234 = (p34 - p23) * t + p23
p1234 = (p234 - p123) * t + p123
return np.array([p1, p12, p123, p1234]), np.array([p1234, p234, p34, p4])
def approx_arc_length(bez):
c0, c1, c2, c3 = bez
v0 = norm(c1-c0)*0.15
v1 = norm(-0.558983582205757*c0 + 0.325650248872424*c1 + 0.208983582205757*c2 + 0.024349751127576*c3)
v2 = norm(c3-c0+c2-c1)*0.26666666666666666
v3 = norm(-0.024349751127576*c0 - 0.208983582205757*c1 - 0.325650248872424*c2 + 0.558983582205757*c3)
v4 = norm(c3-c2)*.15
return v0 + v1 + v2 + v3 + v4
def subdivide_bezier(bez, thresh):
stack = [bez]
res = []
while stack:
bez = stack.pop()
l = approx_arc_length(bez)
if l < thresh:
res.append(bez)
else:
b1, b2 = split_cubic(bez, 0.5)
stack += [b2, b1]
return res
def subdivide_bezier_chain(C, thresh):
beziers = chain_to_beziers(C)
res = []
for bez in beziers:
res += subdivide_bezier(bez, thresh)
return beziers_to_chain(res)
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