def get_control_points(self):
n = self.get_degree()
factorials = np.empty((n, ))
f = 1
for i in range(1, n + 1):
factorials[i - 1] = f
f *= (i + 1)
A = np.zeros((n + 1, n + 1))
for t_power in range(n + 1):
for ctrlpt_i in range(t_power + 1):
sign = 1 if (t_power - ctrlpt_i) % 2 == 0 else -1
A[t_power, ctrlpt_i] = binomial(n, ctrlpt_i) * binomial(
n - ctrlpt_i, t_power - ctrlpt_i) * sign
control_points = np.empty((n + 1, 3))
for d in range(3):
B = np.empty((n + 1, 1))
B[0, 0] = self.start[d]
B[1:, 0] = self.derivatives[:, d] / factorials
x = np.linalg.solve(A, B)
control_points[:, d] = x[:, 0]
return control_points
def coeff_deriv3(self, k, t):
n = self.degree
C = binomial(n, k)
if n - k - 2 > 0:
s1 = -(n - k - 2) * (n - k - 1) * (n - k) * (1 - t)**(n - k -
3) * t**k
else:
s1 = np.zeros_like(t)
if k >= 1 and n - k - 2 > 0:
s2 = 3 * k * (n - k - 1) * (n - k) * (1 - t)**(n - k - 2) * t**(k -
1)
elif k >= 1 and n - k == 2:
s2 = 6 * k * t**(k - 1)
else:
s2 = np.zeros_like(t)
if k >= 2 and n - k - 1 > 0:
s3 = -3 * (k - 1) * k * (n - k) * (1 - t)**(n - k - 1) * t**(k - 2)
elif k >= 2 and n - k == 1:
s3 = -3 * (k - 1) * k * t**(k - 2)
else:
s3 = np.zeros_like(t)
if k >= 3:
s4 = (k - 2) * (k - 1) * k * (1 - t)**(n - k) * t**(k - 3)
else:
s4 = np.zeros_like(t)
coeff = s1 + s2 + s3 + s4
return C * coeff
def elevate_bezier_degree(self_degree, control_points, delta=1):
# See "The NURBS book" (2nd edition), p.5.5, eq. 5.36
t = delta
p = self_degree
new_points = []
P = control_points
for i in range(p + t + 1):
j0 = max(0, i - t)
j1 = min(p, i)
js = range(j0, j1 + 1)
c1 = np.array([binomial(p, j) for j in js])
c2 = np.array([binomial(t, i - j) for j in js])
ps = P[j0:j1 + 1, :]
numerator = (c1 * c2)[np.newaxis].T * ps
denominator = binomial(p + t, i)
#print(f"E: p {p}, i {i}, c1 {c1}, c2 {c2}, denom {denominator}, ps {ps}")
point = numerator.sum(axis=0) / denominator
new_points.append(point)
return np.array(new_points)
def coeff_deriv1(self, k, t):
n = self.degree
C = binomial(n, k)
if k >= 1:
s1 = k * (1 - t)**(n - k) * t**(k - 1)
else:
s1 = np.zeros_like(t)
if n - k - 1 > 0:
s2 = -(n - k) * (1 - t)**(n - k - 1) * t**k
elif n - k == 1:
s2 = -t**k
else:
s2 = np.zeros_like(t)
coeff = s1 + s2
return C * coeff
def coeff_deriv2(self, k, t):
n = self.degree
C = binomial(n, k)
if n - k - 2 > 0:
s1 = (n - k - 1) * (n - k) * (1 - t)**(n - k - 2) * t**k
elif n - k == 2:
s1 = 2 * t**k
else:
s1 = np.zeros_like(t)
if k >= 1 and n - k - 1 > 0:
s2 = -2 * k * (n - k) * (1 - t)**(n - k - 1) * t**(k - 1)
elif k >= 1 and n - k == 1:
s2 = -2 * k * t**(k - 1)
else:
s2 = np.zeros_like(t)
if k >= 2:
s3 = (k - 1) * k * (1 - t)**(n - k) * t**(k - 2)
else:
s3 = np.zeros_like(t)
coeff = s1 + s2 + s3
return C * coeff
def coefficient(cls, n, k, ts):
C = binomial(n, k)
return C * ts**k * (1 - ts)**(n - k)
def coeff(self, k, ts):
n = self.degree
C = binomial(n, k)
return C * ts**k * (1 - ts)**(n-k)
