Bézierspline als Parameterkurve
Author: Prof. Dr. Dörte Haftendorn
Date: 15. Februar 2013

t

Ha 2011

Notenbogen

x(t)=x(A)\cdot b0(t)+x(B)\cdot b1(t)+x(C)\cdot b2(t)+x(D)\cdot b3(t)

y(t)=y(A)\cdot b0(t)+y(B)\cdot b1(t)+y(C)\cdot b2(t)+y(D)\cdot b3(t)

P

Notenbogen
in
Capella 

\copyright

A = (0.2631382470232551, 0.4708675123169748)
B = (0.5787358370961055, 1.7367388618949695)
C = (2.421581254620656, 2.1025385667303396)
D = (3.416896786385133, 0.9924172403902491)
b0(x) = (1 - x)^(3)
b1(x) = ((3 * (1 - x)^(2)) * x)
b2(x) = ((3 * (1 - x)) * x^(2))
b3(x) = x^(3)
E = (0, 2.5)
F = (3, 2.5)
a = Segment(E, F)
T = Point(a)
ts = Segment(E, T)
t: ts / a
xbez: (x(A) * b0(t)) + (x(B) * b1(t)) + (x(C) * b2(t)) + (x(D) * b3(t))
ybez: (y(A) * b0(t)) + (y(B) * b1(t)) + (y(C) * b2(t)) + (y(D) * b3(t))
P: (xbez, ybez)
Bézierkurve = Locus(P, T)
c = Segment(B, A)
d = Segment(C, D)
Text2: "t= " + t
b = false
Axb0(x) = (x(A) * b0(x))
Ayb0(x) = (y(A) * b0(x))
Byb1(x) = (y(B) * b1(x))
G = (0.4258609664320656, -1.9669326513131005)
H = (3.4258609664320656, -1.9669326513131005)
Cyb2(x) = (y(C) * b2(x))
Cxb2(x) = (x(C) * b2(x))
Bxb1(x) = (x(B) * b1(x))
Dyb3(x) = (y(D) * b3(x))
Dxb3(x) = (x(D) * b3(x))
kurve = CurveCartesian((x(A) * b0(t)) + (x(B) * b1(t)) + (x(C) * b2(t)) + (x(D) * b3(t)), (y(A) * b0(t)) + (y(B) * b1(t)) + (y(C) * b2(t)) + (y(D) * b3(t)), t, 0, 1)
i: y = 0.98
j: y = 0.84
k: y = 0.8
I = Point(i)
J = Point(i)
K = Point(i)
L = Point(i)
M = Point(i)
N = Point(i)
O = Point(i)
Q = Point(i)
R = Point(j)
S = Point(j)
U = Point(j)
V = Point(j)
W = Point(j)
Z = Point(j)
A_1 = Point(j)
B_1 = Point(j)
C_1 = Point(k)
D_1 = Point(k)
E_1 = Point(k)
F_1 = Point(k)
G_1 = Point(k)
H_1 = Point(k)
I_1 = Point(k)
J_1 = Point(k)
l = Segment(I, J)
m = Segment(K, L)
n = Segment(M, N)
p = Segment(O, Q)
r: 0.081
q = Segment(R, S)
s = Segment(U, V)
f_1 = Segment(W, Z)
g_1 = Segment(A_1, B_1)
h_1 = Segment(E_1, F_1)
i_1 = Segment(G_1, H_1)
j_1 = Segment(I_1, J_1)
u = Vector(A, B)
v = Vector(B, C)
w = Vector(C, D)
e: (t * u)
f: (t * v)
g: (t * w)
A' = Translate(A, e)
h = Vector(A, A')
B' = Translate(B, f)
u_1 = Vector(B, B')
C' = Translate(C, g)
v_1 = Vector(C, C')
w_1 = Vector(A', B')
a_1 = Vector(B', C')
b_1: (t * w_1)
c_1: (t * a_1)
A'' = Translate(A', b_1)
d_1 = Vector(A', A'')
B'' = Translate(B', c_1)
e_1 = Vector(B', B'')
k_1 = Vector(A'', B'')
l_1: (t * k_1)
A''' = Translate(A'', l_1)
m_1 = Vector(A'', A''')
GerüstOrt = Locus(A''', T)
K_1 = (3.6555069767441957, -0.8361074634937856)