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)

Notenbogen in Capella 

\copyright

Ortslinie

Bogen

Parameterkurve

A = (0.2631382470232551, 0.4708675123169748)
B = (0.5787358370961055, 1.7367388618949695)
C = (2.6219361349304555, 2.0540256689513985)
D = (2.9896186284547284, 1.3290179352415692)
b0(x) = (1 - x)^(3)
b1(x) = ((3 * (1 - x)^(2)) * x)
b2(x) = ((3 * (1 - x)) * x^(2))
b3(x) = x^(3)
E = (0.3973246656477143, 2.3129534450631946)
F = (2.6973246656477143, 2.3129534450631946)
a = Segment(E, F)
T = Point(a)
ts = Segment(E, T)
t: ts / a
c = Segment(B, A)
d = Segment(C, D)
Text2: "t= " + t
b = true
H = (3.0211170727156396, -0.4366487818530811)
Bézierkurve = 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)
r = 0.081 (slider from -5 to 5, step 0.1)
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)
P = Translate(A'', l_1)
m_1 = Vector(A'', P)
K_1 = (3.6555069767441957, -0.8361074634937856)
L_1 = (0.09511957981028946, -0.6712759561855446)
M_1 = (0.6864031109121703, -0.6712759561855446)
i = false
G = (1.5137100276882878, -0.5249304123878538)
I = (4.8379740592812475, -0.5404662923959214)
J = (0.05955165893314318, -0.07438989215388836)
j = false
BézierOrt = Locus(P, T)
k = false