CPSC 424 Rational Curves Syllabus - Ugrad.cs.ubc.ca

Rational Curves
Definition:
• A rational curve is of the form
F(
t) :
m
i0
m
B
i0
m
i
B
( t)
w
m
i
i
( t)
w
b
i
i
B
m m
i
m
i0
m
B
j
j0
( t)
w
i
( t)
w
j
b
i
w
0,
w0 w 1
with weights
Note:
i
m
• We can use either of Bézier and B-Spline basis here
© Alla Sheffer & Wolfgang Heidrich
Rational Curves
In homogeneous coordinates:
• Note that we get the original Bezier
curves if all weights are 1!
• Clear from this representation:
the projective image of a rational
curve is again a rational curve
(projective invariance)
F(
t) :
m
i0
m
i0
m
i0
m
B
B
B
i0
m
i
m
i
m
i
B
( t)
w
( t)
w
( t)
w
m
i
i
i
i
z
( t)
w
x
y
i
i
i
i
© Alla Sheffer & Wolfgang Heidrich
3