Numerical Methods in CAGD
FMN011
2016
Carmen Arévalo & Claus Führer
[email protected], [email protected]
Lund University
Lecture 2
Ht1 2016
Arévalo/Führer FMN100 2013
LINEAR INTERPOLATION
A straight line through points p and q is the set of all points of the form
x(t) = (1 − t)p + tq;
t∈R
For 0 ≤ t ≤ 1, the point x is between p and q.
This mapping of the interval [0, 1] to the straight line through two given
points is also called linear interpolation.
Arévalo/Führer FMN100 2013
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RATIOS
q(t) = (1 − t)p0 + tp1
u
t
p0
x
q
1−t
u
p1
The ratio of q with respect to p0 and p1 is t/(1 − t).
Ratios are invariant under affine maps.
Arévalo/Führer FMN100 2013
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DOMAIN TRANSFORMATIONS
The map Φ : [c, d] → [0, 1]
Φ(u) =
u−c
d−c
is an affine map called a domain transformation.
Ψ(t) = (1 − t)p0 + tp1, 0 ≤ t ≤ 1
d−u
u−c
Ψ ◦ Φ(u) =
p0 +
p1 , c ≤ u ≤ d
d−c
d−c
are both the same segment [p0, p1].
Arévalo/Führer FMN100 2013
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BERNSTEIN POLYNOMIALS
Binomial expansion of 1:
1 = (t + (1 − t))n =
n X
n
i=0
i
ti(1 − t)n−i
The Bernstein polynomials of degree n are
n i
n
Bi (t) =
t (1 − t)n−i,
i
Arévalo/Führer FMN100 2013
i = 0, 1, . . . , n
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COEFFICIENTS OF THE BERNSTEIN POLYNOMIALS
The coefficients can be obtained from Pascal’s Triangle:
1
1
1
1
1
1
1
2
3
4
5
Arévalo/Führer FMN100 2013
1
3
6
10
1
4
10
1
5
1
n=0
n=1
n=2
n=3
n=4
n=5
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PROPERTIES OF BERNSTEIN POLYNOMIALS
n−1
• Recursion formula: Bin(t) = (1 − t)Bin−1(t) + tBi−1
(t)
• Partition of unity:
n
X
Bin(t) ≡ 1
i=0
• Linearly independent
n
• Symmetric: Bin(t) = Bn−i
(1 − t)
• Roots only at 0 and 1: Bin(0) = 0
Bin(1) = 0 (i = 0, . . . , n − 1)
(i = 1, . . . , n),
• Positive: Bin(t) > 0 for 0 < t < 1
Arévalo/Führer FMN100 2013
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