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Splines: Interpolation and Approximation

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Title: Splines: Interpolation and Approximation


1
Splines Interpolation and Approximation
  • Interpolating spline - curve passes through
    control points
  • Approximating spline - control points influence
    shape

2
Specifying Splines
  • Control Points - a set of points that influence
    the curve's shape
  • Knots - control points that lie on the curve
  • Interpolating spline - curve passes through the
    control points
  • Approximating spline - control points merely
    influence shape

3
Piecewise Curve Segments
  • Often we will want to represent a curve as a
    series of curves pieced together
  • But we will want these curves to fit together
    reasonably well

4
Problem
  • Define a smooth curve that interpolates the first
    and last point and approximates the others.

p4
p3
p1
p2
5
Possible solution
Consider a set of of n1 points Let us
construct a curve interpolating/approximating
the given points. It is convenient to choose
vector valued f linear in Pk
are called basis functions of blending
functions is called the
control polygon
6
Properties of blending functions
  • Coordinate system independence
  • Shape of control polygon approximates shape of
    curve
  • Geometric characteristics of P(t) are obtained
    from geometric characteristics of its control
    polygon.

If all the control points coincide is
independent of t. Hence
7
Properties of blending functions
Assume that all the control points except Pk
converge to Q
If P(t) is within the line segment QPk then
Together with It implies the convex hull
property P(t) lies with the convex hull of its
control polygon
8
Physical Analogy
Given N1 points with masses Consider their
center of mass Imagine that each mass varies as
a function of some parameter t
9
Other desirable properties
  • Endpoint interpolation
  • Symmetry
  • Linear independence the blending functions
  • are linearly independent otherwise for
    certain control point arrangements , the curve
    collapses to a single point.

10
French Curve (Bézier curve)
  • P. de Casteljau, Citroen, 1959
  • P. Bezier, Renault, 1962

p2
p3
po
p1
The number of k-combinations
from n-set
11
Bernstein polynomials
The number of k-combinations from
n-set
12
Properties of Bernstein polynomials I
  • They are linearly independent
  • They are symmetric
  • They form a partition of unity

13
Properties of Bernstein polynomials II
  • They are positive
  • They satisfy the recursion formula

14
Properties of Bernstein polynomials III
  • Derivative formula

15
Properties of Bernstein polynomials IV
  • Linear precision
  • Extrema

has its maximum at
16
Properties of Bernstein polynomials V
  • Recursion formula again
  • Triangle scheme for computing the Bernstein
    polynomials

17
Bézier curves
18
Bézier curve geometrically - de Casteljau
Algorithm
19
Bézier curve geometrically - de Casteljau
Algorithm
  • Solution for n3

P1
P0
P2
20
Bézier curve geometrically - de Casteljau
Algorithm
  • Solution for n4

P1
P2
P0
P3
21
Bézier curve geometrically - de Casteljau
Algorithm
P2
P1
u0.25
P0
P3
22
Some properties of Bézier curves
  • The tangent vectors at the end points are
    determined by the control points
  • Convex hull property a curve is always enclosed
    in the convex hull formed by the control polygon

23
Some properties of Bézier curves
  • Variation Diminishing Property
  • no straight line intersects a Bézier curve more
    times than it intersects the curve's control
    polyline.
  • It tells us that the
    complexity (i.e., turning

    and
    twisting) of the curve

    is no more complex than

    the control polyline.

24
Degree elevation
  • Used to add more control over a curve
  • Start with
  • Now figure out the Qi
  • Compare coefficients
  • Repeated elevation converges to curve

25
Degree elevation
P1
1/2
Q2
1/2
1/4
P2
Q1
1/4
Q3
3/4
3/4
P0Q0
P3Q4
26
Joining Bézier Curves
Curvature at P0
27
Polar forms
Every polynomial curve of degree
can be associated with a unique n-variate
symmetric polynomial such
that The polynomial is referred to as the polar
form or blossom of
28
Polar forms properties
  • agrees with on its
    diagonal
  • is symmetric in its
    variables which means that for any permutation
    of
  • is affine in each variable

29
Polar forms examples I
30
Polar forms examples II
31
Polar forms examples III
32
Polar forms
Any polynomial curve defined over
interval a,b can be considered as a Bézier
curve with control points For example
for defined over
0,1
33
Polar Forms in de Casteljau Algorithm
p(0,1)p(1,0)
p(0,0) p(0)
p(1,1) p(1)
00
01
11
1-u
u
1-u
u
0u
1u
1-u
u
uu
p(u) p(u,u) (1-u)2 P0 2u(1-u) P1 u2 P2
34
Polar Forms in de Casteljau Algorithm
35
Subdividing Bezier Curve
0½1
001
011
½½½
0½½
½½1
00½
½11
000
111
36
Degree Elevation
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