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

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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

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Problem

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

p4
p3
p1
p2
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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
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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
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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
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Physical Analogy
Given N1 points with masses Consider their
center of mass Imagine that each mass varies as
a function of some parameter t
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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.

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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
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Bernstein polynomials
The number of k-combinations from
n-set
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Properties of Bernstein polynomials I

• They are linearly independent
• They are symmetric
• They form a partition of unity

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Properties of Bernstein polynomials II

• They are positive
• They satisfy the recursion formula

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Properties of Bernstein polynomials III

• Derivative formula

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Properties of Bernstein polynomials IV

• Linear precision
• Extrema

has its maximum at
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Properties of Bernstein polynomials V

• Recursion formula again
• Triangle scheme for computing the Bernstein
  polynomials

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Bézier curves
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Bézier curve geometrically - de Casteljau
Algorithm
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Bézier curve geometrically - de Casteljau
Algorithm

• Solution for n3

P1
P0
P2
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Bézier curve geometrically - de Casteljau
Algorithm

• Solution for n4

P1
P2
P0
P3
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Bézier curve geometrically - de Casteljau
Algorithm
P2
P1
u0.25
P0
P3
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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

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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.

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Degree elevation

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

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Degree elevation
P1
1/2
Q2
1/2
1/4
P2
Q1
1/4
Q3
3/4
3/4
P0Q0
P3Q4
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Joining Bézier Curves
Curvature at P0
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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
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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

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Polar forms examples I
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Polar forms examples II
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Polar forms examples III
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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
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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
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Polar Forms in de Casteljau Algorithm
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Subdividing Bezier Curve
0½1
001
011
½½½
0½½
½½1
00½
½11
000
111
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Degree Elevation