Splines II

1
Splines II Interpolating Curves

• based on
• Michael Gleicher Curves, chapter 15 in
• Fundamentals of Computer Graphics, 3rd ed.
• (Shirley Marschner)
• Slides by Marc van Kreveld

2
Polynomial pieces
Canonical form of a polynomial of degree n
defined with vector coefficients ai
Generalized form of a polynomial defined with
vector coefficients ci that are blended by the m
polynomials bi(t) The degree is the max of the
degrees of the bi(t)
Using blending polynomials is the way to make
splines
3
Recall Basis and constraint matrices

• Specifications of a curve give a constraint
  matrix p0 f(0) a0 0 a1 02
  a2 p1 f(0.5) a0 0.5 a1
  0.52 a2 p2 f(1) a0 1 a1
  12 a2
• Its inverse B C1 is the basis matrix

(quadratic curve)
4
Blending functions

• Blending functions (or basis functions) are
  functions of u and specify how to mix the
  specified constraints (points to pass through,
  derivatives, )
• Let u 1 u u2 u3 un be the powers of
  u
• b(u) u B, a vector whose elements are the
  blending functions

5
Blending functions

• u 1 u u2 u3 un
• b(u) u B so we obtain for the usual quadric
  example with three points specified

6
Blending functions

• We can now blend the control points

7
Blending functions
We see the contributions of each point depending
on u
For fixed u, we linearly interpolate the three
points
8
Blending functions
1
0
1
0.5
u
9
Blending functions

• Note the sum of the contributions

10
Polynomials for interpolation

• Given points p (p0, p1, , pn) and increasing
  para-meter values t (t0, t1, , tn), we can
  make a poly-nomial of degree n that passes
  through pi exactly at parameter value ti so
  f(ti) pi

p1
p2
p5
p4
p0
p3
11
Polynomials for interpolation

• Given points p (p0, p1, , pn) and increasing
  para-meter values t (t0, t1, , tn), we can
  make a poly-nomial of degree n that passes
  through pi exactly at parameter value ti so
  f(ti) pi

p1
p2
p5
p4
p0
p3
the brown curve has t1 gt t1 and t4 lt t4
12
Polynomials for interpolation

• Method
• Set up constraint matrix as before
• Invert to get basis matrix, giving the n1 basis
  functions bi(t) and the polynomial f(t)
• Alternative method (Lagrange form)

13
Why not use polynomials to interpolate 5 or more
points

• Polynomials of higher degree have
• extra wiggles
• overshoots
• non-locality moving the point pn changes the
  curve even near p0 also when we add a point
  at the end

14
Why not use polynomials to interpolate 5 or more
points

• This gets worse with higher degree (more points)

15
Blending again

• A piecewise-linear curve (polygonal line) can
  also be defined using blending functions

1
0 ? u ? 0.5
0.5 ? u ? 1
p1
p0
p2
0
1
u
0.5
16
Piecewise cubic polynomials

• Allows position and derivative at each end
• Allows C2 continuity
• Are in a sense the most smooth curve minimum
  curvature over its length (curvature second
  derivative)

curvature low
curvature 0
curvature high
17
Cubic polynomials

• f(u) a0 a1 u a2 u2 a3 u3 in
  canonical form
• Four control points (points or derivatives on
  curve) needed
• For piecewise cubic curves n pieces require 4n
  control points, but C0 continuity already fixes
  n 1 of them

18
Cubic polynomials

• The ideal properties
• Each piece is cubic
• The curve interpolates the control points
• The curve has local control
• The curve has C2 continuity
• We can have any three but not all four
  properties
• Natural cubics do not have local control
• Cubic B-splines do not interpolate the control
  points
• Cardinal splines are not C2

19
Natural cubics

• The first piece specifies start and end
  positions, and the first and second derivative at
  the start
• For each other piece, the position, first and
  second derivative match with the piece before
  it? only the endpoint can be specified freely
• A curve with n pieces has n3 control points in
  total (n1 points and 2 derivative specifications)

p1
p2
p0
p3
20
Natural cubics

• We get (with f(u) a0 u a1 u2 a2 u3
  a3) p0 f (0) a0 0 a1 02
  a2 03 a3 p1 f(0) 1 a1
  2?0 a2 3?02 a3 p2 f(0)
  2 a2 6?0 a3 p3 f (1)
  a0 1 a1 12 a2 13 a3

constraint matrix
21
Natural cubics

• When you modify your natural cubic, for instance
  by changing the derivative at the start of the
  first piece, then the whole curve changes? not
  local

22
Hermite cubics

• Specifies positions of start and end, and the
  derivatives at these points
• C1 continuous since the start position of a piece
  must be the same as the end position of the
  previous piece, and the same is true for the
  derivatives
• A curve with n pieces has 2n2 control points
  in total

p1
p2
p3
p0
23
Hermite cubics

• Local control changing the position or
  derivative of any point influences only the piece
  before and the piece after that point
• See slides 34-35 of Curves I lecture

24
Cardinal cubics

• Specifies positions only the derivatives at each
  point are determined by the points before and
  after it
• C1 continuous
• A curve with n pieces has n3 control points in
  total

p1
p2
p3
p0
p1 p3
p0 p2
25
Cardinal cubics

• A tension parameter t ? 0,1) determines the
  strength of bending
• Each derivative vector pi-1 pi1 is scaled by (1
  t)/2

p1
p1
p2
p2
p0
p0
p3
p3
t 0.5
t 0 (Catmull-Rom splines)
26
Cardinal cubics
???
scaling factor related, not the tension t
27
Cardinal cubics

• We can set up the constraints f(0)
  p1 f(1) p2 f(0) ½ (1 t) (p2
  p0) f(1) ½ (1 t) (p3 p1)
• Rewrite to get expressions for p0, p1, p2 and p3
• The first and last control points are not
  interpolated no derivative would be specified
• Therefore, n pieces require n3 points

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Summary

• Piecewise cubic curves are popular for modeling
• We want C2 continuity, locality, and
  interpolation we can get only two of these with
  cubics
• Natural cubics do not have locality
• Hermite cubics are only C1 continuous
• Cardinal cubics are only C1 continuous
• The splines that follow next (Bezier and
  B-spline) will not interpolate but approximate
  the control points