Splines II – Interpolating Curves

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Splines II – Interpolating Curves. based on: Michael Gleicher : Curves , chapter 15 in Fundamentals of Computer Graphics, 3 rd ed. (Shirley &amp; Marschner ) Slides by Marc van Kreveld. Polynomial pieces.

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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

Polynomial pieces

Canonical form of a polynomial of degree n defined with vector coefficients ai

Generalized form of a polynomial defined with vector coefficients cithat 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

Recall: Basis and constraint matrices
• Specifications of a curve give a constraint matrixp0 = f(0) = a0 + 0 a1 + 02a2p1 = f(0.5) = a0 + 0.5 a1 + 0.52a2p2 = f(1) = a0 + 1 a1 + 12a2
• Its inverse B = C–1 is the basis matrix

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 uu2u3 … un ] be the powers of u
• b(u) = uB, a vector whose elements are the blending functions
Blending functions
• u = [ 1 uu2u3 … un ]
• b(u) = uB so we obtain for the usual quadric example with three points specified:
Blending functions
• We can now blend the control points
Blending functions

We see the contributions of each point depending on u

For fixed u, we linearly interpolate the three points

Blending functions
• Note the sum of the contributions

+

Polynomials for interpolation
• Given points p = (p0, p1, … , pn) and increasing parameter values t = (t0, t1, … , tn), we can make a polynomial of degree n that passes through pi exactly at parameter value ti so f(ti) = pi

p1

p2

p5

p4

p0

p3

Polynomials for interpolation
• Given points p = (p0, p1, … , pn) and increasing parameter values t = (t0, t1, … , tn), we can make a polynomial 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’ > t1 and t4’ < t4

Polynomials for interpolation
• Method
• Set up constraint matrix as before
• Invert to get basis matrix, giving the n+1 basis functions bi(t) and the polynomial f(t) :
• Alternative method (Lagrange form)
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
Why not use polynomials to interpolate 5 or more points
• This gets worse with higher degree (more points)
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

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

Cubic polynomials
• f(u) = a0 + a1u + a2u2 + a3u3 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
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
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 n+3 control “points” in total (n+1 points and 2 derivative specifications)

p1

p2

p0

p3

Natural cubics
• We get (with f(u) = a0 + u a1 + u2a2+ u3 a3):p0 = f (0) = a0 + 0 a1 + 02a2 + 03a3p1 = f’(0) = 1 a1 + 20 a2 + 302a3p2 = f’’(0) = 2 a2 + 60 a3p3 = f (1) = a0 + 1 a1 + 12a2 + 13a3

constraint matrix

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
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 2n+2 control “points” in total

p1

p2

p3

p0

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
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 n+3 control points in total

p1

p2

p3

p0

p1p3

p0p2

Cardinal cubics
• A tension parameter t [0,1) determines the “strength” of bending
• Each derivative vector pi-1pi+1 is scaled by (1 – t)/2

p1

p1

p2

p2

p0

p0

p3

p3

t = 0.5

t = 0 (Catmull-Rom splines)

Cardinal cubics

???

scaling factor related,not the tension t

p1

p2

p0

p3

Cardinal cubics
• We can set up the constraints:f(0) = p1f(1) = p2f’(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 n+3 points
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
• Hermitecubics 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
Questions
• Assume piecewise linear interpolation by blending functions, where the curve passes through points p0, p1, p2, p3,and p4 at u = 0, 0.25, 0.5, 0.75, and 1. Define the five blending functions bi(u) so that
• Assume that someone defines quadratic blending functions as on slides 5-9, except that b2(u) = u2. Does the resulting curve pass through its defining points. Is the resulting curve translation-invariant?