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Slide 227 October 1999 CS 318 - Computer Graphics (Top Changwatchai)

Spline Basics (1/3) Definitions

Textbook calls a spline curveany composite curve formed

with polynomial sections satisfying specified continuityconditions at the boundary of the pieces - can be 2D or 3D

Spline surfacesformed from two sets of orthogonal spline

curves

From now on spline refers to spline curve

Control points

Used to specify spline curves

Two ways to fit splines

Interpolation - curve passes through each control point

Approximation - curve does not necessarily pass through anycontrol point

Convex hull

minimum convex polygon boundary that encloses set of

control points

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Spline Basics (2/3) Splines are specified using parametric equations

P(u) = [ x(u) y(u) z(u) ] - row vector

P(u) = Au3+ Bu2+ Cu + D - each of A, B, C, and D are row vectors

From now on, we treat P as scalar:

P(u) = au3+ bu2+ cu + d

Unless otherwise specified, all derivatives are with respect to parameter u

Continuity conditions (between two curve sections)

parametric

C0

- same point C1- tangent lines (1stderivatives) equal

C2- 1stand 2ndderivatives equal

etc

geometric

G0= C0

parametric derivatives proportional (in same direction) but not necessarily equal:

G1, G2, etc

note

there exists a case where you can have G1continuity without C1continuity

Local vs. global control

Global control - if one point altered, the entire curve affected

Local control - only that curve section is affected

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Spline Basics (3/3)

Recall from previous slide

P(u) = au3+ bu2+ cu + d

We can write as: P(u) = U Co U = [ u3u2u 1 ] Co = [ a b c d ]T

= U MsplineMgeom

= B Mgeom

U is the 14parameter vector

Co is the 41 coefficient matrix

Three methods of spline specification

Boundary conditions

Description given in English

Actual values placed in 41 geometry vectorMgeom

Basis matrix

Msplineis the 4

4 basis matrix This is constant for a given type of spline

Co = MsplineMgeom

Blending functions

B is a 14 matrix representing the blending functions

B = U Mspline= [ F0(u) F1(u) F2(u) F3(u) ]

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Types of Splines

Why cubic?

Higher order less stable, more complex to calculate Lower order dont look as good

Cubic is smallest that specifies endpoints and derivatives,

and which is nonplanar in 3D

Cubic splines (interpolation)

Natural

Hermite

Cardinal

Kochanek-Bartels

Bzier (approximation)

B-splines (approximation)

Other

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Review of Cubic Splines Features

Cubic splines interpolate - passes through every control point

We fit one curve section at a time between each pair of successive control

points (u=0 to u=1)

We have 4 coefficients (cubic) so for n curve sections need 4n boundary

conditions

Natural

Boundary conditions: endpoints on control points, 1st and 2nd parametric

derivatives match between adjacent curve sections

Bad: exhibits global control

Hermite

Boundary conditions: endpoints on control points, parametric slope

specified at control points

Exhibits local control, but may be inconvenient to specify slopes

Cardinal Same as Hermite, but we compute tangents from neighboring control points

Has tension parameter t

Kochanek-Bartels

Same as cardinal but adds two additional parameters, bias and continuity

Good for modelling abrupt changes

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7/8Slide 727 October 1999 CS 318 - Computer Graphics (Top Changwatchai)

Bzier Curves - Features

Approximation

Global control

Number of control points determines degree of Bzier

curve

Easy to implement and calculate recursively

Always passes through first and last control points

Lies within convex hull

Easy to connect sections together

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Note from 27 October lecture

In class we went over how to derive the following boundary conditions from the definition of Bzier curve:

P(0) = p0 P(1) = pn

P'(0) = -np0+ np1 P'(1) = -npn-1+ npn

Dr. Chawla points out that there is a simpler (though equivalent) way to view our derivation:

For u= 0, we see that every term except the first has a uand therefore goes to 0.

For u= 1, we see that every term except the last has a (1-u) and therefore goes to 0. So we have:

Taking the parametric derivative, we get:

For u= 0, we see that all terms except the first two have us and therefore go to 0.

For u= 1, we see that all terms except the last two have (1-u)s and therefore go to 0. So we have:

n

k

nkk uBEZu

0

, )(p)(P

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n

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n

n

n

nnnu

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n

nuu

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nu

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p)1(2

p)1(2

p)1(1

p)1(0

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2

22

2

1

10

112

1

223

2

322

2

21

1

1

0

p)1()1)(1(1p)2)(1()2()1(2p

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p)1()1()1(1

p)()1(0

p)(P'

n

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nun

n

unuun

n

uunuun

n

nuuuun

nuuun

nun

u

0p)0(P 0

n

n

n

np)1(P

1p)(

0p)0(P' 10

nn

nn

n

n

n

n

nn

p)1(

1p)1(P' 1

10-27-Ch10_2

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