CS 284: CAGD
Lecture #9  Tu 9/26, 2006.
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Preparation:
Review Rockwood pp. 94117: BSplines
Study handout: B+B+B: Effect of Knot Multiplicities
Topic: BSplines (cont.)
Reviewing some key aspects of BSplines

BSpline
Basis Functions
 Strongly overlapping control domains leads to builtin smoothness.

Periodic
(closed) BSpline Curves (p 105)
 Endaround reuse of control points.

Bsplines of different degrees (Applet on p 102)
 How many control segments does it take to make the first curve
segment appear ?
 Note that quadratic Bspline touches control polygon  Why ?

Use of Bsplines:
 What can you do with a given number of segments ?
 How many segments does it take to make a knotted 3D space curve
?
Vertex Multiplicities

Piling deBoor points on top of one another

Effects on parametrization ?

Effects on basis functions ?

Effects on the Bspline curve ?

Experiment with the interactive display panels shown in the book
on page 102.
Extend a cubic curve by 6 more points and then move the de Boor
control points
to study what happens to the Bspline curve when you:

double up two de Boor points at the end ?

tripple a de Boor endpoint ?

give a de Boor endpoint a multiplicity of four ?

double up two internal de Boor points ?

make a tripple internal de Boor point ?

give an internal de Boor point a multiplicity of four ?
Nonuniform BSplines

Changing the Knot Values
 can assume arbitrary, monotonically ordered tvalues
 does this affect only the parametrization, or also the shape of
the curve ?
 study their influence with applet on p105.

Effect on BSpline Curve (Applet on p106)
 What is the effect on reducing the knot interval ?
 What happens when we doubleup knots 1 and 2, or knots 3 and 4
(Applet on p 107)

Effect on Basis Functions (Applet on p110)
 Study effect of shifting knots for degree 1 ... 4 basis functions.
 How far does the effect of a changed knot value spread ?
Knot Multiplicities

Gives additional design freedom

Effects on the basis functions (BBB p162166)

Effects on the Bspline (BBB p167172)
==> see handout.
Knot Insertion (Curve Refinement)
Topics: Surface Patches
Preparation: Rockwood Chapter 8: pp 133151: Surfaces
From Curves to Surfaces Patches

Do in "u" and in "v" directions what we have learned in "t" direction ...

Bilinear Bezier patch = Coons Patch

Cubic
tensorproduct Bezier patch

Symmetry in u,v: interchange roles of "rails" and "curves"

Biquintic
Bezier patch

DeCasteljau evaluation of tensor product patches (p140)

Patch subdivision, degree elevation ... All still works as in the 1D case!

Putting Bezier patches together with G1 or better continuity becomes difficult
and painful.

If you want a high degree of continuity, consider the approximating Bspline
surfaces:

Bicubic
and biquintic Bspline patches

Rectangular Bspline surfaces

Comparison, tradeoffs between Bezier and BSpline surfaces
Triangular Surfaces Patches
We can also deal with triangular patches, but need a different
interpolation scheme:
Barycentric coordinates: three numbers, but with the constraint that they
must sum to 1.0.
DeCasteljau evaluation technique can also be applied to triangular
patches.
Reading Assignment:
Read the seminal paper by Catmull and Clark on subdivision surfaces
Current Homework Assignment: Construct a Parameterized Bell
Your task is to design a curved thin (metal) surface of finite thickness
that has a chance of emitting a nice sound when struck and used as a bell.
There are many different ways in which such a surface could be defined: Some possible approaches to composing a Bell shape.
For this assignment you should concentrate of using a sweep in some
form and using some spline for either the cross secition or for the
guide curve, or for both.
{ Check out the code segment circle_sweep.txt in the CODE directory, for a practical way to make a circlar/cylindrical sweep around the zaxis}.
There should be from 3 to 5 parameters with wich you can change the
shape of the bell interactively without loosing continuity in the
overall surface.
Let yourself be inspired by this collection of
pictures and references,
but keep your shape simple!
Create a SLIDE file with all parameters set to their preferred values.
Capture your design pictorially using the screen saver.
 Email to me your SLFfile and a captured picture (JPG, GIF, PNG)
DUE: Thu. 9/28/2006, 2:10pm.
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