# CS 284: CAGD  Lecture #9 -- Tu 9/26, 2006.

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## Preparation:

Review Rockwood pp. 94-117: B-Splines
Study handout: B+B+B: Effect of Knot Multiplicities

# Topic:  B-Splines (cont.)

### Reviewing some key aspects of B-Splines

• B-Spline Basis Functions
• Strongly overlapping control domains leads to built-in smoothness.
• Periodic (closed) B-Spline Curves (p 105)
• End-around re-use of control points.
• B-splines of different degrees (Applet on p 102)
• How many control segments does it take to make the first curve segment appear ?
• Note that quadratic B-spline touches control polygon -- Why ?
• Use of B-splines:
• 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 B-spline 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 B-spline curve when you:
1. double up two de Boor points at the end ?
2. tripple a de Boor end-point ?
3. give a de Boor end-point a multiplicity of four ?
4. double up two internal de Boor points ?
5. make a tripple internal de Boor point ?
6. give an internal de Boor point a multiplicity of four ?

### Non-uniform B-Splines

• Changing the Knot Values
• can assume arbitrary, monotonically ordered t-values
• does this affect only the parametrization, or also the shape of the curve ?
• study their influence with applet on p105.
• Effect on B-Spline Curve (Applet on p106)
• What is the effect on reducing the knot interval ?
• What happens when we double-up 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

• Effects on the basis functions (BBB p162-166)
• Effects on the B-spline (BBB p167-172)
==> see handout.

# Topics: Surface Patches

Preparation:  Rockwood Chapter 8: pp 133-151: 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 tensor-product 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 B-spline surfaces:
• Bicubic and biquintic B-spline patches
• Rectangular B-spline surfaces
• Comparison, trade-offs between Bezier and B-Spline 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.

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 z-axis}.
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.

• E-mail to me your SLF-file and a captured picture (JPG, GIF, PNG)

DUE: Thu. 9/28/2006, 2:10pm.

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