CS 284: CAGD 
Lecture #9 -- We 9/24, 2003.

Preparation:

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 a multiplicity of four ?

Non-uniform B-Splines

• Changing the Knot Values, (p105)

- - does this affect only the parametrization, or also the shape of the curve ?
• 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 many knots have to be considered ?
• The global parametrization of the curve

Differences in parametrization between Blossoming and other texts (see bottom of fig).
- - Rockwood:   Bi peaks at t=i
- - B+B+B:        Bi starts at u=i

Knot Multiplicities

• Gives additional design freedom
• Effects on the basis functions (BBB p162-166)
• Effects on the B-spline (BBB p167-172)

==> see handout.

Knot Insertion (Curve Refinement)

• Review of knot insertion and B-spline subdivision

- - The general approach using Blossoming for an arbitrary value for t.
• Degree elevation of a B-spline segment

- - Using Blossoming and detour via Bezier control points.
• Review of some facts about behavior of knots

B-Spline Derivatives

• See handout (BBB p 393-398)

Monday --> Wednesday: Take-home Quiz on Curves.

Then we will jump to Surface Patches (2D Manifolds).
We will deal with rational curves (and surfaces) at a later time.

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Current Homework Assignment: (to be done individually)

Program deCasteljau

DUE: Mon. 9/29/2003, 9:10am.

Reading Assignment: