CS 284: CAGD
Lecture #8 -- Mo 9/25, 2000.

Preparation:

Topic: 3D Curves

Review: Differential Geometry of (3D) Curves

Construction of the Frenet Frame

Serret - Frenet Relations

• What is Curvature ?
  - - rotation around b
• What is Torsion ?
  - - rotation around t
• Which unit vectors serve local rotation axis ?
  - - see above
• Which unit vectors serve as indicators ?
  - - for curvature: use -n; {n would crowd the inside of the bend}
  - - for torsion: use b; { allows both signs; t would crowd the tangent vector}

Puzzle

• Given: curvature as a function of arc-length,
  - - does this define the shape of a curve ?

Remarks on last Homework Assignment

• How to calculate the minimum-torsion geometry.
  - - project from one bisector plane to the next one

Topic: Knot Insertion (Curve Refinement)

The Role of the Knots in B-Splines

• 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.
• Some facts about behavior of knots

Knot Multiplicities

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

Vertex Multiplicities

• Piling deBoor points on top of one another
• Effects on the B-spline
• Effects on parametrization
• 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 ?

On Wednesday we will start with an introduction to Surface Patches.
We will deal with rational curves (and surfaces) at a later time.

Next Reading Assignment:

Current Homework Assignment: (to be done individually)

Programming de Casteljau in Java

DUE: WED 9/27/00, 9:10am.

On line: Follow submission instructions on the instructional pages.