CS 284: CAGD
Lecture #7 -- We 9/20, 2000.

Preparation:

Main Lecture Topics

Review: Multi Segmented B-Splines

• Finding de Boor Points from Bezier Points
• Interpretation: Change to a new set of basis functions -- some linear combination of old ones.

B-Spline Basis Functions

• Concentrate on one dimension of a B-Spline curve: Y(t)= piecewise m-degree polynomial.
  Assemble that function from m-degree polynomial pieces, joined with C-(m-1) continuity.
  How to construct such basis functions: Repeated convolution
  m=1 : triangular hat functions
  m=2 : three parabolic pieces
  m=3 : four cubic pieces
• The limited support of these basis functions
• Comparison with Bezier Basis Functions

The Behavior of B-Splines

• Reviewing the standard seven properties
• Degrees of Continuity
• Comparing B-Splines of different degrees
  - - Study their behavior using the panes on pages 101,102.
• The Use of B-Splines
  - - B-Spline Curves of degree 3 (p 97)
  - - Periodic (closed) B-Spline Curves (p 105)
  - - What can you do with a given number of segments ?
  - - How many does it take to make a knot ?
• The legal range for a parameter
  - - Page 103, bullet 3.
• What do we gain from this restriction ?
• What do we pay -- if anything ?
  - - Page 103, bullet 4.

Non-uniform B-Splines

• Changing the Knot Values (p105)
• Effect on Basis Functions (p106)
• Effect on B-Spline Curve (p110)
• 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).

Discussion of Homework Assignment

Differential Geometry of (3D) Curves (Handout: Ch5, Mortenson)

• Analysis of Curves (as compared to "Design" or "Data Fitting")
• Intrinsic Curve Properties -- defined at each local point on the corve)
  - - Vectors: Tangent, Normal, Binormal;
  - - Planes: Osculating, Normal, Rectifying;
  - - Scalars: Curvature, Torsion.
• Extrinsic Curve Properties (overall, global values)
  - - Arc-length;
  - - Planarity, Linearity;
  - - Closedness, Turning Number, Knot Type;
  - - Highest polynomial degree, lowest continuity type, # of inflection points.

Construction of the Frenet Frame

• Finding the Tangent
• Normal Plane
• Osculating Plane
• Normal vector
• Binormal
• Difference between Normal and second derivative
• The three coordinate planes and their relations to the 3 vectors.

Serret - Frenet Relations

• What is Curvature ? -- 1/r of the osculating circle
• What is Torsion ? -- rate of rotation of osculating plane (around tangent)
• Inflection points -- where curvature = 0.
• Which unit vectors serve as indicators ?
• Which unit vectors serve as local rotation axis ?

New 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.

Next Reading Assignment: