(Smooth) CURVES and SURFACES

Lecture #2  for CS 274

Wednesday  3/16, 2005 -- 12:40-2:00pm, -- 203 McLaughlin

Differential Geometry of Curves

REFERENCE -- TO LEARN MORE:

Chapter 5 in "Geometric Modeling" by Michael E. Mortenson, John Wiley &Sons, 1985

"Analysis" of Curves (as compared to "Design" or "Data Fitting")

Construction of the Frenet Frame (Ref. frame for intrinsic properties)

Serret - Frenet Relations

Puzzle

Derivatives of Splines

REFERENCES -- TO LEARN MORE:

"Interactive Curves and Surfaces," (with Multimedia Tutorial on CAGD), A. Rockwood and P. Chambers, Morgan Kaufman Publishers, Inc.


Introduction to Subdivision

Different from curve-subdivision based on DeCasteljau's algorithm.
Review of the DeCasteljau construction algorithm.

Main idea: stepwise, iterative refinement: (e.g., double the number of line segments in every generation).
Demonstrations:
 -- "Cutting of Corners" (start from a quadrilateral piece of paper) -- (see: Chaikin's Algorithm)
 -- "Segment Breaking" (start with a triangular loop of cardboard) -- (good & bad properties ?)

Interpolating mid-segment subdivisions
 -- Find new mid-points for each linear segment
     -- properly bulging out to lie on curve implied by nearest neighbors;
     -- new points, once placed, stay in same location forever.

How should we do this interpolation ?:
Wish-list for a good subdivision scheme:
 -- Find final points, derivatives, tangents, normals ... without needing to do an infinite number of iteration steps.
 -- How can this be made possible ? (Remember convergent series of numbers: e.g., 1/2 + 1/4 + 1/8 + ... + 1/2n + ... = ?)
 -- Use a "stationary" scheme: The linear transformation applied to the vertices is the same in all generations.
 

An Approximating Curve Subdivision Scheme

 -- Algorithm based on cubic B-Splines

Surface Subdivision Schemes (Two-Manifolds)

The four most useful ones:
 -- Catmull Clark
 -- Doo Sabin
 -- Loop's Triangle Subdivision
 -- Zorin's Interpolating Triangle Subdivision
 

REFERENCES -- TO LEARN MORE:

E. Catmull and J. Clark: "Recursively generated B-spline surfaces on arbitrary topological surfaces," Computer-Aided Design 10(6): 350-355, Nov. 1978.

D. Doo and M. Sabin: "Behaviour of recursive division surfaces near extraordinary points," Computer-Aided Design 10(6): 356-360, Nov. 1978.

C. Loop: "
Smooth Subdivision Surfaces Based on Triangles" MS Thesis, (Chapters 3 and 5), Univ. of Utah 1987.

www.subdivision.org (Textbook site: J. Warren and H. Weimer: "Subdivision Methods for Geometric Design," Morgan Kaufman 2002.


Page Editor: Carlo H. Séquin