CS 284: CAGD (SPLINES)
Lecture #17 -- We 10/21, 1998.
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Preparation:
Read handout: "Local surface interpolation with shape parameters
between adjoining Gregory patches," by Shirman and Sequin.
Lecture Topics
Surfaces
The real challenge is to design surfaces
Need to stitch together multiple patches
- - this is harder than stitching together curve segments
- - need to worry about continuity in both directions
- - corners are tricky !
Simple Surface "Connectivity"
- - single, non-end-around-conncted surfaces not too hard.
Arbitrary Surface "Connectivity"
- - things get more difficult as we close surfaces into "tubes", and "blobs",
- - they get really hard when we have objects of higher genus.
Need a quick survey of the types of connectivity we may encounter:
Topology
Topological properties stay invariant under
smooth deformations of the embedding space.
Topological Issues in Curves
- - open vs. closed
- - knotted ? -- what type ?
Topological Issues in Surfaces
- - open vs. closed (what does this mean ?)
- - double-sided vs. single sided (Moebius band)
- - how many rims ? how many holes ? Genus ?
A brief summary on Genus and Euler Characteristics
- - -
( X = Euler Characteristic ).
- - -
( G = Genus of a Surface ).
What we can do with a single rectangular patch ...
Joining one set of opposite ends straight:
Cylinder: 2 rims, double-sided.
Joining one set of opposite ends flipped:
Moebius Band: 1 rim, single-sided.
Joining both pairs of opposite ends straight:
Torus: closed, double-sided, X=0, G=1.
Joining one pair straight; the other flipped:
Klein Bottle: closed, single-sided, X=0, G=2.
- -
Another picture of a Klein bottle blown in glass by Alan Bennet.
Joining both pairs flipped:
The Crosscap
- - - one possible finite model of
the Projective Plane: closed, single-sided, X=1, G=1.
Representation of Surfaces
and some of the tricks familiar from Curves
- Tensor-product Bezier patches
- Rectangular B-spline surfaces
- Decasteljau evaluation
- Degree elevation
- Triangular Bezier patches
- Gregory patches
A System to Create G-1 Continuous Interpolating Surfaces
- Background of the UniCubix System
- Triangular patches that can be joined to qudrilateral tensor product patches
- Triangular Gregory patches
- Making G-1 continuous seams between patches
Continue next time ...
Programming Assignment #5:
Modified specs; due Wed. 10/21/98
Programming Assignment #6:
Due in about three weeks ... TBA
Next Reading Assignment:
Again: "Local surface interpolation with shape parameters
between adjoining Gregory patches," by Shirman and Sequin.
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