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

    A System to Create G-1 Continuous Interpolating Surfaces

    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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