CS 284: CAGD
Lecture #10 -- Mo 10/2, 2000.

Preparation:

Topics: Smooth Surfaces

Review: From Curves to Surfaces Patches

• Do in "u" and in "v" directions what we have learned in "t" direction ...
• Cubic tensor-product Bezier patch
• Biquintic Bezier patch
• Biquintic B-spline patch
• Symmetry in u,v
• Rectangular B-spline surfaces
• DeCasteljau evaluation of tensor product patches (p140)
• Comparison, trade-offs between Bezier and B-Spline surfaces

Topological Limitations of the B-spline Control Mesh

• A rectilinear mesh of quadrilaterals can be nicely mapped onto a torus.
• It cannot be mapped onto a sphere without either
  - - crunching the u-v-coordinates together at the N- and S- poles, or
  - - or introducing vertices with valences different from 4.
• It cannot be mapped nicely onto surfaces of genus higher than 1.
• For these knds of surfaces, we need a different, more general scheme.

Remarks on the Topology of Surfaces

• Genus = an integer that indicates how many handles (or tunnels) a body bounded by a closed surface will have
• Examples:
  - - sphere has genus 0;
  - - torus (doughnut) has genus 1;
  - - a "figure 8 shaped pretzel" has genus 2.
• Integrating Gaussian curvature over a closed surface will yield a result that is equal to (1-genus)*720 degrees.
• In particular, if we consider a polyhedral surface and we sum the angle deficits/excesses ayt each of its vertices, we will get the same formula.
  Angle deficit at a vertex is equal to: (360 degrees - summ of all angles of all the facets that come together at that vertex).
• Examples:
  - - polyhedra of genus 0 will have an angle deficit of 720 degrees;
  - - polyhedra of genus 1 will have an angle deficit of 0 degrees;
  - - polyhedra of genus 2 will have an angle excess of 720 degrees;
  - - polyhedra of genus 3 will have an angle excess of 1440 degrees;
• The topology of a control mesh will have to match the topology of the desired surface.
• Consequences for the topology of control meshes:
  Quadrilateral control meshes of genus 0 will have a "valence deficit" of 8 links;
  - i.e., they may have 8 vertices with valence 3.
  Quadrilateral control meshes of genus 1 will have a "valence deficit" of 0;
  - i.e., they can be formed with regular recangular B-spline control meshes.
  Quadrilateral control meshes of genus 2 will have a "valence excess" of 8 links;
  - i.e., they may have 8 vertices with valence 5,
  - - or perhaps, 4 vertices of valence 6,
  - - or any combination of extraordinary points that produce an extra 8 links.

The Classical Subdivision Surfaces by Catmull & Clark

• A generalization of the B-spline scheme
• Can handle vertices with valences different from 4
• Can handle meshes other than quadrilaterals
• One subdivision iteration calculates:
  1.) new vertices at center of faces,
  2.) new vertices at centers of all edges,
  3.) a new vertex position for each old vertex.
• After the first iteration, there will only be quadrilateral meshes.
• There will also be a constant number of extraordinary points:
  vertices of valence <>4 for each such original vertex.
  vertices of valence <>4 for each non quadrilateral mesh in original control polyhedron.
• These extraordinary points will not disappear, but they will become more and more isolated,
  being separated by more and more quadrilateral meshes joining in valence-4 vertices.
• In these ever more dominating "regular" regions, the surface will approach a B-spline.

The Analysis of the Extrordinary Points by Doo & Sabin

• After the fist subdivision step, the neighborhood of the extraordinary points does no longer change fundamentally; the topology stays the same, and the geometry gets roughly halved in every step.
• The next generation of control vertices gets calculated from the current vertices with the same linear interpolation formulas.
• The vector of all the new vertices can thus be expressed from the vector of the old vertices by a matrix multiplication.
• Each subdivision step corresponds to another matrix multiplication.
• The behavior for an infinite number of multiplications by the same matrix can be determined from the eigenproperties of the matrix.
• We can gain more insight about the behavior of the surface at a point by decomposing the general geometrical variability into a sequence of terms with clear geometrical meaning by using a discrete Fourier transform of the behavior around the point of interest.
• The dominant eigenvector is 1, indicating the the surface converges in the local neighborhood and does not run off to infinity or collapse into the origin.
• In particular, we decompose the surface behavior into cyclic periodic functions around the point:
  • The second strongest eigenvalue (=1/2) occurs for w=1, which means that there is one peak and one valley (in the opposite direction) of our wave function of one period around the chosen point. The apmplitude of this function describes the "tilt" of the surface at that point. We can choose our coordinate system for analysis in such a way that we stand normal to this tilted plane and can then ignore it for the rest of the analysis.
  • Next we obtain eigenvalues of strenght 1/4 for w=0 as well as for w=2.
  • For w=0 we obtain the constant term, indicating whether the surface is generally bending up (forming a cup) or bending down (forming a hill) around the point of inspection.
  • For w=2 we obtain a cyclic function around our position with two opposite peaks, and with two opposite valleys at right angles to the peaks; this forms a basic saddle configuration, and the amplitude of this function describes the strength of this saddle.
    -- If the saddle function is superposed onto a constant term above, it might change the shape of the cup or hill from being circularly symmetrical into being more ellipsoidal in nature, with maximum and minimum curvature at right angle to one another.
    -- In any case, those two terms will allow us to define the parameters of a best fitting quadric surface (ellipsoid or hyperboloid) that hugs the surface in all direction for a small distance. This "Dupin indicatrix" is the equivalent to the osculating circle for space curves.
• For vertices of valence 4, this all reduces to the behavior of B-splines.
• For vertices with different valences, we would like to get the same kind of pleasing smooth behavior.
• The original formula designed by Catmull and Clark works well for small n, but seems to deterioriate for higher valances.
• Doo and Sabin, propose to do the above type of analysis for all valences of interest and come up with proper weighting functions for each case that would lead to a graceful behavior of the surface. These weighting values can then be stored in a small look-up table.

Next time: An Interpolating Subdivision Scheme

Current Homework Assignment: (to be done individually)

Experimenting with Curve Subdivision Schemes

For more information see the instructional pages.

DUE: WED 10/4/00, 9:10am.

On line: Follow submission instructions on the instructional pages.

Hand in: Pictures of two interesting curves and descriptions of your subdivision schemes.

Next Reading Assignment:

Monday October 9:
We will have an in-class quiz about the material covered so far.