CS 284: CAGD 
Lecture #14 -- Mo 10/13, 2003.

Preparation:

Collect  Paper Selection Forms

Discussion of Homework: Experimenting with Curve Subdivision Schemes

Discussion of Homework: Designing a Genus-4 Smooth Surface

• It should be a quadrilateral mesh,
• It should be relatively simple,
• It should be highly symmetrical,
• It should have few odd-valence vertices,
• and they should not deviate too much from valence 4;
• also, they should be symmetrically located on the surface.
• It should not be too tedious to derive the vertex coordinates for a critical subdomain of your object,
• which can then be replicated to make the full symmetrical object.
• There should be a convenient way to introduce a few good shape parameters into your design.
• This should lead to a smooth, pleasing surface with no unnecessary wrinkles after a couple of steps of subdivision.

How would these evolve under minimum-energy optimization ?

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Topic: The Subdivision Process (cont.)

Testing of Blending / Subdivision Schemes by Visual Inspection

• Subject your scheme to many tough test cases,
• ideally move control points interactively and continuously,
• because "transition cases" (e.g., extra inflection points) often show the weaknesses of a scheme.

Formal Analysis of Blending / Subdivision Methods

• If curve is formed with analytical functions (e.g., for Bezier, Lagrange, Circle Splines ...)
• Cⁿ continuity can readily be inferred form behavior of the polynomial or trigonometric functions.
• Gⁿ continuity needs a separate analysis; perhpas a bound on curvature can be established;
• or it may be sufficient to show that the velocity cannot get to zero (Circle Spline paper, section 3.4)

 
• Subdivision curves/surfaces are harder to analyze:
• How do you prove that final curve points do not have small fractal oscillations ?
• or that the tangents converge to a well defined value at every point ?

 
• Doo & Sabin, extraorinary points in quadratic B-spline surfaces:
• Do not analyze the behavior of individual points, but of the whole ring of vertices around an extraordinary point.
• Do a discrete Fourier analysis of this ring of vertices; needs frequencies from w=0 to n/2 (n=valence) to capture all DoF.
• Repeated application of the subdivision matrix converges to a vector corresponding to largest eigenvector of the matrix.
• For the regular (valence 4) vertex we observe this behavior:
• Largest eigenvalue for w=0 is 1.0;

this guarantees translation invariance of the process (i.e., does not "run away").
• Largest eigenvalue for w=1 is 0.5;

this implies that the region around this vertex shrinks towards an affinely distorted regular n-gon;
this n-gon is used to define a reference plane (= tangent plane).
• Second eigenvalue for w=0 is 0.25; describes hill/bowl-like behavior at this point.
• Second eigenvalue for w=2 is 0.25; describes the amount of warping (into a saddle) at this point.
• Doo&Sabin found subdivision coefficients for the extraordinary cases that also give these eigenvalues,

and thus also guarantee tangent plane continuity at these points (page 360): w_ij = (3+2cos(2p(i-j)/n))/4n
• Loop thesis, triangular spline N²²², (chapter 4):
• New extraordinary vertex V^k+1 = a_nV^k + (1-a_n)Q^k, where Q^k is the centroid of the surrounding vertices P^k_i
• Pick a_n for best performance; convergence occurs for -5/8 <  a_n < 11/8.
• Convergence proof in two steps: Show:  V^k  --> Q^k,  and also for each i:  P^k_i --> Q^k
• The explicit point of convergence is:  Q^k = b_nV⁰ + (1-b_n)Q⁰, where b_n = 3 / (11 - 8a_n ).
• Tangent Plane Continuity -- gives narrower bounds on a_n : -0.25 cos 2p/N < a_n  <  0.75 + 0.25 cos 2p/N.
• Again, use discrete Fourier transform to capture the behavior of all edges converging in V⁰.
• Tangent plane is defined by ring of neighbors only !
• Curvature Continuity --
• Rather than explicitly develop the periodic normal-curvature function around an extraordinary vertex,

study the rate of change of the tangent function with respect to the subdivision process !
• Analysis shows: No choice of a_n can assure a well-defined curvature function around an extraordinary point !   :-(
• I.e., well-define Gaussian curvature does not exist at extraordinary points !
• A reasonable choice that gives good-looking surfaces:  a_n = (3/8 + 0.25 cos(2p/N))² + 3/8

 
• Warren & Weimer, Chapter 3.2
• Study the differences between sets of coefficients in one generation and the next one:

should go to zero for convergence !
• Smoothness test: look at second differences and see whether they also disappear.

(They do for the cubic interpolatory scheme -- Ch.3.2.4)

Next Homework Assignment: (to be done individually)

Design the Control Mesh for a Genus-4 (minimum energy) Surface  -- Part 3.

• WED 10/8: Hand in a sketch of the rough geometry of the object that you plan to construct,

and a paragraph that outlines your plan for constructing the actual control mesh.
• MON 10/13: Hand in a printout of a simple symmetrical mesh of quadrilaterals.

List the control parameters that you have for your mesh.
• WED 10/15: Complete assignment due. Hand in a printout of a smooth Catmull-Clark surface;

and submit your SLIDE file electronically, with parameters initialized to give the best looking results.

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