# CS 284: CAGD  Lecture #13 -- Tu 10/10, 2006.

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## Preparation:

Zorin et al: "Interpolating Subdivision Meshes with Arbitrary Topology"

Chapters 4 and 5 from: C. Loop, "Smooth Subdivision Surfaces Based on Triangles"

# Topic: Subdivision (cont.)

## Interpolating Subdivision Schemes for Surfaces

### Discussion of the Zorin paper:

• What are the key ideas: (Section 1, six bullets in right column)
• Triangle stencil (Figure 3).
• The construction of new edge midpoints: (Section 3.2)
• what are the actual regions of influence ?
• why does it make sense to ignore the neighbors to the right of S1 and S6 in Figure 3b ?
• The modified subdivision scheme: (Section 3.3: four cases)
• Improvements over Butterfly scheme: (Fig. 4a)
• How can such interpolated regions be stitched together ?
• Treatment of boundary edges.

### Discussion of new Homework Assignnment: Devising a Subdivision Scheme

How does your scheme deal with the above issues ?

## Evaluation / Analysis of Subdivision Schemes

How do we know whether a particular interpolation or subdivision scheme is any good ?

### Testing / Evaluation 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.
• Largest 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): wij = (3+2cos(2p(i-j)/n))/4n
• Loop thesis, triangular spline N222, (chapter 4):
• New extraordinary vertex Vk+1 = anVk + (1-an)Qk, where Qk is the centroid of the surrounding vertices Pki
• Pick an for best performance; convergence occurs for -5/8 <  an < 11/8.
• Convergence proof in two steps: Show:  Vk  --> Qk,  and also for each i:  Pki --> Qk
• The explicit point of convergence is:  Qk = bnV0 + (1-bn)Q0, where bn = 3 / (11 - 8an ).
• Tangent Plane Continuity -- gives narrower bounds on an : -0.25 cos 2p/N < an  <  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 an 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:  an = (3/8 + 0.25 cos(2p/N))2 + 3/8

Labsik and Greiner: "Interpolatory Root-3 Subdivision".

## Current Homework Assignment: Devise an Interpolating Subdivision Scheme for Quad-Meshes.

Continue with the Warm-up Exercise in the same five groups.
Exchange phone numbers, e-mails, available time slots ...  NOW.
Meet as a group at least twice between now and  Tuesday to discuss this task.
As a team, e-mail me an interim report (less than 1 page) describing your scheme and its preliminary evaluation.

DUE:  TUESDAY 10/10/2006, Noon!:

2. ## Compare several different subdivision schemes in SLIDE.

DUE:  TUESDAY 10/17/2006, 2:10pm.

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