CS 284: CAGD 
Lecture #12 -- Wed 10/7, 2009.

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

Self-test: Did you get the important pieces of information from the paper?

How to efficiently and effectively read a paper to prepare for these discussions:

• Read: "Introduction," "Conclusions;" look at Figures and read captions.
  This should typically tell you WHAT the authors have done, and WHY.
• Now try to find out HOW they did it:
  Look for the appropriate sections that describe the key techniques.
• Also try to get an understanding what the limitations and caveats are of the described approach.
  Might there be things that the authors do NOT tell you ?
• You can probably skip the section on "Previous Work" or just skim it.
  
  
• Write down a few of the key points you learned,
  as well as some questions that are good for discussion in class.

Topic: Subdivision (cont.) 

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.
  
  

Differential Geometry of Surfaces

"Intrinsic Properties of a Surface" by M. E. Mortenson  (handout)

• We are concerned with 2-manifolds p(u,w),
  
• thus need 2 paramters u, w, 
• 2 derivatives, dp/du, dp/dw (= velocity along parameter lines)
• First Fundamental Form: dp * dp = E du du + 2F du dw + G dw dw
• with E=p^u p^u,  F=p^u p^w,  G=p^w p^w;
• describes metric properties of surface (stretching and shearing).
• Second Fundamental Form: -dp * dn = L du du + 2M du dw + N dw dw
• with L=p^uu n, M=p^uw n, N=p^ww n, where n is the normal;
• describes curving and twisting of surface, assuming a "good" parametrization.
• Descriptive Trihedron: Darboux Frame
• Normal vector
• Tangent plane
• Principal directions
• Normal curvature (curvature of intersection with normal plane)
• Principal curvatures (max. and min. of normal curvature, k₁ and k₂, orthogonal to each other)
• Gaussian curvature: K=k₁*k₂
• K > 0 ==> spherical curvature (dome or bowl);
• K = 0 ==> flat, no curvature (plane, cylinder, or cone);
• K < 0 ==> hyperbolic curvature (saddle points);
• Mean curvature: H=(k₁+k₂)/2
• H > 0 ==> mostly bowl shaped;
• H = 0 ==> a balanced saddle point; minimal surface;
• H < 0 ==> mostly bowl shaped;
• Osculating paraboloid
• best-fitting quadric surface
• corresponds to osculating circle for a curve.
• Dupin indicatrix
• scaled conics obtained from slicing the osculating paraboloid parallel to the tangent plane.
• Curves on a surface
• Geodesic curvature
• Geodesic lines
• Meusnier's sphere (collection of osculating circles of all curves with same tangents through a point)
  

Differential Geometry of Surfaces -- Selected Formulas by Jordan Smith

Any questions about your genus-4 designs ?

Evaluation / Analysis of Subdivision Schemes (repeat)

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): 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-defined 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

Reading Assignments:

Current Homework Assignment:

Design the Control Mesh for a Smooth Genus-4 Surface.

• MON 10/5:  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 10/12:  Complete assignment due. Hand in a printout of a smooth Loop surface;
  
and submit your SLIDE file electronically.

Paper Presentation Assignments (for the future)