CS 284: CAGD
Lecture #12  Wed 10/7, 2009.
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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 2manifolds 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, k_{1}
and k_{2}, orthogonal to each other)

Gaussian curvature: K=k_{1}*k_{2}
 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=(k_{1}+k_{2})/2
 H > 0 ==> mostly bowl shaped;
 H = 0 ==> a balanced saddle point; minimal surface;
 H < 0 ==> mostly bowl shaped;

Osculating paraboloid
 bestfitting 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)
Any questions about your genus4 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 ...)

C^{n} continuity can readily be inferred form behavior of the polynomial
or trigonometric functions.

G^{n }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 Bspline 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 ngon;
this ngon is used to define a reference plane (= tangent plane).

Second eigenvalue for w=0 is 0.25; describes
hill/bowllike 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(ij)/n))/4n

Loop thesis, triangular spline N^{222}, (chapter 4):

New extraordinary vertex V^{k+1} = a_{n}V^{k}
+ (1a_{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_{n}V^{0}
+ (1b_{n})Q^{0}, 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 V^{0}.

Tangent plane is defined by ring of neighbors only !

Curvature Continuity 

Rather than explicitly develop the periodic normalcurvature 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 welldefined curvature function around an extraordinary point
! :(

I.e., welldefined Gaussian curvature does not exist at extraordinary points
!

A reasonable choice that gives goodlooking surfaces: a_{n}
= (3/8 + 0.25 cos(2p/N))^{2} + 3/8
Reading Assignments:
Leif Kobbelt: "Root3 Subdivision", Siggraph 2000.
Current Homework Assignment:
Design the Control Mesh for a Smooth Genus4 Surface.
The goal is to design a highlysymmetrical control mesh for a closed genus4
subdivision surface
based on triangles
(which could be later used for experiments in surfaceenergy minimization
studies).
Following an iterative
design process, we will do this in stages:
 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)
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