CS 284: CAGD
Lecture #11  Thu 9/27, 2012.
PREVIOUS
<     > CS
284 HOME <     > CURRENT
<     > NEXT
Preparation: Read:
Zorin
et al: "Interpolating Subdivision Meshes with Arbitrary Topology"
Interpolating Subdivision 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.
Other Surface Subdivision Schemes:
There are many other subdivision schemes! Here is one example of a scheme with a slower growth rate:
"Root3 Subdivision" by Leif Kobbelt (SIGGRAPH 2000).
"Interpolatory Root3 Subdivision" by Labsik and Greiner.
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)
For review and for when you really need to see that detailed math:
Assignments for Tue. Oct. 2, 2012:
Review: "Intrinsic Properties of a Surface" by M. E. Mortenson (handout).
Select six papers from Paper List and email me your preferences.
Optionally: Read: "Root3 Subdivision" by Leif Kobbelt (SIGGRAPH 2000).
Even more optionally: Read: "Interpolatory Root3 Subdivision" by Labsik and Greiner.
Assignments for Thu. Oct. 4, 2012:
Read: SS'98: Subdivision Surfaces in Character Animation
Sketch a control mesh for an interesting mug to be used as test object in future subdivision assignments.
Start thinking about possible course projects.
PREVIOUS
<     > CS
284 HOME <     > CURRENT
<     > NEXT
Page Editor: Carlo H. Séquin