# CS 284: CAGD  Lecture #11 -- Thu 9/27, 2012.

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# 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:

"Root-3 Subdivision" by Leif Kobbelt (SIGGRAPH 2000).

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

# 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=pu pu,  F=pu pw,  G=pw pw;
• describes metric properties of surface (stretching and shearing).
• Second Fundamental Form: -dp * dn = L du du + 2M du dw + N dw dw
• with L=puu n, M=puw n, N=pww 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
• 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 e-mail me your preferences.

"Root-3 Subdivision" by Leif Kobbelt (SIGGRAPH 2000).
Even more optionally: Read:  "Interpolatory Root-3 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.

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