# CS 284: CAGD  Lecture #10 -- Th 9/28, 2006.

PREVIOUS < - - - - > CS 284 HOME < - - - - > CURRENT < - - - - > NEXT

## Preparation:

Read the seminal paper by Catmull and Clark on subdivision surfaces

# Topics: Subdivision

### Introduction to "General Subdivision (with modification)" Techniques

• Conceptual introduction via iterated refinement
• cutting a rounded shape from paper by "repeated corner cutting."
• carving a rounded object from wood by cutting away edges.
• adding extra "bulges" on the segments of a linear spline (see warm-up).
• re-adjusting the mid-points of all control segments (see warm-up).
• averaging vertices among their neighbors.

• Key points about useful subdivision schemes:
• There are two components to any subdivision scheme, a topological and a geometrical one:
• topology: in a fixed way split the parametric domain of an edge or a face;
• geometry: move some of the old and newly created vertices to new locations (that promise to yield a smoother shape).
• The number of points (line segments) must grow at a geometrical rate with each generation.
• The newly introduced points should have a smoothing effect and converge towards a limit function.
• This can typically be achieved with affine mapping schemes described with a subdivision matrix.
• The infinite application of this matrix then leads directly to a point on the curve or surface.

• Special case study:
Interpolating four points with a cubic polynomial to find a new mid point for subdivision
is the same as averaging two quadratic interpolants through three points each.
• The subdivision approach failed with the circle splines, because there is no affine invariance.

### Conceptual Bridges to Splines, Fractals, and Affine Transformations

• Iterated affine transformations (Warren+Weimer, p 9)
• Defined by the transformation of three non-collinear points to three other ones;
• everything is then referenced to these three points with barycentric coordinates.
• Fractals: a set of contractive transformations (distance between any two points shrinks). Examples:
• Sierpinski triangle
• Koch snowflake
• Bezier curve (iterated mid-point evaluation) based on de Casteljau  (handout WW pp 15-18)
• Subdivision as the limit of an increasingly faceted sequence of polygons (WW p 19)
• each is related to successor by a simple linear transformation.
• Subdivision as a multiresolution rendering algorithm that generates increasingly dense plots
• by taking convex combinations of points used in previous plots (WW p 24).
• Importance of the subdivision matrix (WW p 25) -- becomes clearer later.

### Topological Limitations of the B-spline Control Mesh

• A rectilinear mesh of quadrilaterals can be nicely mapped onto a torus.
• It cannot be mapped onto a sphere without either
• crunching the u-v-coordinates together at the N- and S- poles, or
• or introducing vertices with valences different from 4.
• It cannot be mapped nicely onto surfaces of genus higher than 1.
• For these kinds of surfaces, we need a different, more general scheme ==> Subdivision surfaces!
• Remarks on the Topology of Surfaces

### The Classical Subdivision Surfaces by Catmull & Clark

= foundation for all modern subdivision algorithms.
• A generalization of the B-spline scheme
• Can handle vertices with valences different from 4
• Can handle meshes with facets other than quadrilaterals
• One subdivision iteration calculates:
1. new vertices at the centers of all faces,
2. new vertices "below" the centers of all edges,
3. a new vertex position for each old vertex.
(This could be described as a matrix transformation on all the old vertices.)
• After the first iteration step, there will only be quadrilateral mesh facets;
• and from then on, there will be a constant number of extraordinary points:
• vertices of valence <>4 for each such original vertex.
• vertices of valence <>4 for each non quadrilateral mesh in the original control polyhedron.
• After the second iteration, any facet contains at most one extraordinary vertex.
• The extraordinary points will not disappear, but they will become more and more "isolated",
• being surrounded by ever shrinking irregular regions,
• and being separated by more and more quadrilateral meshes joining in valence-4 vertices,
• In these ever more dominating "regular" regions, the surface will approach a B-spline.

### Another Subdivision Surface

• A more detailed analysis what happens at irregular points is in the paper by Doo and Sabin.
• this paper also introduces matrices into the subdivision process,
• and the analysis of the eigenstructure to understand the behavior of the limit surface.

Paper by Doo and Sabin on subdivision surfaces

## New Homework Assignment: Create a Subdivision Surface

Create a doubly or triply spiralling surface.
Similar to the Creative Thinking Exercise on Koch's Snowflake Curve, you should try to find a surface in 3D space that is inspired by a logarithmic spiral in the plane.
However, you must not just extrude a logarithmic spiral in a direction perpendicular to the spiral plane. You should create a surface that shows some spiralling cut lines when cut in as many different orientaions as possible. The surface will probably have to have some (spiralling?) edges -- which is good, because this will define some windows through which one can look inwards to the inner parts of the surface.

Keep your surface modular. Model as little as absolutely needed; then put multiple copies suitably re-oriented together to make the complete surface.
In addition to exploiting symmetry at one (spherical) level of the surface, you should then extend the surface inward or outward by simply making suitably scaled copies of one layer of that "onion-like" assembly. The use of one or two parameters to optimize the look of the surface is encouraged.

I have put some SLIDE starting file "spiral.slf" into the CODE directory. It has most of the basic elements that you will need to build such a surface and shows how to do mirroring, scaled instanciation, and subdivision. I also have included some token parameters so you can get started with something that already works and then do incremental modifications.

PHASE I --DUE: Tu. 10/3/2006, 2:10pm:

Plan the topology (connectivity and rims)
of your surface to get the desired spiral patterns locked in. Give me something by Tuesday at the latest that allows me to give you feedback whether you are on a good track. You can either give me a paper and pencil sketch, or a rudimentary slide file that shows the basic geometry, even though not all pieces fit together seamlessly yet. Feel free to e-mail me images or SLF files before Tuesday.

PHASE II --DUE: Th. 10/5/2006, 2:10pm:

The final  surface: SLF file and captured image.

NICE JOB ON THE LAST ASSIGNMENT !!   See my Bell Selection.

PREVIOUS < - - - - > CS 284 HOME < - - - - > CURRENT < - - - - > NEXT
Page Editor: Carlo H. Séquin