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 warmup).
 readjusting the midpoints of all control segments (see warmup).
 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 noncollinear 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 midpoint evaluation) based on de Casteljau (handout WW pp 1518)

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 Bspline 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 uvcoordinates 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 Bspline scheme

Can handle vertices with valences different from 4

Can handle meshes with facets other than quadrilaterals

One subdivision iteration calculates:
 new vertices at the centers of all faces,
 new vertices "below" the centers of all edges,
 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
valence4 vertices,

In these ever more dominating "regular" regions, the surface will approach
a Bspline.
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.
Reading Assignments:
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 reoriented 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 "onionlike"
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 email 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