# CS 284: CAGD  Lecture #9 -- Mon 9/28, 2009.

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

### Preparation:

Rockwood Chapter 8: pp 133-151: Surfaces

Fill in: PAPER-SELECTION FORM (handout)

## Warm-up Problem: How to model a bell

### Possible approaches to composing a Bell shape

A good starter SLIDE file: BellShape.slf

# Topic: Surface Patches

### From Curves to Surfaces Patches

• Do in "u" and in "v" directions what we have learned in "t" direction ...
• Bilinear Bezier patch = Coons Patch
• Cubic tensor-product Bezier patch
• Symmetry in u,v: interchange roles of "rails" and "curves"
• Biquintic Bezier patch
• DeCasteljau evaluation of tensor product patches (p138)
• use DeCasteljau on "control rails" then on "ribs".
• Patch subdivision (p140), degree elevation ... All still work as in the 1D case!
• subdivide one direction of control polygons to get new mid patch boundary, ...
• use convex hull of subdivided control polygons for intersection tests, clipping, ...

• Putting Bezier patches together with G1 or better continuity is difficult and tedious.
• If you want a high degree of continuity, consider the approximating B-spline surfaces:
• Bicubic and biquintic B-spline patches
• Rectangular uniform B-spline surfaces
• Typically same degree in both parameter directions -- but this is not required.
• Comparison, trade-offs between Bezier and B-Spline surfaces
• B-splines guarantee continuity, but have less direct handles.
• Hardware support (e.g. for rendering) exists primarily for Bezier.
• May do your design as B-splines, then represent data as Bezier patches (use blossoming to get control points).
(This is what Raph Levien does with his Euler-spiral splines).

### Triangular Surfaces Patches

We can also deal with triangular patches, but need a different interpolation scheme:
Barycentric coordinates: three numbers, but with the constraint that they must sum to 1.0.
DeCasteljau evaluation technique can also be applied to triangular patches.

# Topic: Subdivision

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

• Conceptual introduction via iterated refinement
• Cutting a rounded shape from paper by "repeated corner cutting." (DEMO)
• Carving a rounded object from wood or styrofoam by cutting away edges. (DEMO)
• Smoothing by (weighted) averaging vertices among their neighbors.
• Calculating intermediate data points half-way between the given interpolation points (not the whole curve segment).
• Adding extra "bulges" on the segments of a linear spline (as in a fractal construction, e.g. Koch snowflake curve).

• Special case study: Cubic interpolation:
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.
•
• 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.

• Subdivision of cubic B-spline
• Calculate a control polygon with twice as many control points.
• Doing this in 2-dimensions is the basic idea behind the Catmull-Clark subdivision scheme for surfaces.

Read the seminal paper by Catmull and Clark on subdivision surfaces

## Current Homework Assignments:

### Construct a Parameterized Goblet

A goblet is:
1.  A drinking vessel, such as a glass, that has a stem and base.
2. Archaic:  A drinking bowl without handles.
Your task is to design a curved thin (metal) surface of finite thickness that could serve as a goblet.
There are many different ways in which such a surface could be defined: Some possible approaches to composing a Bell shape also apply to forming a goblet.
For this assignment you should concentrate on using a sweep in some form and using some spline for either the cross secition or for the guide curve, or for both.
There should be from 3 to 5 parameters with wich you can change the shape of the goblet interactively without loosing continuity in the overall surface.
Let yourself be inspired by this collection of images that you get when you do a Google image search for "goblet", -- but keep your shape simple!
You may start from this SLIDE file: BellShape.slf

Create a SLIDE file with all parameters set to their preferred values.
Capture your design pictorially using the screen saver.

• E-mail to me your SLF-file and a captured picture (in JPG, GIF, or PNG)

DUE:  Wed. 9/30/2006, 10:40am.

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