CS 284: CAGD
Lecture #9  Mon 9/28, 2009.
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Preparation:
Rockwood Chapter 8: pp 133151: Surfaces
Fill in: PAPERSELECTION FORM (handout)
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
tensorproduct 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 Bspline
surfaces:

Bicubic
and biquintic Bspline patches

Rectangular uniform Bspline surfaces
 Typically same degree in both parameter directions  but this is not required.

Comparison, tradeoffs between Bezier and BSpline surfaces
 Bsplines guarantee continuity, but have less direct handles.
 Hardware support (e.g. for rendering) exists primarily for Bezier.
 May do your design as Bsplines, then represent data as Bezier patches (use blossoming to get control points).
(This is what Raph Levien does with his Eulerspiral splines).
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 halfway 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 Bspline
 Calculate a control polygon with twice as many control points.
 Doing this in 2dimensions is the basic idea behind the CatmullClark subdivision scheme for surfaces.
Reading Assignment:
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.
 Email to me your SLFfile and a captured picture (in JPG, GIF, or PNG)
DUE: Wed. 9/30/2006, 10:40am.
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