CS 284: CAGD
Lecture #4 -- Mo 9/8, 2003.

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Preparation:

Read: Rockwood: pp 59-73 (Lagrange Interpolation)

Main Lecture Topic

How to make interesting, complex, smooth curves that interpolate given points.

Last lecture: How to use Bezier curve segments, properly stitched together, to make an interpolating spline.
This lecture: Another approach: Lagrange Interpolation

Stitching Bezier Curves Together -- what choices do we have ?

Remember Experiment from Class #1: (closed) interpolating curve through N (2D) points.

Let's do this with N cubic Bezier segments -- one each between two subsequent points.

Counting Degrees of Freedom (DoF)

Each control point in 2D has 2 DoF.
Each constraint imposed removes one DoF.
How many DoF are remaining ?
(3N for G¹; or 2N for G²)

How to use these DoF's ?

Optimize the curve !
Make it as pleasing as possible: --> See Programming Assignment #1.
Find a good automated heuristic to place the extra control points.

Global Optimization (approximation)

Define a cost function:
E.g., Bending Energy = arc-length integral of square of curvature.
(Approximate this as best possible with Bezier segments).
How would you do this ?

Find out how each DoF affects the total Energy (e.g., by doing finite differences),
All these components form an overall gradient vector in 3N space,
Move along gradient vector towards local minumum in energy.

A specific example: Make a "pleasing" (low bending energy) figure 8 curves from just two cubic Bezier segments.

How to set up the first guess ?
How many DoF?

First looks like 2 ...
but because of global scaling instability, there is only one -- e.g., the angle athe central crossing.

How to use them ...

Evaluate the energy for some values of this angle,
do a binary search for minimum.

Advanced approaches to optimization ... later!

Not discussed yet, but useful in this context:

Degree Elevation

Choosing higher-order basis to obtain more control points,
e.g., in order to match degrees of adjacent Bezier patches,
or to allow matching adjacent segments with curvature continuity.
The general formula for new control points;
Interpretation: think about redistributing the needed control points uniformaly along a straight line.

Comments on SLIDE and Tcl

Q&A with Jordan Smith.

Current Homework Assignment: G¹-Stitching of Bezier Curves

In this first programming assignment you will be introduced (gently) to SLIDE and to the Tcl language. Your actual programming will be less than ten lines of code (most of the expressions you will need have already been provided), but it encourages experimentation and thinking.

The goal is to learn how to stitch cubic Bezier segments together to make a smooth, pleasing-looking, interpolating curve that behaves well even for rather ragged control polygons with irregularly spaced control points (like the example we did in class by hand).

Your assignment is to find a robust expression for the placement for the inner control points of each Bezier segment, involving only information from the nearest neighbor points, and which guarantees a G¹-continuous overall curve.

DUE: Sept. 10, 2003, 9:10am.
Hand in:

Window snapshot showing your best solution;

The formula you used to place the inner control points;

A one paragraph discussion of your approach, and what you learned from it;

any other comments you would like to make.

On line:

Put your SLIDE file in the proper place {see instructional page};

Set the initial values for the sliders to the preferred value,

so that when we execute your program, your best solution, the one that you handed in, will show up.

The code that you should modify and execute, as well as additional instructions on how to run SLIDE can be found in
http://www-inst.eecs.berkeley.edu/~cs284/FA00/pa1/pa1.htm

Next Reading Assignment:

READ: Rockwood: pp 59-73 (Lagrange Interpolation)
if you have spare time, start reading:
"Fair, G²- and C²-Continuous Circle Splines," by C. H. Séquin, Kiha Lee, and Jane Yen, to be submitted to CAD.
Constructive criticism (specifically about understandability) is most welcome!