CS 284: CAGD 
Lecture #3 -- Thu 8/30, 2012.

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Rockwood: pp 31-58.

Warm-up Exercises: Bézier Curves

Working with Bézier Curves

Discussion of recent homework: What can you do with a single high-order Bézier Curve?

It is very difficult to design much curve details into the middle of the curve. To make visible wiggles, the control polygon must do extreme back and forth motions.
This is because the Bézier basis functions span the range of the whole curve segment; thus a lot of averaging out is going on.
If you compare this with the B-spline basis functions, you can see that those only have limited support; for the cubic case each hump only spans the range of four knots.
The mutual overlap of these basis functions still guarantees unconditional C2-continuity for cubic basis functions.
On the other hand, this implies that with a multi-segment B-spline curve one typically sees some extra "unused" control points at each end.
The domain over which the B-spline is defined
is somewhat shorter than the extent of the control polygon seem to imply (more later).

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

Options that we will discuss:
-- Stitching together Bézier curve segments:  ==> Requires careful adjustments of control points!
-- Hermite Splines:  ==> Almost like Bézier curves, but more explicit knowledge about derivatives at the ends.

-- Lagrange Interpolation:  ==> One continuous polynomial through all the given points; -- but danger of overshoots!.
All these splines are polynomial based; all of them are invariant under affine transformations,
thus for all of them each coordinate component can be dealt with individually.
All of these spline types have some unconstrained degrees of freedom (DoF) that can be used for optimization.

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

New 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 hard part is to actually get SLIDE to load and to function properly on your computer. SLIDE is very particular about having all the paths set up properly, the variables in the right place, and using exactly the right version of Tcl. But it will be worth the effort, because we can then do many instructive experiements quite quickly. Good luck!

The first such experiment using SLIDE is to learn how to stitch cubic Bézier 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 Bézier segment, involving only information from the nearest neighbor points, and which guarantees a G1-continuous overall curve.

Information on how to install SLIDE on your computer can be found at:

The first file that you should try to run is:
Make sure that you have in the same directory the files:  MATH.tcl,  MOVIE.tcl,  SLFCONSTS.tcl,  SLIDEUI.tcl
Then you may try other files in that same directory:  Instancing.slf,  IcosaGen.slf,  SweepDemol.slf,  GearMovie.slf

DUE: Tue Sept. 4, 2012:   Have SLIDE installed and the provided version of "pa1.slf" running.

For your first programming assignment, the code that you should modify and execute, can be found at:

DUE: Thu Sept. 6, 2012:   Have your modification of "pa1.slf" running. Hand in:

Comments on SLIDE and Tcl

Basic SLIDE concepts:


Hints on SLIDE programming:

New Reading Assignment:

Rockwood: pp 59-73 (Lagrange Interpolation)

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