CS 284: CAGD
Lecture #2  Mon 8/31, 2009.
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Preparation:
read Rockwood: pp 2048:
You should find access to a Windows machine and download the CAGD_LAB from:
http://www.cs.berkeley.edu/%7Esequin/CS284/CAGD_LAB/CAGD_LAB.rar
Brain Teaser of the Day
Homework Discussion
Quick Review of Some Important Concepts

Hodograph:
 plot of parameterized derivative vector in its own coordinate system.
 if hodograph goes through origin, curve velocity goes to zero; this often leads to a cusp.

Winding Number of a (closed, oriented) Curve around a Point:
 how many times does it loop around that point ?
 a crucial quantity to determine the "inside" of selfintersecting polygons.
 only curves with the same winding number can be smoothly transformed into one another in the plane.

Turning Number of a (closed) Curve:
 counts the (signed) number of turns made by the tangent vector of the curve.
 if v<>0 it is equal to the winding number of hodograph around the origin. 
C^{n} Parametric Continuity:
 first n derivatives are continuous;
curve is nth order differentiable,
 but the curve may still have cusps where v=0.

G^{n} Geometric Continuity:
 is determined entirely by the visual appearance of the shape of the curve (ignoring parametrization).
 first norder geometric approximations
(tangent, curvature, ...) vary smoothly with arclength.
How to Draw Smooth Curves

Recall the results of "Connect the Dots"
 look at students' results.

What is a Spline (physical, mathematical) ?
 a piece of physical material, such as a furring strip or a vinyl hose.

Interpolating spline; goes through the dots.
 mostly used in design tasks.

Approximating spline; is "pulled towards" the dots.
 mostly use in fitting to (noisy) data, e.g., from a scanner.
Administrative Comments
 Class Roster, Accounts, etc.  everybody on the wating list will be admitted.
 Just "auditing" this course is _not_ a great idea!
 to get something out of this course you really need to do the work  and then you may as well get credit for it!
Definition of Cubic Bezier Curve

A Very Simple Spline ...

The Defining Control Points

The General Behavior

Quadratic Case

Cubic Case

nth degree Case

How much can we do with a curve of a particular degree ?

See new homework !
(to be continued)
New Homework Assignment:
Use Rockwood's Interactive Curve Editor CAGD_LAB. Open the applet shown on page 52 of the
book, labelled "Higher Degree Bezier Curves" for the following tasks:

Using a heptic Bezier curve {this is degree 7, order 8; using 8 ctrl pts;
==> different ways of saying the same thing},
model G1 continuous {continuous tangent directions} closed loops of as many
different turning numbers
{the # of times the tangent vector sweep around
360 degrees} as possible  at least for turning numbers 0, 1, 2.

Using the minimum number of control points (=minimum order Bezier), make
a G^{1}continuous "figure8" Bezier curve with overall C2pointsymmetry
{= 2fold rotational symmetry around a point that will bring the figure
back onto itself after a 180degree rotation around this point}.
DUE: WED 9/2/09, 10:40am.
Hand in: window snapshots showing your solutions;
label your figures with their turning numbers;
put your name on your handins;
add explanatory comments as necessary.
On the PCs you can hit Alt+PrnScrn to capture the current active window
to the clipboard.
You can then paste the clipboard into a program such
as "paint" and from there readily send it to the printer.
"SnagIt" is another
great screen/window/regionsaving application that you can download.
Next Reading Assignment:
Rockwood: pp 3158.
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