CS 284: CAGD
Lecture #2  Th 8/31, 2006.
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Preparation:
Read: RC pp 3148.
Lecture Topics
Brain Teaser of the Day
Homework Discussion

How to Build a Genus2 Object
Quick Review of Some Important Concepts

Hodograph:
 plot of parameterized derivative vector

Winding Number of a (closed, oriented) Curve around a Point:
 how many times does it loop around that point ?

Turning Number of a (closed) Curve:
 if v<>0 it is equal to winding numper of hodograph around origin.

C^{n} Parametric Continuity:
 first n derivatives are continuous;
curve is nth order differentiable, but may have cusps where v=0.

G^{n} Geometric Continuity:
 first norder geometric approximations
(tangent, curvature, ...) vary smoothly with t (ignoring parametrization).
How to Draw Smooth Curves

Recall the results of "Connect the Dots"

What is a Spline (physical, mathematical) ?

Interpolating spline; goes through the dots.

Approximating spline; is "pulled towards" the dots.
Administrative Intermezzo
 Class Roster, Accounts, etc.  everybody on the wating list will be admitted.

Student Introductions:  share background, interests; (you may have to work in pairs later on).
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 !
Bernstein Basis Functions
Understanding the Properties of Bezier Curves

Endpoint Interpolation

Look at basis functions; check cases for t=0 and t=1.

Tangent Condition

Look at the basis functions; differentiate.

Convex Hull

Partition of unity; only interpolation  not extrapolation.

Linear Precision

Spacial case of convex hull property.

Affine Invariance

Splines are based on linear operators.
 Backtofront Symmetry
 If control polygon is used in reverse order, the same shape results..

Variation Diminishing

# of line intersections with control polygon <= intersections with spline
curve.
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.

What order Bezier curve is needed to make a (G^{1}smooth) loop
of turning number 3 ?

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: TU 9/5/06, 2:10pm.
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 4973.
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