# CS 284: CAGD  Lecture #2 -- Mon 8/31, 2009.

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## Preparation:

http://www.cs.berkeley.edu/%7Esequin/CS284/CAGD_LAB/CAGD_LAB.rar

### 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 self-intersecting 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.
• Cn Parametric Continuity:
-- first n derivatives are continuous; curve is n-th order differentiable,
-- but the curve may still have cusps where v=0.
• Gn Geometric Continuity:
-- is determined entirely by the visual appearance of the shape of the curve (ignoring parametrization).
-- first n-order 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.

• 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
• Cubic Case
• n-th 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:
1. Using a heptic Bezier curve {this is degree 7, order 8; using 8 ctrl pts; ==> different ways of saying the same thing},
2. model G-1 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.
3. Using the minimum number of control points (=minimum order Bezier), make a G1-continuous "figure-8" Bezier curve with overall C2-point-symmetry
{= 2-fold rotational symmetry around a point that will bring the figure back onto itself after a 180-degree 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;