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

Preparation:

Brain Teaser of the Day

• Can a cubic Bezier curve have an inflection point even when its control polygon does NOT have one ?
  

Homework Discussion

• How to Build a Genus-2 Object
  -- Example by Yungil Lee
  

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.
  

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
• n-th degree Case
•  How much can we do with a curve of a particular degree ?
•  See new homework !

New Homework Assignment:

1. Using a heptic Bezier curve {this is degree 7, order 8; using 8 ctrl pts; ==> different ways of saying the same thing},
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.
2. 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}.

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/region-saving application that you can download.

Next Reading Assignment: