CS 284: CAGD 
Lecture #2 -- Th 8/31, 2006.

Preparation:

Lecture Topics

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

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.
• Cⁿ Parametric Continuity:
  -- first n derivatives are continuous; curve is n-th order differentiable, but may have cusps where v=0.
  
• Gⁿ Geometric Continuity:
  -- first n-order 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
• n-th degree Case
•  How much can we do with a curve of a particular degree ?
•  See new homework !

Bernstein Basis Functions

• Formulas  ... in book.
  

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.
• Back-to-front 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:

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. What order Bezier curve is needed to make a (G¹-smooth) loop of turning number 3 ?
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}.

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: