CS 284: CAGD 
Lecture #2 -- We 8/27, 2003.

Preparation:

Lecture Topics

Homework Discussion

• How to Build Genus-2 Object

Quick Review of Some Important Concepts

• Hodograph: Plot of velocity vector as fct of t
• Winding Number of a closed curve around a point
• Turning Number of a closed curve = winding number of hodograph around origin
• Cⁿ Parametric Continuity: first n derivatives are continuous; curve is n-th order differentiable
• Gⁿ Geometric Continuity: first n-order geometric approximations 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.

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 !

Administrative Intermezzo

• Companion Course CS 294-10 ===> makes a nice complement to CS284 !
• Class Roster, Accounts, etc.
• Student Introductions.

Bernstein Basis Functions

• Formula

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.
• Variation Diminishing
• # of line intersections with control polygon <= intersections with spline curve.
• Need a bending reversal in the control polygon to enable a curvature reversal in the 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} 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 form there readily send it to the printer. "SnagIt" is another great screen/window/region-saving application that you can download.

Next Reading Assignment: