CS 284: CAGD 
Lecture #3 -- Tu 9/05, 2006.

Preparation:

Warm-up Exercise: Draw Bézier Curves ...

Lecture Topics:  Bézier Curves

• A very simple and very useful spline ...
• The defining control points
• Quadratic Case
• Cubic Case
• The General Behavior
• n-th degree Case
•  How much can we do with a curve of a particular degree ? -- See new homework !

Homework Discussion: What can you do with Bézier Curves?

• What can we do with an eighth-order Bezier curve (p52)?
• What order is needed for a G1-smooth loop of turning number 3 ?  --> order 9.
• What order is needed for a figure-8 shape with C2 point-symmetry ?  --> order 6.

Bernstein Basis Functions

• Formula (p35)
• Geometric View (p36)

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
• Number of line intersections with the control polygon <= intersections with spline curve.

Drawing Bézier Curves:  de Casteljau Algorithm

• How NOT to evaluate a Bezier curve
• Finding curve points by interpolation
• Recursive linear interpolation (p46).
• An efficient pipelining scheme, fast and robust!
• Finding Tangent Directions at various locations
• Use derivatives.
  
• Subdivision: Split curve at value t into two segments; find control polygons for both segments.
  
• New control points come directly from de Casteljau algorithm.
  
• Use of subdivision for clipping to a boundary.
• Use of subdivision for curve refinement.

Working with Bézier Curves

• Derivatives
• Hodograph is just another Bezier curve;
• its control points have the coordiantes of the original connecting segments.
• Degree Elevation
• Choosing higher-order basis to obtain more control points:
• either to edit fine features into the curve,
• or to match the degrees of adjacent Bezier patches.
• The general formula for finding the new control points;
• think about redistributing the needed control points uniformaly along a straight line.
• Review of Continuity:
• Individual segment is always C-infinity continuous.
  
• It also has G-infinity continuity, unless v=0 at some point;
• then it has a cusp and is only G0 continuous!

Stitching Bezier Curves Together

• Aiming for more control over complicated curves.
• High-order Bezier curve does not allow much real control.
• Basis functions are to broad; affect all points of the curve.
• Making Interpolating Curves from several Bezier segments.
• Cubic Bezier segments for G1 or C1 continuity.
• Quintic Bezier segments for G2 or C2 continuity.
• Tangent Continuity Conditions at the joints.
• Curvature Continuity Conditions at the joints.

(more next time ...)

Preview of Lagrange Interpolation

• Another way to make a smooth interpolating curve.
• The Goal: To interpolate all data (control) points with ONE function.
• A Set of basis functions that will achieve this (p.61, eqn 4.2)
• The concept of "knots" {here: t-values at control points, i.e., at what "time" do we pass the given points.}
• Example of the Cubic Lagrange Basis with uniform knots.
• Effect of changing the knot values: Squeezing "more" of the curve between some knot pair --> yields bigger bulge.

New Reading Assignment:

New Homework Assignment: G¹-Stitching of Bezier Curves

The goal is to learn how to stitch cubic Bezier segments together to make a smooth, pleasing-looking, interpolating curve that behaves well even for rather ragged control polygons with irregularly spaced control points (like the example we did in class by hand).

Your assignment is to find a robust expression for the placement for the inner control points of each Bezier segment, involving only information from the nearest neighbor points, and which guarantees a G¹-continuous overall curve.

DUE: Sept. 12, 2006, 2:10pm. -- Hand in:

• Window snapshot showing your best solution;
• The formula you used to place the inner control points;
• A one paragraph discussion of your approach, and what you learned from it;
• Any other comments you would like to make.

The code that you should modify and execute, can be found at:
http://www.cs.berkeley.edu/~sequin/CS284/CODE/pa1.slf