CS 284: CAGD 
Lecture #4 -- Th 9/07, 2006.

Preparation:

Warm-up Exercises: Bézier Curves

Main Lecture Topic:

How to make interesting, complex, smooth curves that interpolate given points.

Last lecture: How to use Bézier curve segments, properly stitched together, to make an interpolating spline.
This lecture: Other approaches: Hermite Splines; Lagrange Interpolation.
All types of splines thus far, are polynomial based; all of them are invariant under affine transformations,
thus for all of them each coordinate component can be dealt with individually.
For a given curve interpolation problem, all of these spline types have some unconstrained DoF that can be used for optimization.

Stitching Bézier Curves Together -- what choices do we have ?

• Remember Experiment from Lecture #1: (closed) interpolating curve through N (2D) points.
• Let's do this with N cubic Bézier segments -- one each between two subsequent points.
• Why is this better than using a single high-order Bézier curve ?
  
• Counting Degrees of Freedom (DoF)
• Each control point in 2D has 2 DoF.
• Each constraint imposed removes one DoF.
• How many DoF are remaining ?
• (3N for G¹;  or 2N for G²;  or 2N for C¹).
  
• How to use these DoF's ?
• Optimize the curve !
• Make it as pleasing as possible: --> See Programming Assignment #1.
• Find a good automated heuristic to place the extra control points.
• Global Optimization (approximation)
• Define a cost function:
• E.g., Bending Energy = arc-length integral of square of curvature.
• (Approximate this as best possible with Bezier segments).
• How would you do this ?
• Find out how each DoF affects the total Energy (e.g., by doing finite differences),
• All these components form an overall gradient vector in 3N space,
• Move along gradient vector towards local minumum in energy.
• A specific example: Make a "pleasing" (low bending energy) figure 8 curves from just two cubic Bezier segments.
• How to set up the first guess ?
• How many DoF?
• First looks like 2 ... one for shape and one for scale ...
  
• but there may be a scaling instability for MECs (minimum energy curves)!
  
• How to use them ...
• Evaluate the energy for some values of this angle,
• do a binary search for minimum.
• Advanced approaches to optimization ... later!
• Greiner et al
• Friedel et al

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Comments on SLIDE and Tcl -- Q&A

(Cubic) Hermite Splines

• Another - practical - way of making interpolating curves.
• Review Basis Functions (page 70)
• If all direction vectors are given at all points,
  
-- job is done ... just like connected Bezier segments.
• If we don't care explicitly about the directions at the joints (and leave this open for shape optimization),
  
-- Can we "buy" something else ? e.g. G² or C² continuity ?
-- Let's count DoF; assuming N points in D-dimensional space, cubic curve segments between them:
• each cubic function has 4 parameters for each of  D coordinates: a + bt + ct² + dt³; this yields 4D(N-1)  DoF overall;
• at each of the (N-2) inner joints, we loose some DoF for matching:  D*( t-value, position, first derivative, second derivative, ...)
• for an open-ended curve, 2D  DoF are left over.
• Things to do with the spare DoF:
  
-- Pick initial velocity
-- Pick initial and final tangent direction
-- Pick initial and final curvature ...

• often chosen to be zero  at free ends  ==> "natural" spline.
• What if curve is closed ("periodic") ?
  
-- Adds  2D  more constraints.
-- NO  DoF left !
-- Exercise in equation solving. {see chapter 3 in B+B+B "Splines" book, pp 9-17}

Higher-Order Hermite Splines

• They allow to specify higher-order derivatives at the joints, e.g., acceleration, or curvature;
• Quintic Hermite Splines are quite useful; they allow to specify position, velocity, acceleration(curvature) at each joint.
• To make "good-looking" C²-continuous interpolating curves, use some heuristic to set those values based on "local" information.

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.

Current Homework Assignment: G¹-Stitching of Bézier 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

Reading Assignment: