CS 284: CAGD
Lecture #5 -- We 9/13, 2000.

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Preparation:

Rockwood: pp 69-92

Main Lecture Topics

Making smooth, interpolating curves.

Iterated linear interpolation: generalization and graphical interpretation.

Homework Discussion

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,
- - Can we "buy" something else ? e.g. G-2 or C-2 continuity ?
- - Let's Count DOF's
Things to do with the spare DOF's
- - Pick initial velocity
- - Pick initial and final tangent direction
- - Pick initial and final curvature ...
- - - often chosen to be zero ==> "natural" spline.
What if curve is closed (periodic) ?
- - Adds two more constraints.
- - NO DOF's left !
- - Exercise in equation solving. {see chapter 3 in B+B+B "Splines" book}

Higher-Order Hermite Splines

Specifying higher-order derivatives at joints, e.g., curvature
Counting Degrees of Freedom
What to do with these DoF's.

Review: Evaluation of the Lagrange Curve

Iterated Linear Interpolation: Aitken Algorithm

Blossoms

Basic Notation and Machinery
- - Create N-digit labels for system with N iterative steps.
- - Write down interpolation value used in one step as one digit.
General Properties of Blossoms
- - e.g. Symmetry
Bezier Curves in Blossom Form
- - Control point labelling (p82).
Generalization
What can we do with this machinery ?
- - Extension of a quadratic Bezier curve.

New Homework Assignment:

Smooth Sweeps along Complex 3D Curves

For details see notes for Lecture #6.

DUE: WED 9/20/00, 9:10am.
On line:
Put your SLIDE file in the proper place {see instructional page};

Set the initial values for the sliders to the proper values, so that when we execute your program, we will see a nice "prismatic" sweep of a 3-pointed star along your best composite Bezier curve. (You may want to change some of the parameters from the values that you gave them in pa1.) Capture a hardcopy print-out of this sweep.

Hand in: A hardcopy print-out of your sweep; and a page or two of text, answering the questions raised above.

Next Reading Assignment:

Rockwood: pp 93-104 (Uniform B-splines)