PPT4: Rational B-spline Curves and Surfaces CAP 6736 Geometric Modeling PPT4: Rational B-spline Curves and Surfaces PPT and video are due no later than February 8 Submit to: lpiegl@gmail.com This template file is just an outline of the presentation that you need to complete. Additional pages may be necessary to fully explore the topic above. Each page should contain adequate text as well as illustrations. You are free to use all publicly available information (text as well as graphics) as long as the sources are properly acknowledged.

Team members’ contributions Member [name]:

Part I: Technical details For this part you will need an equation editor. You may use: MS equation editor, MathType, LaTeX, or Handwritten equations if all else fails

Rational B-spline Curves Suggested content: Rational B-spline curves Definition: basis functions, control point, knots

Rational B-spline Curves Suggested content: Properties of rational B-spline basis functions non-negativity partition of unity differentiability generalization local support

Rational B-spline Curves Suggested content: Properties of rational B-spline curves end point interpolation affine invariance differentiability convex hull local support

Rational B-spline Curves Suggested content: Impact of weights on basis functions Examples

Rational B-spline Curves Suggested content: Impact of weights on basis functions and curves Examples

Rational B-spline Curves Suggested content: Impact of weights on basis functions Zero and small weights

Rational B-spline Curves Suggested content: Geometric significance of weights Examples

Derivatives of NURBS Curves Suggested content: Derivatives of NURBS curves First and higher derivatives End derivative examples

Derivatives of NURBS Curves: Circle Example Suggested content: Derivatives of NURBS curves: the circle First and second derivative examples

Derivatives of NURBS Curves Suggested content: Derivatives of NURBS curves: algorithm First, second and third derivative examples

Derivatives of NURBS Curves with respect to a Knot Suggested content: Derivatives of NURBS curves with respect to a knot First derivative example

Rational B-spline Surfaces Suggested content: Rational B-spline surfaces Knots, control net and basis functions

Rational B-spline Basis Functions Suggested content: Rational basis function properties non-negativity partition of unity differentiability generalization local support maximum value

Rational B-spline Surfaces Suggested content: Rational surface properties end point interpolation affine invariance differentiability convex hull local support

The Effect of one Weight Suggested content: Rational B-spline surfaces: the impact of the weights

Surface Reaction to Weight Changes Suggested content: Geometric meaning of the weights

NURBS Surface Evaluation Example Suggested content: NURBS surface evaluation An example using matrix notation

Derivatives of NURBS Surfaces Suggested content: Derivatives of NURBS surfaces First partial derivatives Generalization to arbitrary derivatives

Derivative Example Suggested content: Derivative example of NURBS surfaces First partial derivatives Second partial derivatives

Derivative Algorithm Suggested content: Derivative of NURBS surface algorithm Pseudocode

Part II: Design examples

Design Examples Suggested content: Add design examples: images and/or videos Give credit to the designers

Part III: GM lab For this part of the assignment you may use an existing system, such as Blender, or write the code and visualize the result using graphics tools like Processing.

Geometric Modeling Lab Suggested project: Design rational B-spline curves using an interactive tool Design common rational B-spline surfaces