1. 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:
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.

2. Team members’ contributions
Member [name]:

3. 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

23. Derivative Algorithm Suggested content:
Derivative of NURBS surface algorithm
Pseudocode

24. Part II: Design examples

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

26. 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.

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