1. PPT2: B-spline Basics Functions
CAP 6736
Geometric Modeling
PPT2: B-spline Basics Functions
PPT and video are due no later than January 25
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. Father of B-splines
A draft-man’s spline
Isaac J. Schoenberg

5. Piecewise Polynomials
Suggested content:
Shortcomings of polynomials
Use piece-wise polynomials

6. Piecewise Polynomials in Bezier Form
Suggested content:
Piecewise Bezier polynomial functions
Local support

7. B-splines Defined Suggested content:
Recursive definition of B-spline functions
Knots, degree zero and arbitrary degree

8. B-splines Defined Suggested content: Triangular scheme for a knot span
List spline from degree zero to degree 3

9. Example: Degree Zero Suggested content: Degree zero B-splines
Define explicitly all from index 2 to 7

10. Example: Degree One Suggested content: Degree one B-splines
Define explicitly all from index 1 to 7

11. Example: Degree Two Suggested content: Degree two B-splines
Define explicitly all from index 0 to 7

12. N3,2 as Piecewise Polynomial
Suggested content:
N3,2 as piecewise polynomial
Define explicitly on each span

13. Properties: Local Support
Suggested content:
Illustration of local support
Use N1,3 as an example

14. Properties: Local Impact
Suggested content:
Illustration of local impact
Use N3,0 as an example

15. Properties Suggested content: Properties of B-splines non-negativity
partition of unity
differentiability
maximum value

16. Derivative Formulas Suggested content: Derivatives of B-splines
First and higher derivatives

17. Derivative Formulas Suggested content:
Derivatives of B-splines in terms of original B-splines
Recursive definition

18. B-spline Derivatives Suggested content:
Derivatives of B-splines with multiple knots
Degree three example

19. B-spline Derivatives Suggested content: Derivatives of B-splines
All derivatives of a degree three spline

20. Derivatives with respect to a Knot
Suggested content:
Derivatives with respect to a knot
Knot vector notation

21. Derivatives with respect to a Knot
Suggested content:
Derivatives with respect to a knot
Left and right derivatives
Steps to evaluate derivatives

22. Derivatives with respect to a Knot
Suggested content:
Derivatives with respect to a knot
Example of left and right derivatives

23. Computational Algorithms
Suggested content:
Evaluate one basis function at a given parameter value
Evaluate all non-vanishing basis functions at a given parameter value
Evaluate all basis functions and their derivatives at a given parameter value
Evaluate all the derivatives of a single basis function at a given parameter value
Compute the derivative with respect to a knot

24. Computational Algorithms
Suggested content:
Evaluate all basis functions
Write out all non-zero elements
Pseudocode algorithm

25. Computational Algorithms: all Derivatives
Suggested content:
Evaluate all derivatives
Write out all non-zero elements for degree three

26. Part II: Design examples

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

28. 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.

29. Geometric Modeling Lab
Suggested project:
Play around with spline functions
Use different degrees and continuities