1. PPT3: B-spline Curves and Surfaces
CAP 6736
Geometric Modeling
PPT3: B-spline Curves and Surfaces
PPT and video are due no later than February 1
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. B-spline Curves Suggested content: B-spline curve definition
Knot vector, control points, basis functions

5. B-spline Curves Suggested content: Curve point evaluation algorithm
Example

6. Affine Invariance Suggested content: Affine invariance of curves
Math details and example

7. Strong Convex Hull Property
Suggested content:
Strong convex hull property of curves
Math details and example

8. Local Scheme
Suggested content:
Local effect of control points
Example

9. Piecewise Polynomials
Suggested content:
Bezier vs B-spline
Shape fidelity to control polygon

10. Piecewise Linear Approximation
Suggested content:
Control polygon approximation
Example with degrees 1 through 5

11. Various Continuities Suggested content: Various continuities
Variation is knots and control points

12. Derivatives of a Curve Suggested content: Curve derivative formulas
Pseudocode algorithm

13. Derivatives of a Curve Suggested content: Curve derivative formula
Math details with control point differencing
Examples

14. Derivative Curve: Degree two
Suggested content:
Curve derivative: degree two
Derivative curve

15. Derivative Curve: Degree Three
Suggested content:
Curve derivative: degree three
Derivative curve

16. Higher Derivatives Suggested content: Higher curve derivatives
Examples

17. Derivatives with respect to a Knot
Suggested content:
Derivatives with respect to a knot
Knot vectors, left and right derivatives 17

18. Derivatives with respect to a Knot
Suggested content:
Derivatives with respect to a knot
Examples 18

19. B-spline Surfaces Suggested content: B-spline surfaces defined
Knots, control points, basis functions

20. B-spline Surfaces Suggested content: B-spline surfaces defined
Matrix notation

21. Bi-variate Basis Functions
Suggested content:
Bivariate basis functions
generalization
non-negativity
partition of unity
local support
differentiability
maximum value

22. B-spline Surfaces Suggested content: Surface properties
generalization of Bezier
end point interpolation
affine invariance
strong convex hull property
control net forms an approximation
single control points affects the surface locally
surface is continuously differentiable

23. B-spline Surface Suggested content: B-spline surface examples
Use different degrees

24. B-spline Surface Suggested content: B-spline surface examples
Impact of control point movement
Iso-parametric curves

25. B-spline Surface Suggested content: B-spline surface examples
Surfaces with planned discontinuities

26. Surface Derivatives Suggested content: B-spline surface derivatives
Recursive derivative formula
Pseudocode algorithm

27. Derivative Surface Suggested content: Derivative surface
Recursive formula with control point differencing
Examples for lower degree partial derivatives

28. Part II: Design examples

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

30. 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.

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