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: 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

Father of B-splines A draft-man’s spline Isaac J. Schoenberg 1903-1990

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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: Play around with spline functions Use different degrees and continuities