1. Constructing Bezier Curves on the Surface of a Sphere By Reza Ali Fundamentals of Spatial Computing UCSB MAT 594CM Spring 2009

2. Presentation Outline Purpose/Goal Spherical Coordinates & Properties SLERP (spherical linear interpolation) OpenGL Bezier Curves Particle Systems Voronoi & Shaders

3. Purpose/Goal Partition the surface of a sphere using the voronoi algorithm Allow the points that define the voronoi to be an interactive magnetic particle system Real Time manipulate of particles

4. Spherical Voronoi

5. Spherical Coordinates Sphere Equations: Cartesian Equivalents:

6. Sphere Properties Equation of a sphere – Cartesian Coordinates: – Spherical Coordinates: 2 values define a sphere – Center & Radius Geodesic: curve that is the shortest distance between two points

7. Sphere Properties Antipodal: points are located directly opposite of each other on a sphere (no geodesic) Great Circle: the intersection of a plane containing the origin and the unit sphere

8. Spherical Linear Interpolation A method of interpolating between two points on a sphere Estimation: Not good enough this will traverse the geodesic at non- constant rate

9. Spherical Linear Interpolation Z=slerp(x,y,α) (constant rate) Watch for the case Ω=180° (antipodal case) Related to Quaternion

10. Bezier Curves Develop a set of parametric cubic equations to represent curves and surfaces using only a small set of control points (4)

11. Bezier Curves & OpenGL OpenGL evaluator functions allow you to use polynomial equations to produce vertices, normals, textures coordinates, and colors Evaluator functions define a Bezier Curve (also the basis for NURBS)

12. Bezier Curves & OpenGL Function: glMap1f() Data Glfloat ctlpts[4][3] glMap1f( target type Lower t range Higher t range Stride Number of points Reference to points) glEnable(GL_MAP1_VERTEX_3) glMapGrid1d(20,0,1) glEvaMesh1(GL_LINE,0,20) t=(0,1/20,2/20,…1) 20 = number of points to evaluate

13. Particle Systems Developing 3D Particle System The particles will distribute themselves along the surface of a sphere Electromagnetic repulsion Voronoi Pattern Creation based of particle system

14. Voronoi & Shaders Create a voronoi curves that will define a sphere and use these curves as points where light escapes like -> Allow user to interactive with system via GUI (GLV) Real Time, maybe?

15. Inspiration

16. References Principles of Computer Graphics (Shalini Govil-Pai) 3D Computer Graphics (Samuel R. Buss) Wikipedia: Spherical CoordinatesSpherical Coordinates Wikipedia: SphereSphere Google Image Search Efficient Reconstruction of Functions on the Sphere from Scattered Data (Keiner, Kunis, Potts) Efficient Reconstruction of Functions on the Sphere from Scattered Data Vimeo: Mass_InsMass_Ins Spherical Centroid Voronoi Tesselation Distributing Points on a Sphere Voronoi Diagram on the sphere Voronoi Diagram of Curves Objects Voronoi diagrams on the sphere (Na, Lee, Cheong)