1. ENDS 375 Foundations of Visualization Geometric Representation 10/5/04

2. Geometric representation is the fundamental basis for describing or modeling the data, objects and scenes to be visualized.

3. 3D Representation Points - x, y and z coordinates Lines - same a 2D but with three components Vectors - directed line segments x,y and z components (x,y,z)

4. Space curves Analytic line shapes –Equation based –Circles, ellipses,... Splines

5. parametric forms for x, y, and z x = f(t), y = g(t) and z = h(t) order of equations - quadratic, cubic,... f(t) = at 2 + bt + c or f(t) = at 3 + bt 2 + ct + d control points and basis functions interpolating vs approximating

6. Splines number of control points - 2 for linear, 3 for quadric, 4 for cubic,...

7. Splines locality of control continuity issues

8. Surfaces Analytic surfaces –spheres, tori, ellipsoids –conic sections - parabolic, hyperbolic,...

9. Surfaces Surfaces of revolution Extrusions

10. Polygons Concave vs convex "dual" form using planar equations ax + by + cz +d = 0 intersection of planes - inside vs outside polyhedra - convex objects

11. Polygonal Surfaces Approximate curved surfaces Planarity an issue if polygon has more than 3 vertices

12. Polygonal Surfaces Polygonal - vertices and topology networks - points-polygons meshes - regular topology

13. Surface "normals" vectors perpendicular to the surface

14. Bi-parametric surfaces x, y, and z functions of two parameters U and V x = f(U,V), y = g(U,V) and z = h(U,V)

15. Bi-parametric surfaces order of the functions bi-quadratic, bi-cubic,... surface continuity issues

16. Bi-parametric surfaces control points and basis functions approximating - B-splines interpolating - Catmull-Rom number of control points 3x3 for quadratic 4x4 for cubic

17. Bi-parametric surfaces Bezier patches - "Coons" patches hermite polynomial basis points and tangents NURB surfaces - "non-uniform rational b-splines" as opposed to uniform non- rational b-splines

18. Subdivision Surfaces Start with a polygon mesh Subdivide the mesh into a finer mesh Creates smaller and smaller polygons This process converges to the same kind of surfaces as created by spline surfaces

19. Volume Descriptions Volumes rather than boundaries Voxels Boolean Set Operators –usually on primitive shapes union intersection difference

20. Voxels Voxel oct-trees Density functions –CAT scans, MRI data,... –find isosurfaces –marching cubes algorithm

21. Implicit functions Thresholded analytic functions - "blobby" objects are common example density = f(x,y,z), find isosurface where f(x,y,z) = (some value) use concatenation of simple functions to define overall density function

22. “Blobby’s” “blobbys” use sums of exponential radial functions, for example

23. Stochastic surfaces Probabilistic –randomness Fractals –subdivision –self-similar