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ENDS 375

ENDS 375. Foundations of Visualization Geometric Representation 10/5/04. Geometric representation is the fundamental basis for describing or modeling the data, objects and scenes to be visualized. 3D Representation. Points - x, y and z coordinates

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ENDS 375

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  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. Splines • parametric forms for x, y, and z   x = f(t), y = g(t) and  z = h(t) • order of equations - quadratic, cubic, ...     f(t) = at2 + bt + c  or f(t) = at3 + bt2 + 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 • 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

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