ENDS 375

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ENDS 375

• Foundations of Visualization
• Geometric Representation II
• 10/6/05

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

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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)

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Space curves

• Analytic line shapes
• Equation based
• Circles, ellipses, . . .
• Splines

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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

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  Splines

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

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Splines

• locality of control
• continuity issues

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Surfaces

• Analytic surfaces
• spheres, tori, ellipsoids
• conic sections - parabolic, hyperbolic, . . .

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Surfaces

• Surfaces of revolution
• Extrusions

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Polygons

• Concave vs convex
• "dual" form using planar equations 
• ax by cz d 0
• intersection of planes - inside vs outside
• polyhedra - convex objects

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Polygonal Surfaces

• Approximate curved surfaces
• Planarity an issue if polygon has more than 3
  vertices

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Polygonal Surfaces

• Polygonal - vertices and topology
• networks - points-polygons
• meshes - regular topology

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Surface "normals"

• vectors perpendicular to the surface

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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)

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Bi-parametric surfaces

• order of the functions
• bi-quadratic, bi-cubic, . . .
• surface continuity issues

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Bi-parametric surfaces

• control points and basis functions
• approximating - B-splines
• interpolating    - Catmull-Rom
• number of control points
• 3x3 for quadratic
• 4x4 for cubic

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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

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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

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Volume Descriptions

• Volumes rather than boundaries
• Voxels
• Boolean Set Operators
• usually on primitive shapes
• union
• intersection
• difference

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Voxels

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

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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

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Blobbys

• blobbys use sums of exponential radial
  functions, for example

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Stochastic surfaces

• Probabilistic
• randomness
• Fractals
• subdivision
• self-similar