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ENDS 375 Foundations of Visualization Geometric Representation 10/5/04
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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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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
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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 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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“Blobby’s” “blobbys” use sums of exponential radial functions, for example
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Stochastic surfaces Probabilistic –randomness Fractals –subdivision –self-similar
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