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ENDS 375 Foundations of Visualization Geometric Representation 9/30/04

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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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2D Representation Points the most basic geometric primitive (x,y) coordinate pairs point "clouds", scatter plots,...

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Lines - line equations slope/intercept form y = ax + b parametric form x = at + b and y = ct + d where t is the parameter (t is usually in the range 0.0 to 1.0)

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Lines between endpoints P a and P b P = P a (1-t) + P b t where 1.0 t 0.0

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Vectors Directed line segments x and y components - (x,y)

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Polylines connected line segments connect points or vertices can approximate curves

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Analytic shapes Circles - center point and radius Rectangles – corner points, or height and width others,...

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Splines parametric forms for x and y single parameter t x = f(t) and y = g(t) order of equations - quadratic, cubic,... f(t) = at 2 + bt + c or f(t) = at 3 + bt 2 + ct + d or f(t) = at 4 + bt 3 + ct 2 + dt +e

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Splines number of control points (per span) 2 for linear, 3 for quadric, 4 for cubic,... locality of control Local vs infinite depends on basis function – bezier, b-splines,… Extent of basis functions Useful splines are usually local

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

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Splines

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continuity issues first order C 1, second order C 2,... no continuity C 0 continuity tangents C 1 continuity

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Splines control points and basis functions –B-splines, Catmull-Rom, Bezier, … interpolating vs approximating interpolating approximating

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Polygons closed area defined by connected set of vertices concave vs convex convex concave

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Pixel Arrays Filled polygons or analytic forms "painted" images 2 1/2 D - layers quad-trees Quadtree Subdivision

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