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

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

3. 2D Representation Points the most basic geometric primitive (x,y) coordinate pairs point "clouds", scatter plots,...

4. 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)

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

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

7. Polylines connected line segments connect points or vertices can approximate curves

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

9. 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

10. 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

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

12. Splines

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

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

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

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