DEFINING OBJECTS - 3D REPRESENTATIONS

1. DEFINING OBJECTS - 3D REPRESENTATIONS Surface representations Polygon tables 3D curves and curved surfaces Sweep representations Constructive solid geometry Octrees

2. Two classes of representations Surface Solid

3. Surface representations • Polygon surfaces

4. Surface representations Object triangulation

5. Succession of models Wireframe Surface patch

6. Surface patch Surface shading

7. Surface representation Surface Vertex Edge

8. Vertices

9. Wire-frame

10. V1 V4 V1 V4 V3 V3 V2 V2 V2 V2 V5 V5 V8 V8 V6 V7 V6 V7 Surface

11. Vertex table

12. V8 V7 V6 V5 V4 V3 V2 V1 Define vertex table(vertices and edges) Y Z 8 6 4 2 2 X 2 4 6 8

13. Consistency checking • every vertex is listed as an endpoint of at least two lines • every polygon is closed • each polygon has at least one shared edge • if edge table contains pointers to polygons

14. Hierarchical definitions • interaction: groups of primitives can be interactively manipulated together, irrespective of a method of construction. • storage efficiency: definition of repeated symbol needs to be stored only once • common visual properties: groups of primitives can be given the same attributes, for example colour, transparency, visibility etc. • common transformation: groups of primitives can be all transformed together (for example rotated)

16. Advanced representations for curves and curved surfaces (3D) • parametric equations • Bézier curves and surfaces • splines and spline surfaces • B-splines • ß-splines • fractal surfaces

17. Advanced representations for solids • Sweep representations • Constructive Solid Geometry (CSG) • Octrees

18. Sweep representations

19. Sweep representations

20. A B A - B A  B Constructive solid geometry Volume = set of points

21. Octrees 0 1 0 1 2 3 3 2 0 1 2 3 0 1 3 2 0 1 2 3

22. Octrees 5 4 1 0 6 2 3 7

23. V8 V7 V6 V5 V4 V3 V2 V1 Define vertex table(vertices and edges) Y Z V10 V18 V17 8 V15 V16 6 4 V9 V20 2 V19 V14 2 V13 V12 V11 X 2 4 6 8