1. Vertices and Fragments I CS4395: Computer Graphics 1 Mohan Sridharan Based on slides created by Edward Angel

2. Objectives Chapter 7. Implementation of rendering pipeline. – Clipping: viewing volume. – Rasterization: fragments from valid objects. – Hidden-surface removal: occlusions, blocked surfaces. This lecture: – Clipping. – Scan conversion. CS4395: Computer Graphics 2

3. Overview At end of the geometric pipeline, vertices have been assembled into primitives. Must clip out primitives that are outside the view frustum: – Algorithms represent primitives by lists of vertices. Must compute pixels that can be affected by each primitive: – Fragment generation. – Rasterization or scan conversion. CS4395: Computer Graphics 3

4. Required Tasks Clipping. Rasterization or scan conversion. Transformations. Some tasks deferred until fragment processing: – Hidden surface removal. – Anti-aliasing. CS4395: Computer Graphics 4

5. Rasterization Meta Algorithms Consider two approaches to render a scene with opaque objects. For every pixel, determine object that projects on the pixel and is closest to the viewer. Compute the shade of this pixel. – Ray tracing paradigm. For every object, determine pixels it covers and shade them: – Pipeline approach. – Must keep track of depths. CS4395: Computer Graphics 5

6. Clipping 2D against clipping window. 3D against clipping volume. Easy for line segments and polygons. Hard for curves and text: – Convert to lines and polygons first CS4395: Computer Graphics 6

7. Clipping 2D Line Segments Brute force approach: compute intersections with all sides of clipping window. – Inefficient: one division per intersection  CS4395: Computer Graphics 7

8. Cohen-Sutherland Algorithm Idea: eliminate as many cases as possible without computing intersections. Start with four lines that determine the sides of the clipping window. CS4395: Computer Graphics 8 x = x max x = x min y = y max y = y min

9. The Cases Case 1: endpoints of line segment inside all four lines. – Draw (accept) the line segment. Case 2: endpoints outside all lines and on same side of a line. – Discard (reject) the line segment. CS4395: Computer Graphics 9 x = x max x = x min y = y max y = y min

10. The Cases Case 3: one endpoint inside, one outside. – Must do at least one intersection. Case 4: both outside. – May have part inside. – Must do at least one intersection. CS4395: Computer Graphics 10 x = x max x = x min y = y max

11. Defining Outcodes For each endpoint, define an outcode: Outcodes divide space into 9 regions. Computation of outcode requires at most 8 subtractions per line segment. CS4395: Computer Graphics 11 b0b1b2b3b0b1b2b3 b 0 = 1 if y > y max, 0 otherwise b 1 = 1 if y < y min, 0 otherwise b 2 = 1 if x > x max, 0 otherwise b 3 = 1 if x < x min, 0 otherwise

12. Using Outcodes Consider the 5 cases below. AB: outcode(A) = outcode(B) = 0000. – Accept line segment. CS4395: Computer Graphics 12

13. Using Outcodes CD: outcode (C) = 0000, outcode(D)  0 (0010). – Compute intersection. – Location of 1 in outcode(D) determines which edge to intersect with. – Note: if there were a segment from A to a point in a region with 2 ones in outcode, we might have to do two intersections. CS4395: Computer Graphics 13

14. Using Outcodes EF: outcode(E) logically ANDed with outcode(F) (bitwise)  0. – Both outcodes have a 1 bit in the same place (0010, 0010). – Line segment is outside of corresponding side of clipping window. – Reject. CS4395: Computer Graphics 14

15. Using Outcodes GH, IJ have similar outcodes: (1001, 1000) and (0001, 1000). – Reject GH but what about IJ (logical AND returns zero) ? Shorten line segment by intersecting with sides of window. Compute outcode of intersection (new endpoint of shortened line segment). Re-execute algorithm. CS4395: Computer Graphics 15

16. Efficiency In many applications, the clipping window is small relative to the size of the entire data base: – Most line segments are outside one or more side of the window and can be eliminated based on their outcodes. Inefficiency when code has to be re-executed for line segments that must be shortened in more than one step. CS4395: Computer Graphics 16

17. Cohen Sutherland in 3D Use 6-bit outcodes. When needed, clip line segment against planes. CS4395: Computer Graphics 17

18. Liang-Barsky Clipping Consider the parametric form of a line segment: As parameter  changes from 0 to 1, move from p 1 to p 2. We can distinguish between the cases by looking at the values of  where the extended line segment crosses the lines that determine the window. CS4395: Computer Graphics 18 p1p1 p2p2

19. Liang-Barsky Clipping In (a):  4 >  3 >  2 >  1 – Intersect right, top, left, bottom: shorten. In (b):  4 >  2 >  3 >  1 – Intersect right, left, top, bottom: reject. CS4395: Computer Graphics 19

20. Advantages Can accept or reject as easily as with Cohen- Sutherland. Using values of , we do not have to use algorithm recursively as with C-S. Extends to 3D. CS4395: Computer Graphics 20

21. Clipping and Normalization General clipping in 3D requires intersection of line segments against arbitrary plane. Example: oblique view. CS4395: Computer Graphics 21

22. Plane-Line Intersections CS4395: Computer Graphics 22

23. Normalized Form CS4395: Computer Graphics 23 before normalizationafter normalization Normalization is part of viewing (pre-clipping), but after normalization clip against sides of right parallelepiped. Typical intersection calculation now requires only a floating point subtraction, e.g. is x > x max ? top view

24. Polygon Clipping Not as simple as line segment clipping: – Clipping a line segment yields at most one line segment. – Clipping a polygon can yield multiple polygons. However, clipping a convex polygon can yield at most one other polygon. CS4395: Computer Graphics 24

25. Tessellation and Convexity Replace non-convex (concave) polygons with a set of triangular polygons (a tessellation). Also makes fill easier. Tessellation code in GLU library. CS4395: Computer Graphics 25

26. Line Clipping as a Black Box Consider line segment clipping as a process that takes in two vertices and produces either no vertices or the vertices of a clipped line segment. CS4395: Computer Graphics 26

27. Pipeline Clipping of Line Segments Clipping against each side of window is independent of other sides. Can use four independent clippers in a pipeline. CS4395: Computer Graphics 27

28. Pipeline Clipping of Polygons Three dimensions: add front and back clippers! Strategy used in SGI Geometry Engine. Small increase in latency. CS4395: Computer Graphics 28

29. Bounding Boxes Rather than doing clipping on a complex polygon, we can use an axis-aligned bounding box or extent. – Smallest rectangle aligned with axes that encloses the polygon. – Simple to compute: max and min of x and y. CS4395: Computer Graphics 29

30. Bounding Boxes Can usually determine accept or reject decisions based only on bounding box. CS4395: Computer Graphics 30 reject accept requires detailed clipping

31. Clipping and Visibility Clipping has much in common with hidden-surface removal. In both cases, we are trying to remove objects that are not visible to the camera. Often we can use visibility or occlusion testing early in the process to eliminate as many polygons as possible before going through the entire pipeline. CS4395: Computer Graphics 31

32. Summary More information: Sections 7.4-7.7. What next: – Rasterization. – Hidden surface removal. CS4395: Computer Graphics 32