1. Drawing with Thick Primitives Drawing with Thick Primitives • How do we thicken the line stroke width? • Ideas: – Place the center of the circular “brush” on the pixel – Place the upper corner of the square “marker” on the pixel (issues of orientation) – Then do scan conversion algorithm
2. Why Clip? Rasterization is very expensive Approximately linear with number of fragments created Math and logic per pixel If we only rasterize what is actually viewable, we can save a lot A few operations now can save many later
4. 159.235 Graphics 4 Viewing & Clipping - Outline Viewing in 2D Clipping in 2D Cohen-Sutherland Algorithm Cyrus & Beck Algorithm Clipping Polygons Sutherland-Hodgman Algorithm
5. 159.235 Graphics 5 Viewing in 2D - Viewport 250 45 Window in world coordinates. 250 x 250 Pixels. Viewport in Device coords
6. 159.235 Graphics 6 Clipping in 2D Need to clip primitives (eg lines) against the sides of the viewing window e.g lines or polygons We only see what is inside the window
7. 159.235 Graphics 7 Trivial Clipping Acceptance If all a line’s vertices lie inside box we “accept” it
8. 159.235 Graphics 8 Trivial Vertex Rejection All line vertices lie outside and on same side reject.
9. 159.235 Graphics 9 This is an efficient method of accepting or rejecting lines that do not intersect the window edges. Assign a binary 4 bit code to each vertex : First bit : above top of window, y > ymax Second bit : below bottom, y < ymin Third bit : to right of right edge, x > xmax Fourth bit : to left of left edge, x < xmin 4-bit code called: Outcode Cohen-Sutherland Clipping Algorithm
10. Cohen-Sutherland Line Clipping Divide 2D space into 3x3 regions. Middle region is the clipping window. Each region is assigned a 4-bit code. Bit 1 is set to 1 if the region is to the left of the clipping window, 0 otherwise. Similarly for bits 2, 3 and 4. 4 3 2 1 Top Bottom Right Left
12. 159.235 Graphics 12 Cohen-Sutherland Algorithm 1001 1000 1010 0000 0001 0010 0100 0110 0101 Both endpoint codes 0000, trivial acceptance, else: Do logical AND of Outcodes (reject if non-zero)
13. Cohen-Sutherland Line Clipping For those lines that we cannot immediately determine, we successively clip against each boundary. Then check the new endpoint for its region code. How to find boundary intersection: To find y coordinate at vertical intersection, substitute x value at the boundary into the line equation of the line to be clipped. Other algorithms: Faster: Cyrus-Beck Even faster: Liang-Barsky Even faster: Nichol-Lee-Nichol
14. 159.235 Graphics 14 Clipping Polygons Clip polygons by clipping successively against 4 sides.
15. 159.235 Graphics 15 Clipping Polygons Clip polygons by clipping successively against all 4 sides Can implement as pipelined algorithm (ie special hardware can do the work) Recursively test each edge. Form new edge with next vertex Call with new edge
16. 159.235 Graphics 16 Four cases of polygon clipping: Inside Outside Inside Outside Inside Outside Inside Outside Output Intersection Second Output Case 3 No output. Case 1 Case 2. Case 4 Sutherland-Hodgman Clipping Algorithm First Output Output Vertex
17. 159.235 Graphics 17 Sutherland-Hodgman Algorithm Loop round vertices, test each against all 4 clipping planes in sequence Call algorithm again with new edge formed by next vertex -re-entrant No storage requirement between stages Easy to implement in hardware
18. 18 Viewing & Clipping - Summary Window and Viewport Clipping cuts out what we do not “see” Also reduces unnecessary computation Can be done at various levels Java 2D system does a lot of the clipping for us if we use that rendering pipeline
