CS 445 / 645: Introductory Computer Graphics

1. CS 445 / 645: Introductory Computer Graphics Polygon Rasterization and Clipping

2. Assignment 1 • Grader has released a version of the code that you can review • Go to class web page • Click on Assignment 1 • Find link to sample code

3. Triangle Rasterization Issues • Exactly which pixels should be lit? • A: Those pixels inside the triangle edges • What about pixels exactly on the edge? • Draw them: order of triangles matters (it shouldn’t) • Don’t draw them: gaps possible between triangles • We need a consistent (if arbitrary) rule • Example: draw pixels on left or top edge, but not on right or bottom edge

4. Triangle Rasterization Issues • Sliver

5. Triangle Rasterization Issues • Moving Slivers

6. Triangle Rasterization Issues • Shared Edge Ordering

7. General Polygon Rasterization • Now that we can rasterize triangles, what about general polygons? • We’ll take an edge-walking approach

8. General Polygon Rasterization • Consider the following polygon: • How do we know whether a given pixel on the scanline is inside or outside the polygon? D B C A E F

9. Polygon Rasterization • Inside-Outside Points

10. Polygon Rasterization • Inside-Outside Points

11. General Polygon Rasterization • Basic idea: use a parity test for each scanline edgeCnt = 0; for each pixel on scanline (l to r) if (oldpixel->newpixel crosses edge) edgeCnt ++; // draw the pixel if edgeCnt odd if (edgeCnt % 2) setPixel(pixel);

12. General Polygon Rasterization • Count your vertices carefully • Details… G F I H E C J D A B

13. Faster Polygon Rasterization • How can we optimize the code? for each scanline edgeCnt = 0; for each pixel on scanline (l to r) if (oldpixel->newpixel crosses edge) edgeCnt ++; // draw the pixel if edgeCnt odd if (edgeCnt % 2) setPixel(pixel); • Big cost: testing pixels against each edge • Solution: active edge table (AET)

14. Active Edge Table • Idea: • Edges intersecting a given scanline are likely to intersect the next scanline • The order of edge intersections doesn’t change much from scanline to scanline

15. Active Edge Table • Algorithm: scanline from bottom to top… • Go over example… • Sort all edges by their minimum y coord • Starting at bottom, add edges with Ymin= 0 to AET • For each scanline: • Sort edges in AET by x intersection • Walk from left to right, setting pixels by parity rule • Increment scanline • Retire edges with Ymax < Y • Add edges with Ymin< Y • Recalculate edge intersections (how?) • Stop when Y > Ymax for last edges

16. Next Topic: Clipping • We’ve been assuming that all primitives (lines, triangles, polygons) lie entirely within the viewport • In general, this assumption will not hold:

17. Clipping • Analytically calculating the portions of primitives within the viewport

18. Why Clip? • Bad idea to rasterize outside of framebuffer bounds • Also, don’t waste time scan converting pixels outside window

19. Clipping • The naïve approach to clipping lines: for each line segment for each edge of viewport find intersection points pick “nearest” point if anything is left, draw it • What do we mean by “nearest”? • How can we optimize this? B D C A

20. Trivial Accepts • Big optimization: trivial accept/rejects • How can we quickly determine whether a line segment is entirely inside the viewport? • A: test both endpoints.

21. Trivial Rejects • How can we know a line is outside viewport? • A: if both endpoints on wrong side of same edge, can trivially reject line

22. Clipping Lines To Viewport • Discard segments of lines outside viewport • Trivially accept lines with both endpoints inside all edges of the viewport • Trivially reject lines with both endpoints outside the same edge of the viewport • Otherwise, reduce to trivial casesby splitting into two segments

23. Cohen-Sutherland Line Clipping • Divide viewplane into regions defined by viewport edges • Assign each region a 4-bit outcode: 1001 1000 1010 0001 0000 0010 0101 0100 0110

24. Cohen-Sutherland Line Clipping • Assign an outcode to each vertex of line • If both outcodes = 0, trivial accept • bitwise AND vertex outcodes together • If result  0, trivial reject

25. Cohen-Sutherland Line Clipping • If line cannot be trivially accepted or rejected, subdivide so that one or both segments can be discarded • Pick an edge that the line crosses (how?) • Intersect line with edge (how?) • Discard portion on wrong side of edge and assign outcode to new vertex • Apply trivial accept/reject tests; repeat if necessary

26. Cohen-Sutherland Line Clipping • If line cannot be trivially accepted or rejected, subdivide so that one or both segments can be discarded • Pick an edge that the line crosses • Check against edges in same order each time • For example: top, bottom, right, left E D C B A

27. E D C B A Cohen-Sutherland Line Clipping • Intersect line with edge (how?)

28. Cohen-Sutherland Line Clipping • Discard portion on wrong side of edge and assign outcode to new vertex • Apply trivial accept/reject tests and repeat if necessary D C B A

29. Solving Simultaneous Equations • Equation of a line • Slope-intercept (explicit equation): y = mx + b • Implicit Equation: Ax + By + C = 0 • Parametric Equation: Line defined by two points, P0 and P1 • P(t) = P0 + (P1 - P0) t, where P is a vector [x, y]T • x(t) = x0 + (x1 - x0) t • y(t) = x0 + (y1 - y0) t

30. Parametric Line Equation • Describes a finite line • Works with vertical lines (like the viewport) • 0 <=t <= 1 • Defines line between P0 and P1 • t < 0 • Defines line before P0 • t > 1 • Defines line after P1

31. Parametric Lines and Clipping • Define each line in parametric form: • P0(t)…Pn-1(t) • Define each edge of viewport in parametric form: • PL(t), PR(t), PT(t), PB(t) • When using Cohen-Sutherland perform intersection calculation for appropriate viewport edge and line

32. Computing Intersections • Line 0: • x0 = x00 + (x01 - x00) t0 • y0 = y00 + (y01 - y00) t0 • Viewport Edge L: • xL = xL0 + (xL1 - xL0) tL • yL = yL0 + (yL1 - yL0) tL • x00 + (x01 - x00) t0 = xL0 + (xL1 - xL0) tL • y00 + (y01 - y00) t0 = yL0 + (yL1 - yL0) tL • Solve for t0 and tL • Truth in advertising… Cohen-Sutherland doesn’t exactly do it this way. You don’t need to solve for t0 and tL

33. Cohen-Sutherland Line Clipping • Outcode tests and line-edge intersects are quite fast • But some lines require multiple iterations • Fundamentally more efficient algorithms: • Cyrus-Beck uses parametric lines to clip against arbitrary polygons (not just rectangles) • Liang-Barsky optimizes this for upright volumes