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ICS 415 Computer Graphics Clipping

ICS 415 Computer Graphics Clipping. Dr. Muhammed Al-Mulhem March 1, 2009. Transform. Illuminate. Transform. Clip. Project. Rasterize. Model & Camera Parameters. Rendering Pipeline. Framebuffer. Display. The Rendering Pipeline. Model World. World Camera. Why clip?.

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ICS 415 Computer Graphics Clipping

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  1. ICS 415Computer Graphics Clipping Dr. Muhammed Al-Mulhem March 1, 2009 Dr. Muhammed Al-Mulhem

  2. Transform Illuminate Transform Clip Project Rasterize Model & CameraParameters Rendering Pipeline Framebuffer Display The Rendering Pipeline ModelWorld WorldCamera Dr. Muhammed Al-Mulhem

  3. Why clip? • We don’t want to waste time rendering objects that are outside the viewing window (or clipping window) Dr. Muhammed Al-Mulhem

  4. What is clipping? • Analytically calculating the portions of primitives within the view window Dr. Muhammed Al-Mulhem

  5. View Window Eye position (focal point) Clip to what? Viewing Frustum Dr. Muhammed Al-Mulhem

  6. Transform Illuminate Transform Clip Project Rasterize Model & CameraParameters Rendering Pipeline Framebuffer Display Why illuminate before clipping? ModelWorld WorldCamera Dr. Muhammed Al-Mulhem

  7. Transform Illuminate Transform Clip Project Rasterize Model & CameraParameters Rendering Pipeline Framebuffer Display Why WorldCamera before clipping? ModelWorld WorldCamera Dr. Muhammed Al-Mulhem

  8. Why? • Why illuminate before clipping? • Why WorldCamera before clipping? • Efficiency Dr. Muhammed Al-Mulhem

  9. View Window Eye position (focal point) Clip to what? Dr. Muhammed Al-Mulhem

  10. Transform Illuminate Transform Clip Project Rasterize Model & CameraParameters Rendering Pipeline Framebuffer Display Why clip before project and rasterize? ModelWorld WorldCamera Dr. Muhammed Al-Mulhem

  11. Why Clip? • Bad idea to rasterize outside of framebuffer bounds • Also, don’t waste time scan converting pixels outside window Dr. Muhammed Al-Mulhem

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

  13. Trivial Accepts • Big optimization: trivial accept/rejects • How can we quickly determine whether a line segment is entirely inside the view window? • A: test both endpoints. Dr. Muhammed Al-Mulhem

  14. Trivial Rejects • How can we know a line is outside view window? • A: if both endpoints on wrong side of same edge, can trivially reject line Dr. Muhammed Al-Mulhem

  15. Clipping Lines To View Window • Combining trivial accepts/rejects • Trivially accept lines with both endpoints inside all edges of the view window • Trivially reject lines with both endpoints outside the same edge of the view window • Otherwise, reduce to trivial casesby splitting into two segments Dr. Muhammed Al-Mulhem

  16. Cohen-Sutherland Line Clipping • Divide view window into regions defined by window edges • Assign each region a 4-bit outcode: Bit 1 indicates y-value of points are above ymax ymax 1001 1000 1010 xmax 0001 0000 0010 0101 0100 0110 Dr. Muhammed Al-Mulhem

  17. Cohen-Sutherland Line Clipping • For each line segment • Assign an outcode to each vertex • If both outcodes = 0, trivial accept • Same as performing if (bitwise OR = 0) • Else • bitwise AND vertex outcodes together • if result  0, trivial reject Dr. Muhammed Al-Mulhem

  18. 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 of view window that the line crosses (how?) • Intersect line with edge (how?) • Discard portion on wrong side of edge and assign new outcode to new vertex • Apply trivial accept/reject tests; repeat if necessary Dr. Muhammed Al-Mulhem

  19. 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 Dr. Muhammed Al-Mulhem

  20. E D C B A Cohen-Sutherland Line Clipping • Intersect line with edge (how?) – Chapter 6. Dr. Muhammed Al-Mulhem

  21. 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 Dr. Muhammed Al-Mulhem

  22. 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 C B A Dr. Muhammed Al-Mulhem

  23. View Window Intersection Code • (x1, y1), (x2, y2) intersect with vertical edge at xright • yintersect = y1 + m(xright – x1) • where m=(y2-y1)/(x2-x1) • (x1, y1), (x2, y2) intersect with horizontal edge at ybottom • xintersect = x1 + (ybottom – y1)/m • where m=(y2-y1)/(x2-x1) Dr. Muhammed Al-Mulhem

  24. Cohen-Sutherland Review • Use Outodes to quickly eliminate/include lines • Best algorithm when trivial accepts/rejects are common • Must compute viewing window clipping of remaining lines • Non-trivial clipping cost • Redundant clipping of some lines • More efficient algorithms exist Dr. Muhammed Al-Mulhem

  25. 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) = y0 + (y1 - y0) t Dr. Muhammed Al-Mulhem

  26. Parametric Line Equation • Describes a finite line • Works with vertical lines (like the view window edge) • 0 <=t <= 1 • Defines line between P0 and P1 • t < 0 • Defines line before P0 • t > 1 • Defines line after P1 Dr. Muhammed Al-Mulhem

  27. Parametric Lines and Clipping • Define each line in parametric form: • P0(t)…Pn-1(t) • Define each edge of view window in parametric form: • PL(t), PR(t), PT(t), PB(t) • Perform Cohen-Sutherland intersection tests using appropriate view window edge and line Dr. Muhammed Al-Mulhem

  28. Line / Edge Clipping Equations • Faster line clippers use parametric equations • Line 0: • x0 = x00 + (x01 - x00) t0 • y0 = y00 + (y01 - y00) t0 • x00 + (x01 - x00) t0 = xL0 + (xL1 - xL0) tL • y00 + (y01 - y00) t0 = yL0 + (yL1 - yL0) tL • Solve for t0 and/or tL • View Window Edge L: • xL = xL0 + (xL1 - xL0) tL • yL = yL0 + (yL1 - yL0) tL Dr. Muhammed Al-Mulhem

  29. Cyrus-Beck Algorithm • We wish to optimize line/line intersection • Start with parametric equation of line: • P(t) = P0 + (P1 - P0) t • And a point and normal for each edge • PL, NL Dr. Muhammed Al-Mulhem

  30. PL P(t) Inside NL Cyrus-Beck Algorithm P1 • Find t such that • NL [P(t) - PL] = 0 • Substitute line equation for P(t): • NL [P0 + (P1 - P0) t - PL] = 0 • Solve for t • t = NL [PL – P0] / -NL [P1 - P0] P0 Dr. Muhammed Al-Mulhem

  31. Cyrus-Beck Algorithm • Compute t for line intersection with all four edges • Discard all (t < 0) and (t > 1) • Classify each remaining intersection as • Potentially Entering (PE) • Potentially Leaving (PL) • NL [P1 - P0] > 0 implies PL • NL [P1 - P0] < 0 implies PE Dr. Muhammed Al-Mulhem

  32. P1 PL PL PE PE P0 Cyrus-Beck Algorithm • Compute PE with largest t • Compute PL with smallest t • Clip to these two points Dr. Muhammed Al-Mulhem

  33. Cyrus-Beck Algorithm • Because of horizontal and vertical clip lines: • Many computations reduce • Normals: (-1, 0), (1, 0), (0, -1), (0, 1) • Pick constant points on edges • solution for t: • -(x0 - xleft) / (x1 - x0) • (x0 - xright) / -(x1 - x0) • -(y0 - ybottom) / (y1 - y0) • (y0 - ytop) / -(y1 - y0) Dr. Muhammed Al-Mulhem

  34. Comparison • Cohen-Sutherland • Repeated clipping is expensive • Best used when trivial acceptance and rejection is possible for most lines • Cyrus-Beck • Computation of t-intersections is cheap • Computation of (x,y) clip points is only done once • Algorithm doesn’t consider trivial accepts/rejects • Best when many lines must be clipped • Liang-Barsky: Optimized Cyrus-Beck • Nicholl et al.: Fastest, but doesn’t do 3D Dr. Muhammed Al-Mulhem

  35. Clipping Polygons • Clipping polygons is more complex than clipping the individual lines • Input: polygon • Output: Original polygon, new polygon, or nothing • Optimization • The biggest line optimizer we had was trivial accept or reject. • When can we trivially accept/reject a polygon as opposed to the line segments that make up the polygon? Dr. Muhammed Al-Mulhem

  36. Why Is Clipping Hard? • What happens to a triangle during clipping? • Possible outcomes: triangle  quad triangle  5-gon triangle  triangle • How many sides can a clipped triangle have? Dr. Muhammed Al-Mulhem

  37. How many sides? • Seven… Dr. Muhammed Al-Mulhem

  38. Why Is Clipping Hard? • A really tough case: Dr. Muhammed Al-Mulhem

  39. Why Is Clipping Hard? • A really tough case: concave polygon  multiple polygons Dr. Muhammed Al-Mulhem

  40. Sutherland-Hodgman Clipping • Basic idea: • Consider each edge of the view window individually • Clip the polygon against the view window edge’s equation Dr. Muhammed Al-Mulhem

  41. Sutherland-Hodgman Clipping • Basic idea: • Consider each edge of the viewport individually • Clip the polygon against the edge equation Dr. Muhammed Al-Mulhem

  42. Sutherland-Hodgman Clipping • Basic idea: • Consider each edge of the viewport individually • Clip the polygon against the edge equation Dr. Muhammed Al-Mulhem

  43. Sutherland-Hodgman Clipping • Basic idea: • Consider each edge of the viewport individually • Clip the polygon against the edge equation Dr. Muhammed Al-Mulhem

  44. Sutherland-Hodgman Clipping • Basic idea: • Consider each edge of the viewport individually • Clip the polygon against the edge equation Dr. Muhammed Al-Mulhem

  45. Sutherland-Hodgman Clipping • Basic idea: • Consider each edge of the viewport individually • Clip the polygon against the edge equation Dr. Muhammed Al-Mulhem

  46. Sutherland-Hodgman Clipping • Basic idea: • Consider each edge of the viewport individually • Clip the polygon against the edge equation Dr. Muhammed Al-Mulhem

  47. Sutherland-Hodgman Clipping • Basic idea: • Consider each edge of the viewport individually • Clip the polygon against the edge equation Dr. Muhammed Al-Mulhem

  48. Sutherland-Hodgman Clipping • Basic idea: • Consider each edge of the viewport individually • Clip the polygon against the edge equation Dr. Muhammed Al-Mulhem

  49. Sutherland-Hodgman Clipping • Basic idea: • Consider each edge of the viewport individually • Clip the polygon against the edge equation • After doing all edges, the polygon is fully clipped Dr. Muhammed Al-Mulhem

  50. Sutherland-Hodgman Clipping • Input/output for algorithm: • Input: List of polygon vertices in order. • Output: List of clipped polygon vertices consisting of old vertices (maybe) and new vertices (maybe). • Note: this is exactly what we expect from the clipping operation against each edge Dr. Muhammed Al-Mulhem

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