1. From Vertices to Fragments Chapter 7

2. Part I Objectives Introduce basic implementation strategies Introduce basic implementation strategies Clipping Clipping Scan conversion Scan conversion CS 480/680 2 Chapter 7 -- From Vertices to Fragments

3. Overview At end of the geometric pipeline, vertices have been assembled into primitives At end of the geometric pipeline, vertices have been assembled into primitives Must clip out primitives that are outside the view frustum Must clip out primitives that are outside the view frustum Algorithms based on representing primitives by lists of vertices Algorithms based on representing primitives by lists of vertices Must find which pixels can be affected by each primitive Must find which pixels can be affected by each primitive Fragment generation Fragment generation Rasterization or scan conversion Rasterization or scan conversion CS 480/680 3 Chapter 7 -- From Vertices to Fragments

4. Required Tasks Clipping Clipping Rasterization or scan conversion Rasterization or scan conversion Transformations Transformations Some tasks deferred until fragment processing Some tasks deferred until fragment processing Hidden surface removal Hidden surface removal Antialiasing Antialiasing CS 480/680 4 Chapter 7 -- From Vertices to Fragments

5. Rasterization Meta Algorithms Consider two approaches to rendering a scene with opaque objects Consider two approaches to rendering a scene with opaque objects For every pixel, determine which object that projects on the pixel is closest to the viewer and compute the shade of this pixel For every pixel, determine which object that projects on the pixel is closest to the viewer and compute the shade of this pixel Ray tracing paradigm Ray tracing paradigm For every object, determine which pixels it covers and shade these pixels For every object, determine which pixels it covers and shade these pixels Pipeline approach Pipeline approach Must keep track of depths Must keep track of depths CS 480/680 5 Chapter 7 -- From Vertices to Fragments

6. Clipping 2D against clipping window 2D against clipping window 3D against clipping volume 3D against clipping volume Easy for line segments polygons Easy for line segments polygons Hard for curves and text Hard for curves and text Convert to lines and polygons first Convert to lines and polygons first CS 480/680 6 Chapter 7 -- From Vertices to Fragments

7. Clipping 2D Line Segments Brute force approach: compute intersections with all sides of clipping window Brute force approach: compute intersections with all sides of clipping window Inefficient: one division per intersection Inefficient: one division per intersection CS 480/680 7 Chapter 7 -- From Vertices to Fragments

8. Cohen-Sutherland Algorithm Idea: eliminate as many cases as possible without computing intersections Idea: eliminate as many cases as possible without computing intersections Start with four lines that determine the sides of the clipping window Start with four lines that determine the sides of the clipping window CS 480/680 8 Chapter 7 -- From Vertices to Fragments x = x max x = x min y = y max y = y min

9. The Cases Case 1: both endpoints of line segment inside all four lines Case 1: both endpoints of line segment inside all four lines Draw (accept) line segment as is Draw (accept) line segment as is Case 2: both endpoints outside all lines and on same side of a line Case 2: both endpoints outside all lines and on same side of a line Discard (reject) the line segment Discard (reject) the line segment CS 480/680 9 Chapter 7 -- From Vertices to Fragments x = x max x = x min y = y max y = y min

10. The Cases Case 3: One endpoint inside, one outside Case 3: One endpoint inside, one outside Must do at least one intersection Must do at least one intersection Case 4: Both outside Case 4: Both outside May have part inside May have part inside Must do at least one intersection Must do at least one intersection CS 480/680 10 Chapter 7 -- From Vertices to Fragments x = x max x = x min y = y max

11. Defining Outcodes For each endpoint, define an outcode For each endpoint, define an outcode Outcodes divide space into 9 regions Outcodes divide space into 9 regions Computation of outcode requires at most 4 subtractions Computation of outcode requires at most 4 subtractions CS 480/680 11 Chapter 7 -- From Vertices to Fragments 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 Consider the 5 cases below AB: outcode(A) = outcode(B) = 0 AB: outcode(A) = outcode(B) = 0 Accept line segment Accept line segment CS 480/680 12 Chapter 7 -- From Vertices to Fragments

13. Using Outcodes CD: outcode (C) = 0, outcode(D)  0 CD: outcode (C) = 0, outcode(D)  0 Compute intersection Compute intersection Location of 1 in outcode(D) determines which edge to intersect with 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 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 CS 480/680 13 Chapter 7 -- From Vertices to Fragments

14. Using Outcodes EF: outcode(E) logically ANDed with outcode(F) (bitwise)  0 EF: outcode(E) logically ANDed with outcode(F) (bitwise)  0 Both outcodes have a 1 bit in the same place Both outcodes have a 1 bit in the same place Line segment is outside of corresponding side of clipping window Line segment is outside of corresponding side of clipping window reject reject CS 480/680 14 Chapter 7 -- From Vertices to Fragments

15. Using Outcodes GH and IJ: same outcodes, neither zero but logical AND yields zero GH and IJ: same outcodes, neither zero but logical AND yields zero Shorten line segment by intersecting with one of sides of window Shorten line segment by intersecting with one of sides of window Compute outcode of intersection (new endpoint of shortened line segment) Compute outcode of intersection (new endpoint of shortened line segment) Reexecute algorithm Reexecute algorithm CS 480/680 15 Chapter 7 -- From Vertices to Fragments

16. Efficiency In many applications, the clipping window is small relative to the size of the entire data base 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 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 Inefficiency when code has to be re-executed for line segments that must be shortened in more than one step CS 480/680 16 Chapter 7 -- From Vertices to Fragments

17. Cohen Sutherland in 3D Use 6-bit outcodes Use 6-bit outcodes When needed, clip line segment against planes When needed, clip line segment against planes CS 480/680 17 Chapter 7 -- From Vertices to Fragments

18. Liang-Barsky Clipping Consider the parametric form of a line segment Consider the parametric form of a line segment We can distinguish between the cases by looking at the ordering of the values of  where the line determined by the line segment crosses the lines that determine the window We can distinguish between the cases by looking at the ordering of the values of  where the line determined by the line segment crosses the lines that determine the window CS 480/680 18 Chapter 7 -- From Vertices to Fragments p(  ) = (1-  )p 1 +  p 2 1  0 p1p1 p2p2

19. Liang-Barsky Clipping In (a):  4 >  3 >  2 >  1 In (a):  4 >  3 >  2 >  1 Intersect right, top, left, bottom: shorten Intersect right, top, left, bottom: shorten In (b):  4 >  2 >  3 >  1 In (b):  4 >  2 >  3 >  1 Intersect right, left, top, bottom: reject Intersect right, left, top, bottom: reject CS 480/680 19 Chapter 7 -- From Vertices to Fragments

20. Advantages Can accept/reject as easily as with Cohen- Sutherland Can accept/reject as easily as with Cohen- Sutherland Using values of , we do not have to use algorithm recursively as with C-S Using values of , we do not have to use algorithm recursively as with C-S Extends to 3D Extends to 3D CS 480/680 20 Chapter 7 -- From Vertices to Fragments

21. Clipping and Normalization General clipping in 3D requires intersection of line segments against arbitrary plane General clipping in 3D requires intersection of line segments against arbitrary plane Example: oblique view Example: oblique view CS 480/680 21 Chapter 7 -- From Vertices to Fragments

22. Plane-Line Intersections CS 480/680 22 Chapter 7 -- From Vertices to Fragments

23. Normalized Form Normalization is part of viewing (pre clipping) but after normalization, we clip against sides of right parallelepiped Normalization is part of viewing (pre clipping) but after normalization, we clip against sides of right parallelepiped Typical intersection calculation now requires only a floating point subtraction, e.g. is x > x max ? Typical intersection calculation now requires only a floating point subtraction, e.g. is x > x max ? CS 480/680 23 Chapter 7 -- From Vertices to Fragments before normalizationafter normalization top view

24. CS 480/680 24 Chapter 7 -- From Vertices to Fragments

25. Part II Objectives Introduce clipping algorithms for polygons Introduce clipping algorithms for polygons Survey hidden-surface algorithms Survey hidden-surface algorithms CS 480/680 25 Chapter 7 -- From Vertices to Fragments

26. Polygon Clipping Not as simple as line segment clipping Not as simple as line segment clipping Clipping a line segment yields at most one line segment Clipping a line segment yields at most one line segment Clipping a polygon can yield multiple polygons Clipping a polygon can yield multiple polygons However, clipping a convex polygon can yield at most one other polygon However, clipping a convex polygon can yield at most one other polygon CS 480/680 26 Chapter 7 -- From Vertices to Fragments

27. Tessellation and Convexity One strategy is to replace nonconvex (concave) polygons with a set of triangular polygons (a tessellation) One strategy is to replace nonconvex (concave) polygons with a set of triangular polygons (a tessellation) Also makes fill easier Also makes fill easier Tessellation code in GLU library Tessellation code in GLU library CS 480/680 27 Chapter 7 -- From Vertices to Fragments

28. Clipping as a Black Box Can 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 Can 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 CS 480/680 28 Chapter 7 -- From Vertices to Fragments

29. Pipeline Clipping of Line Segments Clipping against each side of window is independent of other sides Clipping against each side of window is independent of other sides Can use four independent clippers in a pipeline Can use four independent clippers in a pipeline CS 480/680 29 Chapter 7 -- From Vertices to Fragments

30. Pipeline Clipping of Polygons Three dimensions: add front and back clippers Three dimensions: add front and back clippers Strategy used in SGI Geometry Engine Strategy used in SGI Geometry Engine Small increase in latency Small increase in latency CS 480/680 30 Chapter 7 -- From Vertices to Fragments

31. Bounding Boxes Rather than doing clipping on a complex polygon, we can use an axis-aligned bounding box or extent 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 Smallest rectangle aligned with axes that encloses the polygon Simple to compute: max and min of x and y Simple to compute: max and min of x and y CS 480/680 31 Chapter 7 -- From Vertices to Fragments

32. Bounding boxes Can usually determine accept/reject based only on bounding box Can usually determine accept/reject based only on bounding box CS 480/680 32 Chapter 7 -- From Vertices to Fragments reject accept requires detailed clipping

33. Clipping and Visibility Clipping has much in common with hidden- surface removal 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 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 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 CS 480/680 33 Chapter 7 -- From Vertices to Fragments

34. Hidden Surface Removal Object-space approach: use pair-wise testing between polygons (objects) Object-space approach: use pair-wise testing between polygons (objects) Worst case complexity O(n 2 ) for n polygons Worst case complexity O(n 2 ) for n polygons CS 480/680 34 Chapter 7 -- From Vertices to Fragments partially obscuringcan draw independently

35. Painter’s Algorithm Render polygons a back to front order so that polygons behind others are simply painted over Render polygons a back to front order so that polygons behind others are simply painted over CS 480/680 35 Chapter 7 -- From Vertices to Fragments B behind A as seen by viewer Fill B then A

36. Depth Sort Requires ordering of polygons first Requires ordering of polygons first O(n log n) calculation for ordering O(n log n) calculation for ordering Not every polygon is either in front or behind all other polygons Not every polygon is either in front or behind all other polygons Order polygons and deal with Order polygons and deal with easy cases first, harder later CS 480/680 36 Chapter 7 -- From Vertices to Fragments Polygons sorted by distance from COP

37. Easy Cases A lies behind all other polygons A lies behind all other polygons Can render Can render Polygons overlap in z but not in either x or y Polygons overlap in z but not in either x or y Can render independently Can render independently CS 480/680 37 Chapter 7 -- From Vertices to Fragments

38. Hard Cases CS 480/680 38 Chapter 7 -- From Vertices to Fragments Overlap in all directions but one is fully on one side of the other cyclic overlap penetration

39. Back-Face Removal (Culling) face is visible iff 90    -90 face is visible iff 90    -90 equivalently cos   0 equivalently cos   0 or v n  0 or v n  0 plane of face has form ax + by +cz +d =0 plane of face has form ax + by +cz +d =0 but after normalization n = ( 0 0 1 0) T but after normalization n = ( 0 0 1 0) T need only test the sign of c need only test the sign of c In OpenGL we can simply enable culling but may not work correctly if we have nonconvex objects In OpenGL we can simply enable culling but may not work correctly if we have nonconvex objects CS 480/680 39 Chapter 7 -- From Vertices to Fragments

40. Image Space Approach Look at each projector ( nm for an n x m frame buffer) and find closest of k polygons Look at each projector ( nm for an n x m frame buffer) and find closest of k polygons Complexity O (nmk) Complexity O (nmk) Ray tracing Ray tracing z -buffer z -buffer CS 480/680 40 Chapter 7 -- From Vertices to Fragments

41. z-Buffer Algorithm Use a buffer called the z or depth buffer to store the depth of the closest object at each pixel found so far Use a buffer called the z or depth buffer to store the depth of the closest object at each pixel found so far As we render each polygon, compare the depth of each pixel to depth in z buffer As we render each polygon, compare the depth of each pixel to depth in z buffer If less, place shade of pixel in color buffer and update z buffer If less, place shade of pixel in color buffer and update z buffer CS 480/680 41 Chapter 7 -- From Vertices to Fragments

42. Efficiency If we work scan line by scan line as we move across a scan line, the depth changes satisfy a  x+b  y+c  z=0 If we work scan line by scan line as we move across a scan line, the depth changes satisfy a  x+b  y+c  z=0 CS 480/680 42 Chapter 7 -- From Vertices to Fragments Along scan line  y = 0  z = -  x In screen space  x = 1

43. Scan-Line Algorithm Can combine shading and hsr through scan line algorithm Can combine shading and hsr through scan line algorithm CS 480/680 43 Chapter 7 -- From Vertices to Fragments scan line i: no need for depth information, can only be in no or one polygon scan line j: need depth information only when in more than one polygon

44. Implementation Need a data structure to store Need a data structure to store Flag for each polygon (inside/outside) Flag for each polygon (inside/outside) Incremental structure for scan lines that stores which edges are encountered Incremental structure for scan lines that stores which edges are encountered Parameters for planes Parameters for planes CS 480/680 44 Chapter 7 -- From Vertices to Fragments

45. Visibility Testing In many real-time applications, such as games, we want to eliminate as many objects as possible within the application In many real-time applications, such as games, we want to eliminate as many objects as possible within the application Reduce burden on pipeline Reduce burden on pipeline Reduce traffic on bus Reduce traffic on bus Partition space with Binary Spatial Partition (BSP) Tree Partition space with Binary Spatial Partition (BSP) Tree CS 480/680 45 Chapter 7 -- From Vertices to Fragments

46. Simple Example CS 480/680 46 Chapter 7 -- From Vertices to Fragments consider 6 parallel polygons top view The plane of A separates B and C from D, E and F

47. BSP Tree Can continue recursively Can continue recursively Plane of C separates B from A Plane of C separates B from A Plane of D separates E and F Plane of D separates E and F Can put this information in a BSP tree Can put this information in a BSP tree Use for visibility and occlusion testing Use for visibility and occlusion testing CS 480/680 47 Chapter 7 -- From Vertices to Fragments

48. CS 480/680 48 Chapter 7 -- From Vertices to Fragments

49. Part III Objectives Survey Line Drawing Algorithms Survey Line Drawing Algorithms DDA DDA Bresenham Bresenham CS 480/680 49 Chapter 7 -- From Vertices to Fragments

50. Rasterization Rasterization (scan conversion) Rasterization (scan conversion) Determine which pixels that are inside primitive specified by a set of vertices Determine which pixels that are inside primitive specified by a set of vertices Produces a set of fragments Produces a set of fragments Fragments have a location (pixel location) and other attributes such color and texture coordinates that are determined by interpolating values at vertices Fragments have a location (pixel location) and other attributes such color and texture coordinates that are determined by interpolating values at vertices Pixel colors determined later using color, texture, and other vertex properties Pixel colors determined later using color, texture, and other vertex properties CS 480/680 50 Chapter 7 -- From Vertices to Fragments

51. Scan Conversion of Line Segments Start with line segment in window coordinates with integer values for endpoints Start with line segment in window coordinates with integer values for endpoints Assume implementation has a write_pixel function Assume implementation has a write_pixel function CS 480/680 51 Chapter 7 -- From Vertices to Fragments y = mx + h

52. DDA Algorithm Digital Differential Analyzer Digital Differential Analyzer DDA was a mechanical device for numerical solution of differential equations DDA was a mechanical device for numerical solution of differential equations Line y=mx+ h satisfies differential equation Line y=mx+ h satisfies differential equation dy/dx = m =  y/  x = y 2 -y 1 /x 2 -x 1 dy/dx = m =  y/  x = y 2 -y 1 /x 2 -x 1 Along scan line  x = 1 Along scan line  x = 1 CS 480/680 52 Chapter 7 -- From Vertices to Fragments For(x=x1; x<=x2,ix++) { y+=m; write_pixel(x, round(y), line_color) }

53. Problem DDA = for each x plot pixel at closest y DDA = for each x plot pixel at closest y Problems for steep lines Problems for steep lines CS 480/680 53 Chapter 7 -- From Vertices to Fragments

54. Using Symmetry Use for 1  m  0 Use for 1  m  0 For m > 1, swap role of x and y For m > 1, swap role of x and y For each y, plot closest x For each y, plot closest x CS 480/680 54 Chapter 7 -- From Vertices to Fragments

55. Bresenham’s Algorithm DDA requires one floating point addition per step DDA requires one floating point addition per step We can eliminate all fp through Bresenham’s algorithm We can eliminate all fp through Bresenham’s algorithm Consider only 1  m  0 Consider only 1  m  0 Other cases by symmetry Other cases by symmetry Assume pixel centers are at half integers Assume pixel centers are at half integers If we start at a pixel that has been written, there are only two candidates for the next pixel to be written into the frame buffer If we start at a pixel that has been written, there are only two candidates for the next pixel to be written into the frame buffer CS 480/680 55 Chapter 7 -- From Vertices to Fragments

56. Candidate Pixels CS 480/680 56 Chapter 7 -- From Vertices to Fragments 1  m  0 last pixel candidates Note that line could have passed through any part of this pixel

57. Decision Variable CS 480/680 57 Chapter 7 -- From Vertices to Fragments d =  x(b-a) d is an integer d > 0 use upper pixel d < 0 use lower pixel

58. Incremental Form More efficient if we look at d k, the value of the decision variable at x = k More efficient if we look at d k, the value of the decision variable at x = k For each x, we need do only an integer addition and a test For each x, we need do only an integer addition and a test Single instruction on graphics chips Single instruction on graphics chips CS 480/680 58 Chapter 7 -- From Vertices to Fragments d k+1 = d k –2  y, if d k <0 d k+1 = d k –2(  y-  x), otherwise

59. Polygon Scan Conversion Scan Conversion = Fill Scan Conversion = Fill How to tell inside from outside How to tell inside from outside Convex easy Convex easy Nonsimple difficult Nonsimple difficult Odd even test Odd even test Count edge crossings Count edge crossings Winding number Winding number CS 480/680 59 Chapter 7 -- From Vertices to Fragments odd-even fill

60. Winding Number Count clockwise encirclements of point Count clockwise encirclements of point Alternate definition of inside: inside if winding number  0 Alternate definition of inside: inside if winding number  0 CS 480/680 60 Chapter 7 -- From Vertices to Fragments winding number = 2 winding number = 1

61. Filling in the Frame Buffer Fill at end of pipeline Fill at end of pipeline Convex Polygons only Convex Polygons only Nonconvex polygons assumed to have been tessellated Nonconvex polygons assumed to have been tessellated Shades (colors) have been computed for vertices (Gouraud shading) Shades (colors) have been computed for vertices (Gouraud shading) Combine with z-buffer algorithm Combine with z-buffer algorithm March across scan lines interpolating shades March across scan lines interpolating shades Incremental work small Incremental work small CS 480/680 61 Chapter 7 -- From Vertices to Fragments

62. Using Interpolation CS 480/680 62 Chapter 7 -- From Vertices to Fragments span C1C1 C3C3 C2C2 C5C5 C4C4 scan line C 1 C 2 C 3 specified by glColor or by vertex shading C 4 determined by interpolating between C 1 and C 2 C 5 determined by interpolating between C 2 and C 3 interpolate between C 4 and C 5 along span

63. Flood Fill Fill can be done recursively if we know a seed point located inside (WHITE) Fill can be done recursively if we know a seed point located inside (WHITE) Scan convert edges into buffer in edge/inside color (BLACK) Scan convert edges into buffer in edge/inside color (BLACK) CS 480/680 63 Chapter 7 -- From Vertices to Fragments flood_fill(int x, int y) { if(read_pixel(x,y)= = WHITE) { write_pixel(x,y,BLACK); flood_fill(x-1, y); flood_fill(x+1, y); flood_fill(x, y+1); flood_fill(x, y-1); }

64. Scan Line Fill Can also fill by maintaining a data structure of all intersections of polygons with scan lines Can also fill by maintaining a data structure of all intersections of polygons with scan lines Sort by scan line Sort by scan line Fill each span Fill each span CS 480/680 64 Chapter 7 -- From Vertices to Fragments vertex order generated by vertex list desired order

65. Data Structure CS 480/680 65 Chapter 7 -- From Vertices to Fragments

66. Aliasing Ideal rasterized line should be 1 pixel wide Ideal rasterized line should be 1 pixel wide Choosing best y for each x (or visa versa) produces aliased raster lines Choosing best y for each x (or visa versa) produces aliased raster lines CS 480/680 66 Chapter 7 -- From Vertices to Fragments

67. Antialiasing by Area Averaging Color multiple pixels for each x depending on coverage by ideal line Color multiple pixels for each x depending on coverage by ideal line CS 480/680 67 Chapter 7 -- From Vertices to Fragments original antialiased magnified

68. Polygon Aliasing Aliasing problems can be serious for polygons Aliasing problems can be serious for polygons Jaggedness of edges Jaggedness of edges Small polygons neglected Small polygons neglected Need compositing so color Need compositing so color of one polygon does not totally determine color of pixel CS 480/680 68 Chapter 7 -- From Vertices to Fragments All three polygons should contribute to color

69. CS 480/680 69 Chapter 7 -- From Vertices to Fragments