From Vertices to Fragments Software College, Shandong University Instructor: Zhou Yuanfeng E-mail: [email protected] Review Shading in OpenGL; Lights & Material; From vertex to fragment: 2 Cohen-Sutherland Black box Projection

Fragments Clipping Shading Surface hidden Texture Transparency Objectives Geometric Processing: Cyrus-Beck clipping algorithm Liang-Barsky clipping algorithm Introduce clipping algorithm for polygons Rasterization: DDA & Bresenham

3 Cohen-Sutherland In case IV: o1 & o2 = 0 Intersection: Clipping Lines by Solving Simultaneous Equations 4 Solving Simultaneous Equations Equation of a line: Slope-intercept: y = mx + h difficult for vertical line Implicit Equation: Ax + By + C = 0 Parametric: Line defined by two points, P0 and P1

P(t) = P0 + (P1 - P0) t x(t) = x0 + (x1 - x0) t y(t) = x0 + (y1 - y0) t Parametric Lines and Intersections For L1 : x=x0l1 + t(x1l1 x0l1) y=y0l1 + t(y1l1 y0l1) For L2 : x=x0l2 + t(x1l2 x0l2) y=y0l2 + t(y1l2 y0l2) The Intersection Point:

x0l1 + t1 (x1l1 x0l1) = x0l2 + t2 (x1l2 x0l2) y0l1 + t1 (y1l1 y0l1) = y0l2 + t2 (y1l2 y0l2) 6 Cyrus-Beck Algorithm Cyrus-Beck algorithm (1978) for polygons Mike Cyrus, Jay Beck. "Generalized two- and three-dimensional clipping". Computers & Graphics, 1978: 23-28. Given a convex polygon R: R P(t ) ( P2 P1 )t P1 0t1

N P2 A t)0 ,then A) P (t ) is inside of R; N ( P (t ) N A( P ) ( N P (t ) A cos N ( P (t ) A) 0 ,then P (t ) is on R or extension;

90 cos 0 ) isoutside ,then N ( P(t ) A) 0 90 P (tcos 0 of R. 90 cos 0 P1 para ts para te

How to get ts and te 7 Cyrus-Beck Algorithm Intersection: NL (P(t) A) = 0 Inside P(t) P0 A

Outside NL Substitute line equation for P(t) P(t) = P0 + t(P1 - P0) Solve for t t = NL (P0 A) / -NL (P1 - P0) P1 Cyrus-Beck Algorithm

Compute t for line intersection with all edges; Discard all (t < 0) and (t > 1); Classify each remaining intersection as Potentially Entering Point (PE) Potentially Leaving Point (PL) (How?) NL(P1 - P0) < 0 implies PL NL(P1 - P0) > 0 implies PE Note that we computed this term in when computing t Cyrus-Beck Algorithm For each edge: Ni ( P0 Ai ) Ni ( P1 P0 )ti 0,

L2 L3 t3 t4 L1 t1 t5 Compute PE with largest t Compute PL with smallest t Clip to these two points

P1 L4 L5 t2 P0 0 ti 1 para ts para te ts max{0, max{ti | Ni ( P1 P0 ) 0}}

te min{1, min{ti | N i ( P1 P0 ) 0}} 10 Cyrus-Beck Algorithm When Ni ( P1 P0 ) 0 ; then N i ( P1 P0 ) if Ni ( P0 Ai ) 0 Then P0P1 is invisible; if Ni ( P0 Ai ) 0 Then go to next edge; 11 Programming: for k edges of clipping polygon {

solve Ni(p1-p0); solve Ni(p0-Ai); if ( Ni(p1-p0) = = 0 ) //parallel with the edge { if ( Ni(p0-Ai) < 0 ) break; //invisible else go to next edge; } else // Ni(p1-p0) != 0 { solve ti; if ( Ni(p1-p0) < 0 ) else }

Input: If (P0 = P1 ) Line is degenerate so clip as a point; Output: if ( ts > te ) return nil; else return P(ts) and P(te) as the true clip intersections; te min{1, min{ti | Ni ( P1 P0 ) 0}} ts max{0, max{ti | Ni ( P1 P0 ) 0}} }

12 Liang-Barsky Algorithm (1984) The ONLY algorithm named for Chinese people in Computer Graphics course books Liang, Y.D., and Barsky, B., "A New Concept and Method for Line Clipping", ACM Transactions on Graphics, 3(1):113 22, January 1984. Liang-Barsky Algorithm (1984) Because of horizontal and vertical clip lines: Many computations reduce Normals Pick constant points on edges

PL solution for t: (0, -1) (1, 0) tL=-(x0 - xleft) / (x1 - x0) tR=(x0 - xright) / -(x1 - x0) tB=-(y0 - ybottom) / (y1 - y0) tT=(y0 - ytop) / -(y1 - y0) PE P0

PE (0, 1) PL (-1, 0) P1 Liang-Barsky Algorithm (1984) N ( P1 A) N ( P2 P1 )

Edge Inner normal A Left x=XL 1 0 XL y

x1-XL y1y ( x1 XL) ( x 2 x1 ) Right x=XR -1 0 XR y

x1-XR y1-y ( x1 XR) x 2 x1 Bottom y=YB 0 1 xY B x1-x y1-YB

( y1 YB) ( y 2 y1 ) Top y=YT 01 xY T x1-x y1-YT P1-A

t ( y1 YT ) y 2 y1 15 Liang-Barsky Algorithm (1984) Let x=x2 x1 y=y2 y1: r1 x, s1 x1 xL , r2 x,

s2 xR x1 , r3 y, s3 y1 y B , r4 y, s4 yT y1 , tk sk / rk , k L, R, B, T When rk<0, tk is entering point; when rk>0, tk is leaving point. If rk=0 and sk<0, then the line is invisible; else process other edges

16 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 doesnt consider trivial accepts/rejects Best when many lines must be clipped Liang-Barsky: Optimized Cyrus-Beck

Nicholl et al.: Fastest, but doesnt do 3D 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 18 Pipeline Clipping of Line Segments Clipping against each side of window is independent of other sides Can use four independent clippers in a pipeline

19 Clipping and Normalization General clipping in 3D requires intersection of line segments against arbitrary plane Example: oblique view 20 Plane-Line Intersections (( P 0 ( P1 t ( P 2 P1))) n 0 n ( p 0 p1)

t n ( p 2 p1) 21 Point-to-Plane Test Dot product is relatively expensive 3 multiplies 5 additions 1 comparison (to 0, in this case) Think about how you might optimize or special-case this? Normalized Form top view

before normalization after normalization 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 > xmax 23 Clipping Polygons Clipping polygons is more complex than clipping the individual lines

Input: polygon Output: polygon, or nothing 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 25 Tessellation and Convexity One strategy is to replace nonconvex

(concave) polygons with a set of triangular polygons (a tessellation) Also makes fill easier (we will study later) Tessellation code in GLU library, the best is Constrained Delaunay Triangulation 26 Pipeline Clipping of Polygons Three dimensions: add front and back clippers Strategy used in SGI Geometry Engine Small increase in latency 27

Sutherland-Hodgman Clipping Ivan Sutherland, Gary W. Hodgman: Reentrant Polygon Clipping. Communications of the ACM, vol. 17, pp. 32-42, 1974 Basic idea: Consider each edge of the viewport individually Clip the polygon against the edge equation Sutherland-Hodgman Clipping Basic idea: Consider each edge of the viewport individually Clip the polygon against the edge equation

After doing all planes, the polygon is fully clipped Sutherland-Hodgman Clipping Basic idea: Consider each edge of the viewport individually Clip the polygon against the edge equation After doing all planes, the polygon is fully clipped Sutherland-Hodgman Clipping Basic idea: Consider each edge of the viewport individually

Clip the polygon against the edge equation After doing all planes, the polygon is fully clipped Sutherland-Hodgman Clipping Basic idea: Consider each edge of the viewport individually Clip the polygon against the edge equation After doing all planes, the polygon is fully clipped Sutherland-Hodgman Clipping Basic idea:

Consider each edge of the viewport individually Clip the polygon against the edge equation After doing all planes, the polygon is fully clipped Sutherland-Hodgman Clipping Basic idea: Consider each edge of the viewport individually Clip the polygon against the edge equation After doing all planes, the polygon is fully clipped Sutherland-Hodgman Clipping

Basic idea: Consider each edge of the viewport individually Clip the polygon against the edge equation After doing all planes, the polygon is fully clipped Sutherland-Hodgman Clipping Basic idea: Consider each edge of the viewport individually Clip the polygon against the edge equation After doing all planes, the polygon is fully clipped Sutherland-Hodgman

Clipping Basic idea: Consider each edge of the viewport individually Clip the polygon against the edge equation After doing all planes, the polygon is fully clipped Will this work for non-rectangular clip regions? What would 3-D clipping involve? Sutherland-Hodgman Clipping Input/output for algorithm: Input: list of polygon vertices in order Output: list of clipped poygon vertices consisting

of old vertices (maybe) and new vertices (maybe) Note: this is exactly what we expect from the clipping operation against each edge Sutherland-Hodgman Clipping Sutherland-Hodgman basic routine: Go around polygon one vertex at a time Current vertex has position p Previous vertex had position s, and it has been added to the output if appropriate Sutherland-Hodgman

Clipping Edge from s to p takes one of four cases: (Gray line can be a line or a plane) inside outside inside outside inside s

p p output s i output p outside p inside p

s no output i output p output outside s Sutherland-Hodgman Clipping Four cases: s inside plane and p inside plane

Add p to output Note: s has already been added s inside plane and p outside plane Find intersection point i Add i to output s outside plane and p outside plane Add nothing s outside plane and p inside plane Find intersection point i Add i to output, followed by p Sutherland-Hodgman

Clipping 42 Point-to-Plane test A very general test to determine if a point p is inside a plane P, defined by q and n: (p - q) n < 0: (p - q) n = 0: (p - q) n > 0: q p inside P p on P p outside P

q q n p n p p P

P P n 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 44

Bounding Boxes Can usually determine accept/reject based only on bounding box reject accept requires detailed clipping Ellipsoid collision detection 45 Rasterization Rasterization (scan conversion)

Determine which pixels that are inside primitive specified by a set of vertices Produces a set of fragments Fragments have a location (pixel location) and other attributes such color, depth and texture coordinates that are determined by interpolating values at vertices Pixel colors determined later using color, texture, and other vertex properties. 46 Scan Conversion of Line Segments Start with line segment in window

coordinates with integer values for endpoints Assume implementation has a write_pixel function y = mx + h y m x 47 Scan Conversion of Line Segments One pixel

48 DDA Algorithm Digital Differential Analyzer (1964) DDA was a mechanical device for numerical solution of differential equations Line y=mx+h satisfies differential equation dy/dx = m = y/x = y2-y1/x2-x1 Along scan line x = 1 For(x=x1; x<=x2, x++) { y += m; //note:m is float number write_pixel(x, round(y), line_color); } 49

Problem DDA = for each x plot pixel at closest y Problems for steep lines 50 Using Symmetry Use for 1 m 0 For m > 1, swap role of x and y For each y, plot closest x 51 Bresenhams Algorithm

DDA requires one floating point addition per step We can eliminate all fp through Bresenhams algorithm Consider only 1 m 0 Other cases by symmetry Assume pixel centers are at half integers (OpenGL has this definition) Bresenham, J. E. (1 January 1965). "Algorithm for computer control of a digital plotter". IBM Systems Journal 4(1): 2530. 52 Bresenhams Algorithm Observing:

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 53 Candidate Pixels 1m0 candidates last pixel Note that line could have passed through any part of this pixel

54 Decision Variable d = x(a-b)x(a-b) d is an integer d < 0 use upper pixel d > 0 use lower pixel How to compute a and b? b-a =(yi+1yi,r)-( yi,r+1-yi+1) =2yi+1yi,r(yi,r+1) = 2yi+12yi,r1 A B

C - (xi+1)= yi+1yi,r0.5 =BC-AC=BA=B-A = yi+1(yi,r+ yi,r+1)/2 55 Incremental Form if (xi+1) 0, yi+1,r= yi,r+1, pick pixel D, the next pixel is ( xi+1, yi,r+1) if (xi+1) < 0, yi+1,r= yi,r, pick pixel C, the next pixel is ( xi+1, yi,r) yi,r+1

d1 yi,r xi D D d2 B A yi,r+1 C

yi,r xi+1 d2 A d1 xi C xi+1

56 Improvement - - - anew = alast m anew = alast (m-1) bnew = blast + m

bnew = blast + (m-1) d = x(a-b)x(a-b) 57 Improvement More efficient if we look at dk, the value of the decision variable at x = k dk+1= dk 2x(a-b)y, if dk > 0 dk+1= dk 2(y - x), otherwise For each x, we need do only an integer addition and a test Single instruction on graphics chips

multiply 2 is simple. 58 BSP display Type Tree Tree* front; Face face; Tree *back;

End Algorithm DrawBSP(Tree T; point: w) //w If T is null then return; endif If w is in front of T.face then DrawBSP(T.back,w); Draw(T.face,w); DrawBSP(T.front,w); Else // w is behind or on T.face DrawBSP(T.front,w); Draw(T.face,w); DrawBSP(T. back,w); Endif end 59

Hidden Surface Removal Object-space approach: use pairwise testing between polygons (objects) partially obscuring can draw independently Worst case complexity O(n2) for n polygons 60 Image Space Approach Look at each projector (nm for an n x m frame buffer) and find closest of k

polygons Complexity O(nmk) Ray tracing z-buffer 61 Painters Algorithm Render polygons a back to front order so that polygons behind others are simply painted over B behind A as seen by viewer Fill B then A

62 Depth Sort Requires ordering of polygons first O(n log n) calculation for ordering Not every polygon is either in front or behind all other polygons Order polygons and deal with easy cases first, harder later Polygons sorted by distance from COP 63

Easy Cases (1) A lies behind all other polygons Can render (2) Polygons overlap in z but not in either x or y Can render independently 64 Hard Cases (3) Overlap in all directions but can one is fully on one side of the other

(4) cyclic overlap penetration 65 Back-Face Removal (Culling) face is visible iff 90 -90 equivalently cos 0 or v n 0

plane of face has form ax + by +cz +d =0 but after normalization n = ( 0 0 1 0)T need only test the sign of c In OpenGL we can simply enable culling but may not work correctly if we have nonconvex objects 66 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 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

67 Efficiency If we work scan line by scan line as we move across a scan line, the depth changes satisfy ax+by+cz=0 Along scan line y = 0 a z = - x c In screen space x

=1 68 Scan-Line Algorithm Can combine shading and hsr through scan line algorithm 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 69

Implementation Need a data structure to store Flag for each polygon (inside/outside) Incremental structure for scan lines that stores which edges are encountered Parameters for planes 70 Visibility Testing In many realtime applications, such as games, we want to eliminate as many objects as possible within the application Reduce burden on pipeline

Reduce traffic on bus Partition space with Binary Spatial Partition (BSP) Tree 71 Simple Example consider 6 parallel polygons top view The plane of A separates B and C from D, E and F 72

BSP Tree Can continue recursively Plane of C separates B from A Plane of D separates E and F Can put this information in a BSP tree Use for visibility and occlusion testing 73 Polygon Scan Conversion Scan Conversion = Fill How to tell inside from outside Convex easy

Nonsimple difficult Odd even test Count edge crossings Winding number odd-even fill 74 Winding Number Count clockwise encirclements of point winding number = 1 winding number = 2

Alternate definition of inside: inside if winding number 0 75 Filling in the Frame Buffer Fill at end of pipeline Convex Polygons only Nonconvex polygons assumed to have been tessellated Shades (colors) have been computed for vertices (Gouraud shading) Combine with z-buffer algorithm March across scan lines interpolating shades Incremental work small 76

Using Interpolation C1 C2 C3 specified by glColor or by vertex shading C4 determined by interpolating between C1 and C2 C5 determined by interpolating between C2 and C3 interpolate between C4 and C5 along span C1 C4 scan line C2 C5 span

C3 77 Flood Fill Fill can be done recursively if we know a seed point located inside (WHITE) Scan convert edges into buffer in edge/inside color (BLACK) 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); } } 78 Scan Line Fill Can also fill by maintaining a data structure of all intersections of polygons with scan lines Sort by scan line Fill each span vertex order generated by vertex list desired order

79 Data Structure 80 Aliasing Ideal rasterized line should be 1 pixel wide Choosing best y for each x (or visa versa) produces aliased raster lines 81 Antialiasing by Area Averaging

Color multiple pixels for each x depending on coverage by ideal line antialiased original magnified 82 Polygon Aliasing Aliasing problems can be serious for polygons Jaggedness of edges Small polygons neglected Need compositing so color

of one polygon does not totally determine color of pixel All three polygons should contribute to color 83