1. CS 551 / 645: Introductory Computer Graphics
Rasterizing Triangles

2. The Rock Says This... Midterm Date - before or after reading weekend?
Final Date - Early or late in week
Assignment 1 - Sun vs. SGI vs. PC
Grad school orientation after class

3. Review Line rasterization Basic Incremental Algorithm
Digital Differential Analyzer
Rather than solve line equation at each pixel, use evaluation of line from previous pixel and slope to approximate line equation
Bresenham (Midpoint line algorithm)
Use integer arithmetic and midpoint discriminator to test between two possible pixels (over vs. over-and-up)

4. Rasterizing Polygons In interactive graphics, polygons rule the world
Two main reasons:
Lowest common denominator for surfaces
Can represent any surface with arbitrary accuracy
Splines, mathematical functions, volumetric isosurfaces…
Mathematical simplicity lends itself to simple, regular rendering algorithms
Like those we’re about to discuss…
Such algorithms embed well in hardware

5. Rasterizing Polygons Triangle is the minimal unit of a polygon
All polygons can be broken up into triangles
Convex, concave, complex
Triangles are guaranteed to be:
Planar
Convex
What exactly does it mean to be convex?

6. Convex Shapes
A two-dimensional shape is convex if and only if every line segment connecting two points on the boundary is entirely contained.

7. Triangularization Convex polygons easily triangulated
Concave polygons present a challenge

8. Rasterizing Triangles
Interactive graphics hardware commonly uses edge walking or edge equation techniques for rasterizing triangles

9. Edge Walking Basic idea: Draw edges vertically
Fill in horizontal spans for each scanline
Interpolate colors down edges
At each scanline, interpolate edge colors across span

10. Edge Walking: Notes Order three triangle vertices in x and y
Find middle point in y dimension and compute if it is to the left or right of polygon. Also could be flat top or flat bottom triangle
We know where left and right edges are.
Proceed from top scanline downwards
Fill each span
Until breakpoint or bottom vertex is reached
Advantage: can be made very fast
Disadvantages:
Lots of finicky special cases

11. Edge Walking: Disadvantages
Fractional offsets:
Be careful when interpolating color values!
Beware of gaps between adjacent edges
Beware of duplicating shared edges

12. Edge Equations
An edge equation is simply the equation of the line defining that edge
Q: What is the implicit equation of a line?
A: Ax + By + C = 0
Q: Given a point (x,y), what does plugging x & y into this equation tell us?
A: Whether the point is:
On the line: Ax + By + C = 0
“Above” the line: Ax + By + C > 0
“Below” the line: Ax + By + C < 0

13. Edge Equations
Edge equations thus define two half-spaces:

14. Edge Equations
And a triangle can be defined as the intersection of three positive half-spaces:
A1x + B1y + C1 < 0
A2x + B2y + C2 < 0
A3x + B3y + C3 < 0
A1x + B1y + C1 > 0
A3x + B3y + C3 > 0
A2x + B2y + C2 > 0

15. Edge Equations
So…simply turn on those pixels for which all edge equations evaluate to > 0: - - + + + -

16. Using Edge Equations Which pixels: compute min,max bounding box
Edge equations: compute from vertices
Orientation: ensure area is positive (why?)

17. Computing Edge Equations
Want to calculate A, B, C for each edge from (x1, y1) and (x2, y2)
Treat it as a linear system:
Ax1 + By1 + C = 0
Ax2 + By2 + C = 0
Notice: two equations, three unknowns
What can we solve?
Goal: solve for A & B in terms of C

18. Computing Edge Equations
Set up the linear system:
Multiply both sides by matrix inverse:
Let C = x0 y1 - x1 y0 for convenience
Then A = y0 - y1 and B = x1 - x0

19. Edge Equations So…we can find edge equation from two verts.
Given three corners C0, C1, C2 of a triangle, what are our three edges?
How do we make sure the half-spaces defined by the edge equations all share the same sign on the interior of the triangle?
A: Be consistent (Ex: [C0 C1], [C1 C2], [C2 C0])
How do we make sure that sign is positive?
A: Test, and flip if needed (A= -A, B= -B, C= -C)

20. Edge Equations: Code Basic structure of code:
Setup: compute edge equations, bounding box
(Outer loop) For each scanline in bounding box...
(Inner loop) …check each pixel on scanline, evaluating edge equations and drawing the pixel if all three are positive

21. Optimize This! findBoundingBox(&xmin, &xmax, &ymin, &ymax);
setupEdges (&a0,&b0,&c0,&a1,&b1,&c1,&a2,&b2,&c2);
/* Optimize this: */
for (int y = yMin; y <= yMax; y++) {
for (int x = xMin; x <= xMax; x++) {
float e0 = a0*x + b0*y + c0;
float e1 = a1*x + b1*y + c1;
float e2 = a2*x + b2*y + c2;
if (e0 > 0 && e1 > 0 && e2 > 0) setPixel(x,y); }

22. Edge Equations: Speed Hacks
Some speed hacks for the inner loop:
int xflag = 0;
for (int x = xMin; x <= xMax; x++) {
if (e0|e1|e2 > 0) {
setPixel(x,y);
xflag++;
} else if (xflag != 0) break;
e0 += a0; e1 += a1; e2 += a2; }
Incremental update of edge equation values (think DDA)
Early termination (why does this work?)
Faster test of equation values

23. Edge Equations: Interpolating Color
Given colors (and later, other parameters) at the vertices, how to interpolate across?
Idea: triangles are planar in any space:
This is the “redness” parameter space
Note:plane follows form z = Ax + By + C
Look familiar?

24. Edge Equations: Interpolating Color
Given redness at the 3 vertices, set up the linear system of equations:
The solution works out to:

25. Edge Equations: Interpolating Color
Notice that the columns in the matrix are exactly the coefficients of the edge equations!
So the setup cost per parameter is basically a matrix multiply
Per-pixel cost (the inner loop) cost equates to tracking another edge equation value

26. Triangle Rasterization Issues
Exactly which pixels should be lit?
A: Those pixels inside the triangle edges
What about pixels exactly on the edge? (Ex.)
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

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

28. 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

29. General Polygon Rasterization
Does it still work? D B H C A G I E F

30. 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);
Why does this work?
What assumptions are we making?

31. 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)

32. 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

33. Active Edge Table Algorithm: 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

34. 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:

35. Clipping
Analytically calculating the portions of primitives within the viewport

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

37. 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?

38. 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.

39. 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

40. 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

41. Cohen-Sutherland Line Clipping
How do we set the bits in the outcode?
How do you suppose we use them?
1001
1000
1010
0001
0000
0010
0101
0100
0110

42. Cohen-Sutherland Line Clipping
Set bits with simple tests
x > xmax y < ymin etc.
Assign an outcode to each vertex of line
If both outcodes = 0, trivial accept
bitwise AND vertex outcodes together
If result  0, trivial reject

43. 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

44. Cohen-Sutherland Line Clipping
Outcode tests and line-edge intersects are quite fast
But some lines require multiple iterations (see F&vD figure 3.40)
Fundamentally more efficient algorithms:
Cyrus-Beck uses parametric lines
Liang-Barsky optimizes this for upright volumes

45. Coming Up… We know how to clip a single line segment
How about a polygon in 2D?
How about in 3-D?