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Recap: Rasterizing Triangles

- Edge walking:
- Use DDA approach to “walk” down edges of triangle
- At each scanline, walk the span from one edge to another
- Edge equations:
- Compute bounding box
- Edge equations: compute from vertices (numerical issues!)
- Orientation: ensure area is positive

David Luebke 38/20/2014

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);

}

}

David Luebke 48/20/2014

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

David Luebke 58/20/2014

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 formz = Ax + By + C
- Look familiar?

David Luebke 68/20/2014

Edge Equations: Interpolating Color

- Given redness at the 3 vertices, set up the linear system of equations:
- The solution works out to:

David Luebke 78/20/2014

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 (which is?)
- A: 1 add

David Luebke 88/20/2014

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

David Luebke 98/20/2014

General Polygon Rasterization

- Now that we can rasterize triangles, what about general polygons?
- We’ll not take an edge-equations approach (why?)

David Luebke 108/20/2014

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

David Luebke 118/20/2014

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?

David Luebke 138/20/2014

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)

David Luebke 148/20/2014

Active Edge Table

- Idea:
- Edges intersecting a given scanline are likely to intersect the next scanline
- Within a scanline, the order of edge intersections doesn’t change much from scanline to scanline

David Luebke 158/20/2014

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 and resort (how?)
- Stop when Y > Ymax for last edges

David Luebke 168/20/2014

Color

- Next topic: Color

To understand how to make realistic images, we need a basic understanding of the physics and physiology of vision. Here we step away from the code and math for a bit to talk about basic principles.

David Luebke 178/20/2014

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