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Color. CS 445/645 Introduction to Computer Graphics David Luebke, Spring 2003. Admin. Drop deadline imminent!. 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:

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CS 445/645Introduction to Computer Graphics

David Luebke, Spring 2003

David Luebke 18/20/2014


  • Drop deadline imminent!

David Luebke 28/20/2014

Recap rasterizing triangles
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
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
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) {



    } 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
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 color1
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 color2
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
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
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 rasterization1
General Polygon Rasterization

  • Consider the following polygon:

  • How do we know whether a given pixel on the scanline is inside or outside the polygon?







David Luebke 118/20/2014

General polygon rasterization2
General Polygon Rasterization

  • Does it still work?










David Luebke 128/20/2014

General polygon rasterization3
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)


  • Why does this work?

  • What assumptions are we making?

David Luebke 138/20/2014

Faster polygon rasterization
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)


  • Big cost: testing pixels against each edge

  • Solution: active edge table (AET)

David Luebke 148/20/2014

Active edge table
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 table1
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


  • 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