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• Checking for degenrate triangles
  • The cross product of two vectors, a and b is defined as a × b = ||a|| ||b|| sin(Θ) n
    • Θ is the angle between the two vectors.
    • n is the normal to the plane containing the two vectors.
    • Let a = [a₁, a₂, a₃]
    • Let b = [b₁, b₂, b₃]
    • a × b = [ a₂b₃-a₃b₂, a₃b₁-a₁b₃, a₁b₂-a₂b₁]
  • Consider the points (10,5), (20, 30), and (10,20)

    a = [20-10, 30-5, 0-0] = [10,25,0]
    b = [10-10, 20-5, 0-0] = [0, 15, 0]
     
             |  25*0  - 0*15 |
    a × b =  |  0*0   - 10*0 |  = [ 0, 0, 150]
             | 10*15  - 0*10 |
    	 

  • Some points in this example.
    • First of all, the triangle is in the xy plane
    • So the z component of each point is 0
    • But the cross product is a vector that is orthogonal to the xy plane.
    • And the cross product is not 0, so the triangle is ok.
  • Consider the points (10,5), (20, 15) and (30,25)

    a = [20-10, 15-5, 0-0] = [10, 10,0]
    b = [30-10, 25-5, 0-0] = [20, 20, 0]
    
             |  10*0  - 0*20 |
    a × b =  |  0*20   - 10*0 |  = [ 0, 0, 200-200] = [0,0,0]
             | 10*20  - 20*10 |
    	 

  • We will use the cross product quite a bit, so it makes sense to build this into any hardware that deals with graphics.
  • Look at your book page 90
    • Polygons (triangles) have a front and back face.
    • You can turn on and off front or back face drawing
    • The front and back faces are determined by the order the trianlge is specified.
    • You can change either, but you might need to specify you are using Clockwise, (Counterclockwise is the default)
• Floodfill
  • This is the last of the raster algorithms.
  • If we have a shape determined by "curved" borders
  • Filling this can not be done as a scan line, we don't have the equations for the lines.
  • Assume that we have a
    • Point inside of the thing to be filled.
    • A color with which we wish to fill.
    • An edge color.
  • This assumes that the area is completely surrounded by the boundary color.

  • FloodFill(x, y, fillColor, borderColor) 
       if not valid(x,y) 
           return
       if FrameBuffer[x,y] == borderColor
          return
       FrameBuffer[x,y] = fillColor
       FloodFill(x+1,y,fillColor,borderColor);
       FloodFill(x-1,y,fillColor,borderColor);
       FloodFill(x,y+1,fillColor,borderColor);
       FloodFill(x,y-1,fillColor,borderColor);
    	

  • 
  • Fill this.