Rasterization

Rasterization

Rasterization is the process of converting geometric primitives (points, lines, triangles) into discrete pixels (fragments) on the screen. It is the most common rendering approach used in real-time graphics.

Triangle Rasterization

Triangles are the fundamental primitive in graphics. Rasterizing triangles involves determining which pixels fall inside the triangle.

Bounding Box Method:

  For triangle with vertices (x0,y0), (x1,y1), (x2,y2):

  1. Compute bounding box:
     min_x = min(x0, x1, x2)
     max_x = max(x0, x1, x2)
     min_y = min(y0, y1, y2)
     max_y = max(y0, y1, y2)

  2. For each pixel (px, py) in bounding box:
     if inside_triangle(px, py):
         rasterize(px, py)

Edge Functions

The edge function determines if a point is to the left or right of a directed edge. A point is inside the triangle if it is on the same side of all three edges.

Edge function for edge (V0, V1):

  E(x,y) = (x - x0)(y1 - y0) - (y - y0)(x1 - x0)

  E > 0 → point is to the left (inside)
  E < 0 → point is to the right (outside)
  E = 0 → point is on the edge

For triangle V0, V1, V2:
  p is inside if:
    E01(p) ≥ 0 AND
    E12(p) ≥ 0 AND
    E20(p) ≥ 0

Barycentric Coordinates

Barycentric coordinates express a point as a weighted combination of triangle vertices. They are used for interpolating attributes across the triangle.

Barycentric coordinates (λ0, λ1, λ2):

  P = λ0*V0 + λ1*V1 + λ2*V2
  λ0 + λ1 + λ2 = 1

λ0 = E12(P) / E12(V0)
λ1 = E20(P) / E20(V1)
λ2 = E01(P) / E01(V2)

Interpolation of attribute A:
  A(P) = λ0*A(V0) + λ1*A(V1) + λ2*A(V2)

Can interpolate: color, normal, texcoord, depth

Z-Buffer (Depth Buffer)

The Z-buffer stores the depth of each rendered pixel. When rendering a new fragment, its depth is compared to the stored depth. Closer fragments replace farther ones.

Z-Buffer Algorithm:

  Initialize depth_buffer[x][y] = 1.0 (far)
  Initialize frame_buffer[x][y] = background

  For each triangle:
      For each pixel (x,y) in triangle:
          z = interpolated depth at (x,y)
          if z < depth_buffer[x][y]:
              depth_buffer[x][y] = z
              frame_buffer[x][y] = color(x,y)