Scan-Line Algorithms & Linear Interpolation

Depth Interpolation, Z-Buffering

Planar polygon with plane equation: Ax + By + Cz + D = 0

==> is LINEAR in all directions !

Solve for z: z = (-A/C)x + (-B/C)y -D --> dz/dx = const; dz/dy = const.

Keep this information with the polygon, or with the edges on the scan-line.

Allows quick updating by addition:

-- calculate initial z-value for edges

-- add dz*/dy along the edges, and

-- add dz/dx along coherent spans.

Updated depth values get used in z-buffer for depth comparison.

Brightness/Color Interpolation, Gouraud Shading

Same general approach used again !
Assume we know brightness of a polyhedral object face at its vertices.
Now we can also linearly interpolate the brightness across the face.

However, if there are more than 3 vertices, there is no guarantee that

the brightness values globally form a linear function of x and y coordinates.

Thus do the interpolation in a bi-linear form in trapezoidal (triangular) areas.

Compute di*/dy along each edge directly from brightness at vertices.

Again, do quick updates by addition:

-- calculate initial i-value for edges

-- add the (di*/dy)'s along the edges.

Need to calculate a new di/dx along each span for each scan-line.

The updated brightness values get used in the i-buffer, if the z-comparison is successful.

There may be tangent discontinuities in the generated "brightness surface" !