Scan-converting the Sphere: Circle In-Out Test

Putting it All Together: Method 2 using only Screen Space
Kenny Hoff 6/17/97

We have problems that we would like to simplify or remove entirely from the previously techniques:

Necessity to convert between so many different coordinate systems through the NDC system
The separate calculation stages for determining points inside the projected sphere's circle and for evaluating the depth with the paraboloid approximation.

To avoid the expensive conversions from screen space to NDC space for computing depth values, we can instead do the entire paraboloid fit in screen space with (x,y,z) values being (x,y) pixel coordinates and z being the actual fixed-point Z-values (already scaled, etc). At the same time, we would like to combine the depth-evaluation with the scan-conversion by simply performing the depth-evaluation and discarding pixels that are "too deep" (beyond the sphere's center screen Z).

Like before, we need to transform the center and a point on the boundary into eye space so that we can compute an eye space radius as the distance between these transformed points.
We still need a silhouette point on the boundary relative to the viewer. The viewer is assumed to be at the origin and a point EyeRadius distance from the center along the direction perpendicular to the viewing direction (-Z) is selected as the silhouette sample point (actually it should be perpendicular to the direction from the eye to the sphere's boundary - trickier, more expensive, and probably unnecessary). SilhouettePt = EyeCenter + (0,1,0)*EyeRadius
We also need a point on the sphere in eye space that is closest to the viewer (the origin in eye space). We should pick the point on the sphere the farthest in the direction opposite of the direction from the eye to the sphere's center; we will also approximate this with a point EyeRadius distance from the center in the direction opposite of the viewing direction (+Z). ClosestPt = EyeCenter + (0,0,1)*EyeRadius
The three eye-space points EyeCenter, SilhouettePt, and ClosestPt must all be transformed into screen space.
Pixel radius is computed as the 2D euclidean distance between EyeCenter and SilhouettePt pixel coordinates.
The Z-values of the EyeCenter and the ClosestPt in screen space is used to know the range of Z-values to fit with the paraboloid approximation. So in the XY-plane, we are fitting the pixel coordinates, but in Z we are fitting the actual Z-values.
We can perform the scan-conversion in-out test using the paraboloid approximation. Since we are only approximating the front hemisphere we can assume that the sphere's center Z-value is the maximum Z for the sphere and the closest point's Z-value is the minimum. This means that boundary pixels will contain the largest Z-values; in fact, if the depth value of a pixel exceeds the max Z-value as determined by the center, it cannot be inside the circular projection.
Using on the depth-evaluation, pixels are disabled if the computed Z is greater than the max Z (outside circle) or if it is greater than the depth already in the Zbuffer.