Like in Pixel-Planes 4 (see "Fast Spheres. . ." by Fuchs et al in SIGGRAPH '85), PixelFlow can evaluate a linear expression of the form Ax+By+C for every pixel in a single tile simultaneously. We will formulate approximations for rasterizing the sphere that will utilize this expression.

Development of the sphere rasterizer proceeds as follows:

• Scan-Converting the Sphere: Circle In-Out Test
• Depth-Evaluation using Paraboloid Approximation
• Linearizing the Paraboloid Depth Approximation
• Perspective Scaling of the Radius
• From Vertices to Pixels: the Transformation Pipeline
• Coordinate Systems for the Sphere Rasterizer
• Putting it All Together: Method 1: using Screen Space to NDC mapping
• Putting it All Together: Method 2: using only Screen Space
• Depth-Evaluation in Screen Space: Combining Circle In-Out Test and Depth Evaluation
• Linearizing the Combined Circle Test and Depth Evaluation
• Calculation of Surface Normals : finding the normals for each pixel for shading calculations
• 2D/3D Texture Coordinate Evaluation : calculation of (u,v) or (u,v,w) texture coords for texture-mapping each pixel

• bruteforce.cpp : nonlinear direct evaluation for scan-conversion and depth-evaluation. Tests circle approximation for scan-conversion and paraboloid approximation for depth-evaluation.
• bruteforce2.cpp : direct evaluation of combined circle in-out test and depth evaluation.
• lineval.cpp : uses PixelFlow-style linear expression evaluator and the Qbuffer.
• lineval2.cpp : linear evaluation of combined method.
• normals.cpp : evaluation of surface normals using floating point.
• normals2.cpp : surface normals using fixed-point scaling.
• fbppm.hpp : portable pixel map (PPM) "mock" framebuffer