.CH 1 "Introduction" 1
.SE 2 "Background" 1 1 
.SE 3 "Previous Work" 1 2 
.SE 4 "Overview of This Research" 1 3 
.SE 6 "Contents of This Paper" 1 4 
.CH 7 "Four Dimensional Geometry" 2
.SE 7 "Vector Operations and Points in Four-Dimensional Space" 2 1 
.SE 12 "Rotations in Four Dimensions" 2 2 
.CH 14 "Overview of Visualization in Three and Four Space" 3
.SE 14 "Viewing in Three-Space" 3 1 
.FG 16 "The 3D Viewing Vectors and From, To Points" 3.1
.FG 16 "The Resulting View From Figure 3.1" 3.2
.FG 18 "The 3D Viewing Vectors and Viewing Frustum" 3.3
.SE 19 "Viewing in Four-Space" 3 2 
.FG 20 "The 4D Viewing Vectors and Viewing Frustum" 3.4
.CH 21 "Wireframe Display of Four Dimensional Objects" 4
.SE 21 "High-Level Overview of 4D to 2D Projection" 4 1 
.SE 22 "Description of 3D to 2D Projection" 4 2 
.FG 23 "3D Eye Coordinates" 4.1
.FG 27 "3D Perspective Projection in Eye Coordinates" 4.2
.SE 29 "Description of 4D to 3D Projection" 4 3 
.FG 31 "4D Eye Coordinates" 4.3
.FG 35 "4D Perspective Projection in Eye Coordinates" 4.4
.SE 38 "Rotations of 4D Wireframes" 4 4 
.SE 39 "User Interaction and Visualization Aids" 4 5 
.SE 40 "Example 4D Wireframe Images" 4 6 
.FG 42 "The 4-Cube with Various 4D and 3D Projections" 4.5
.FG 46 "Another View of The 4-Cube with Various 4D and 3D Projections" 4.6
.FG 50 "4D Rotation of the 4-Cube" 4.7
.FG 52 "The 4-Cube With All Edges Rendered in One Color" 4.8
.FG 52 "The Dual of The 4-Cube With All Edges Rendered in One Color" 4.9
.FG 53 "The 4-Cube Rendered With Multiple Edge Colors" 4.10
.FG 53 "The Dual of The 4-Cube Rendered With Multiple Edge Colors" 4.11
.FG 54 "The 4-Cube Rendered With Depth-Cueing" 4.12
.FG 54 "The Dual of The 4-Cube Rendered With Depth-Cueing" 4.13
.FG 55 "A 4D Curve on a 4-Sphere" 4.14
.FG 55 "A 4D Curve on a 4-Sphere with Poor Parameterization" 4.15
.CH 56 "Raytracing in Four Dimensions" 5
.SE 56 "General Description of the Raytracing Algorithm" 5 1 
.FG 58 "A 2x2x2 4D Raytrace Grid" 5.1
.SE 59 "Generating the Four-Dimensional Ray Grid" 5 2 
.SE 62 "The General Raytrace Algorithm" 5 3 
.SE 63 "Reflection and Refraction Rays" 5 4 
.FG 64 "Ray-Object Reflection" 5.2
.SE 66 "Illumination Calculations" 5 5 
.FG 67 "Components of Illumination" 5.3
.SE 69 "Intersection Algorithms" 5 6 
.SE 69 "Ray - Hypersphere Intersection" 5 6 1
.SE 72 "Ray - Tetrahedron Intersection" 5 6 2
.SE 77 "Ray - Parallelepiped Intersection" 5 6 3
.SE 78 "Display of 4D Raytrace Data" 5 7 
.SE 79 "Example Ray4 Images" 5 8 
.FG 81 "Sliced 4D Image Cube of Random 4-Sphere Distribution" 5.4
.FG 82 "Sliced Image of 16 4-Spheres Placed at 4-Cube Vertices" 5.5
.FG 82 "Sliced Image of 16 4-Tetrahedrons Placed at 4-Cube Vertices" 5.6
.CH 83 "Conclusion" 6
.SE 83 "Research Conclusions" 6 1 
.SE 84 "Future Research Areas" 6 2 
.CH 86 "References"
.AP 88 A "Implementation Notes"
