visualizing the fourth dimension using geometry and light
We explore techniques for visualizing mathematical objects in four-dimensional space that exploit four-dimensional lighting effects. We analyze the geometry of image production, stereography, and shadows in 4D. We examine alternatives for smooth and specular shaded rendering of curves, surfaces, and solids in 4D, and propose a new approach that systematically converts curves or surfaces into uniquely renderable solids in four-dimensional space by attaching spheres or circles to each point. Analogs of 3D shading methods are used to produce volume renderings that distinguish objects whose 3D projections from 4D are identical. Analyzing the procedures needed to justify and evaluate a system such as ours for teaching humans to "see" in four dimensions leads us to propose a generally applicable four-step visualization paradigm.
visualization geometry lighting exploration object space exploit effect image product smooth rendering surface systems point 3d method project procedure evaluation teaching