Point Based Rendering of Implicit 4-Dimensional Surfaces

We present a point based rendering algorithm that uses hyper-cubes to perform spatial subdivision in a 4D volume. A 4D function interval exclusion test is used to speed up the rendering of 4D Implicit surfaces in this hyper-volume. A 4D orthonormal basis function is used to define a 4D camera, which projects isometric or perspective views onto a plane in 4D for viewing. The technique requires evaluations of 4D implicit surface functions and gradients to render shaded images. The technique also allows for hidden surface elimination using a z-buffer modified for use in a 4D space. We give examples of its use in rendering some 4D surfaces and discuss problems with the technique. The algorithm can be generalised to higher dimensions.