Interactive Exploration of 4D Geometry with Volumetric Halos

Halos have been employed as a compelling illustrative hint in many applications to promote depth perception and to emphasize occlusion effects among projected objects. We generalize the application of halo methods from the widely-used domain of 2D projections of 3D objects to the domain of 3D projections of 4D objects. Since 4D imaging involves a projection from 4D geometry (such as a surface with 4D vertices) to a 3D image, such projection typically produces intersecting surfaces, and thus occlusion phenomena result in apparent curves in 3D space. Adding volumetric halos to the surfaces then gives useful information about the spatial relations of intersecting surfaces, and allows a more accurate perception of the geometry. A typical application is knotted spheres embedded in 4D, and the volumetric halos perform the same function as traditional knot diagrams do in 2D drawings of 3D knotted curves. In addition, we design a series of GPU-based algorithms to achieve real-time updating of the halo-enhanced image when the geometry is interactively rotated in 4D.