Visualization of four-dimensional space and its applications

In this thesis a method has been proposed to visualize curves, surfaces and hypersurfaces in four-dimensional space. Objects in 4-space are first projected into the 3D image space and further projected into the 2D image space. Four topics have been investigated: (1) Fundamental Concepts. (2) Visual Phenomena and Their Meaning. (3) System Architecture. (4) Applications. The orientation of the 3D and 2D image spaces can be specified by a set of six Euler angles or a pair of quaternions. Since the quaternion pairs representing 4D rotations are not unique, three useful forms have been discussed. Each form is suitable for rotation combination, for conversion between quaternions and Euler angles, or for user interface. The silhouette point of an m-surface with respect to a projection from n-space to l-space (m $\leq$ l $\u3c$ n) has been defined. The close relationship between silhouette and envelope turned out to be useful in explaining some phenomena with a surface defined by the envelope theorem and in understanding the image of a hypersurface. Several geometric properties and phenomena in 4-space have been demonstrated by computer generated pictures and animations. They include the ambiguity caused by projection, the degeneracy of the silhouette surface of a hypersurface, and the principal curvatures of a hypersurface. The architecture of an interactive 4D visualization system has been presented. The system was built on z-buffer based graphics workstations. Algorithms and data structures of polygonalization, point refinement, merging polygons, and visibility determination in 4-space are discussed. Finally several examples have been presented to illustrate the application of the visualization system: tool path generation for NC machines, collision detection and analysis for robot motion planning, and visualization of electron density data of a virus for molecular biology. Supplementary media for this thesis, a video tape and a set of color slides, is available in the Film Library of Purdue University