This book is based on the authors' research on rendering images of higher dimensional fractals by a distance estimation technique. It is self-contained, giving a careful treatment of both the known techniques and the authors' new methods. The distance estimation technique was originally applied to Julia sets and the Mandelbrot set in the complex plane. It was justified, through the work of Douady and Hubbard, by deep results in complex analysis. In this book the authors generalise the distance estimation to quaternionic and other higher dimensional fractals, including fractals derived from iteration in the Cayley numbers (octonionic fractals). The generalization is justified by new geometric arguments that circumvent the need for complex analysis. This puts on a firm footing the authors' present work and the second author's earlier work with John Hart and Dan Sandin. The results of this book will be of great interest to mathematicians and computer scientists interested in fractals and computer graphics.
Contents: Introduction: Hypercomplex Iteractions in a Nutshell; Deterministic Fractals and Distance Estimation; Classical Analysis: Complex and Quaternionic: Distance Estimation in Complex Space; Quaternion Analysis; Quaternions and the Dirac String Trick; Hypercomplex Iteractions: Quaternion Mandelbrot Sets; Distance Estimation in Higher Dimensional Spaces; Inverse Iteraction, Ray Tracing and Virtual Reality: Inverse Iteraction: An Interactive Visualization; Ray Tracing Methods by Distance Estimation; Quaternion Deterministic Fractals in Virtual Reality.