Hypercomplex iterations: distance estimation and higher dimensional fractals

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 NutshellDeterministic Fractals and Distance EstimationClassical Analysis: Complex and Quaternionic:Distance Estimation in Complex SpaceQuaternion AnalysisQuaternions and the Dirac String TrickHypercomplex Iteractions:Quaternion Mandelbrot SetsDistance Estimation in Higher Dimensional SpacesInverse Iteraction, Ray Tracing and Virtual Reality:Inverse Iteraction: An Interactive VisualizationRay Tracing Methods by Distance EstimationQuaternion Deterministic Fractals in Virtual RealityReadership: Mathematicians and computer scientists.