Isosurfaces: Geometry, Topology, and Algorithms, 1st Edition (Hardback) book cover


Geometry, Topology, and Algorithms, 1st Edition

By Rephael Wenger

A K Peters/CRC Press

488 pages | 228 Color Illus. | 228 B/W Illus.

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pub: 2013-06-24
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Ever since Lorensen and Cline published their paper on the Marching Cubes algorithm, isosurfaces have been a standard technique for the visualization of 3D volumetric data. Yet there is no book exclusively devoted to isosurfaces. Isosurfaces: Geometry, Topology, and Algorithms represents the first book to focus on basic algorithms for isosurface construction. It also gives a rigorous mathematical perspective on some of the algorithms and results.

In color throughout, the book covers the Marching Cubes algorithm and variants, dual contouring algorithms, multilinear interpolation, multiresolution isosurface extraction, isosurfaces in four dimensions, interval volumes, and contour trees. It also describes data structures for faster isosurface extraction as well as methods for selecting significant isovalues.

For designers of visualization software, the book presents an organized overview of the various algorithms associated with isosurfaces. For graduate students, it provides a solid introduction to research in this area. For visualization researchers, the book serves as a reference to the vast literature on isosurfaces.


"Visualization has long needed a solid, standard and detailed text on the algorithmic aspects of isosurface construction and use. This text will become the standard entry point into this vast literature for at least the next decade, even for researchers already accustomed to working with isosurfaces. It belongs on every professional’s shelf."

—Hamish Carr, University of Leeds

"Isosurfaces are one of the most prevalent ways to visualize three-dimensional data. This wonderful book is the first that nicely summarizes the foundations as well as the state of the art on isosurfaces. Everyone, from the novice to the expert, will find something new and interesting in this book. This book's treatment of isosurfaces goes way beyond the surface, deep into the heart and soul of this rich topic situated in between the fields of graphics, visualization, and computational geometry."

—Torsten Möller, University of Vienna (Universität Wien)

"…well written, well illustrated, and extensively referenced."

—Lyuba S. Alboul, Mathematical Reviews Clippings, January 2015

Table of Contents


What Are Isosurfaces?

Applications of Isosurfaces

Isosurface Properties

Isosurface Construction

Limitations of Isosurfaces

Multi-Valued Functions and Vector Fields

Definitions and Basic Techniques

Marching Cubes and Variants


Marching Squares

Marching Cubes

Marching Tetrahedra

Dual Contouring


Surface Nets

Dual Marching Cubes

Comparison with Marching Cubes

Multilinear Interpolation

Bilinear Interpolation: 2D

The Asymptotic Decider: 3D

Trilinear Interpolation

Isosurface Patch Construction

Definitions and Notation

Isosurface Patch Construction

Isosurface Table Construction

Marching Polyhedra Algorithm


Isosurface Generation in 4D

Definitions and Notation

Isosurface Table Generation in 4D

Marching Hypercubes

Marching Simplices

Marching Polytopes

4D Isohull

4D Surface Nets

Interval Volumes

Definitions and Notation


Automatic Table Generation

MCVol Interval Volume Properties

Tetrahedral Meshes

Convex Polyhedral Meshes

Data Structures

Uniform Grid Partitions


Span Space Priority Trees

Seed Sets

Multiresolution Tetrahedral Meshes

Bisection of Tetrahedra

Multiresolution Isosurfaces

Multiresolution Polyhedral Meshes

Multiresolution Convex Polyhedral Mesh

Multiresolution Surface Nets

Multiresolution in 4D


Counting Grid Vertices

Counting Grid Edges and Grid Cubes

Measuring Gradients

Contour Trees

Examples of Contour Trees

Definition of Contour Tree

Join, Split and Merge Trees

Constructing Join, Split and Merge Trees

Constructing Contour Trees

Theory and Proofs

Simplification of Contour Trees


Appendix A: Geometry

Appendix B: Topology

Appendix C: Graph Theory

Appendix D: Notation



Notes and Comments appear at the end of each chapter.

About the Author

Rephael Wenger is an associate professor in the Department of Computer Science and Engineering at the Ohio State University. He earned a Ph.D. from McGill University. He has published over fifty papers in computational geometry, computational topology, combinatorics, geometric modeling, and visualization.

Subject Categories

BISAC Subject Codes/Headings:
COMPUTERS / Computer Graphics
COMPUTERS / Programming / Games
MATHEMATICS / Geometry / General