Synopses & Reviews
Geometric tomography deals with the retrieval of information about a geometric object from data concerning its projections (shadows) on planes or cross-sections by planes. It is a geometric relative of computerized tomography, which reconstructs an image from X-rays of a human patient. The subject overlaps with convex geometry and employs many tools from that area, including some formulas from integral geometry. It also has connections to discrete tomography, geometric probing in robotics and to stereology. This comprehensive study provides a rigorous treatment of the subject. Although primarily meant for researchers and graduate students in geometry and tomography, brief introductions, suitable for advanced undergraduates, are provided to the basic concepts. More than 70 illustrations are used to clarify the text. The book also presents 66 unsolved problems. Each chapter ends with extensive notes, historical remarks, and some biographies. This new edition includes numerous updates and improvements, with some 300 new references bringing the total to over 800.
Synopsis
A research text which makes unsolved problems accessible to advanced undergraduates. It rigorously develops the new subject of geometric tomography, which deals with the retrieval of information about a geometric object from data concerning its projections on planes or cross-sections by planes.
Synopsis
Develops the new field of retrieving information about geometric objects from projections on planes.
Synopsis
Geometric tomography overlaps with convex geometry and employs many tools from that area, including some formulas from integral geometry. This comprehensive study provides a rigorous treatment of the subject. Although meant primarily for researchers and graduate students in geometry and tomography, brief introductions, suitable for advanced undergraduates, are provided to the basic concepts.
Synopsis
A comprehensive, rigorous treatment, with 66 unsolved problems, over 70 illustrations, and over 800 references.
About the Author
'Richard Gardner has been Professor of Mathematics at Western Washington University since 1991. He is the author of 70 papers and founded geometric tomography as a subject in its own right with the publication of the first edition of this book in 1995.'
Table of Contents
0. Background material; 1. Parallel X-rays of planar convex bodies; 2. Parallel X-rays in n dimensions; 3. Projections and projection functions; 4. Projection bodies and volume inequalities; 5. Point X-rays; 6. Chord functions and equichordal problems; 7. Sections, section functions, and point X-rays; 8. Intersection bodies and volume inequalities; 9. Estimates from projection and section functions; A. Mixed volumes and dual mixed volumes; B. Inequalities; C. Integral transforms.