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The main focus of this book is disseminating research results regarding the pencil of ellipses inscribing arbitrary convex quadrilaterals. In particular, the author proves that there is a unique ellipse of maximal area, EA, and a unique ellipse of minimal eccentricity, EI, inscribed in Q. Similar results are also proven for ellipses passing through the vertices of a convex quadrilateral along with some comparisons with inscribed ellipses. Special results are also given for parallelograms.
Researchers in geometry and applied mathematics will find this unique book of interest. Software developers, image processors along with geometers, mathematicians, and statisticians will be very interested in this treatment of the subject of inscribing and circumscribing ellipses with the comprehensive treatment here.
Most of the results in this book were proven by the author in several papers listed in the references at the end. This book gathers results in a unified treatment of the topics while also shortening and simplifying many of the proofs.
This book also contains a separate section on algorithms for finding ellipses of maximal area or of minimal eccentricity inscribed in, or circumscribed about, a given quadrilateral and for certain other topics treated in this book.
Anyone who has taken calculus and linear algebra and who has a basic understanding of ellipses will find it accessible.
List of contents
1. Locus of Centers, Maximal Area, and Minimal Eccentricity. 2. Ellipses inscribed in parallelograms. 3. Area Inequality. 4. Midpoint Diagonal Quadrilaterals. 5.Tangency Points as Midpoints of sides of Q. 6. Dynamics of Ellipses inscribed in Quadrilaterals. 7. Algorithms for Inscribed Ellipses. 8.Non-parallelograms. 9. Parallelograms. 10. Bielliptic Quadrilaterals. 11. Algorithms for Circumscribed Ellipses. 12. Related Research and Open Questions.
About the author
Alan Horwitz holds a Ph.D. in Mathematics from Temple University in Philadelphia, PA, USA and is Professor Emeritus at Penn State University, Brandywine Campus where he served for 28 years. He has published 43 articles in refereed mathematics journals in various areas of mathematics. This is his first book.
Summary
The main focus is disseminating research results regarding the pencil of ellipses inscribing arbitrary convex quadrilaterals. In particular, the author proves that there is a unique ellipse of maximal area, EA, and a unique ellipse of minimal eccentricity, EI, inscribed in Q.
