1st Edition

Ellipses Inscribed in, and Circumscribed about, Quadrilaterals

By Alan Horwitz Copyright 2024
    146 Pages
    by Chapman & Hall

    146 Pages
    by Chapman & Hall

    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.

    Preface

    I Ellipses Inscribed in Quadrilaterals

    1 Locus of Centers, Maximal Area, and Minimal Eccentricity

    1.1 Locus of Centers

    1.2 Maximal Area

    1.3 Minimal Eccentricity

    1.4 Examples

    1.5 Trapezoids

    2 Ellipses inscribed in parallelograms

    2.1 Preliminary Results

    2.2 Maximal Area

    2.3 Minimal Eccentricity

    2.4 Special Result for Rectangles

    2.5  Orthogonal Least Squares

    2.6 Example
    2.7 Tangency Chords and Conjugate Diameters Parallel

    to the Diagonals

    3 Area Inequality

    4 Midpoint Diagonal Quadrilaterals

    4.1 Conjugate Diameters and Tangency Chords

    4.2 Equal Conjugate Diameters and the Ellipse of Minimal Eccentricity

    4.3 Example

    5 Tangency Points as Midpoints of sides of Q

    5.1  Non–Trapezoids

         5.1.1 Example

    5.2 Trapezoids

    6 Dynamics of Ellipses inscribed in Quadrilaterals

        6.0.1 Examples

    7 Algorithms for Inscribed Ellipses

    7.1 Transformations

    7.2 Maximal Area and Minimal Eccentricity

    for Non–parallelograms

    7.3 Maximal Area and Minimal Eccentricity for Parallelograms

    7.4 Dynamics

    II    Ellipses Circumscribed about Quadrilaterals

    8   Non–parallelograms

    8.1 Equation
    8.2 Minimal Eccentricity

    8.3 Minimal Area

    8.4 Examples

    9 Parallelograms

    9.1 Equation

    9.2 Minimal Eccentricity

    9.3 Minimal Area

    9.4 Example

    9.5 Area Inequality

    III Inscribed versus Circumscribed

    10 Bielliptic Quadrilaterals

    11 Algorithms for Circumscribed Ellipses

    11.1 Minimal Area and Minimal Eccentricity for Non–parallelograms

    11.2 Minimal Area and Minimal Eccentricity for Parallelograms   

    12 Related Research and Open Questions

    12.1 Arc Length

    12.2 Bielliptic Quadrilaterals

    12.3 Other Families of Curves

    Bibliography

    Appendix

    (1) General Results on Ellipses

        (a)Coefficient Formulas

               (b) Conjugate Diameters and Tangency Chords

    (2) Proofs of Some Earlier Results

               (a) Proposition 1.1

               (b) Proposition 1.2

               (c) Proposition 1.3

               (d) Lemma 2.1

               (e) Lemma 2.4

    Biography

    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.