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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.
Preface
Ellipses Inscribed in Quadrilaterals
Locus of Centers, Maximal Area, and Minimal Eccentricity
Locus of Centers
Maximal Area
Minimal Eccentricity
Examples
Trapezoids
Ellipses Inscribed in Parallelograms
Preliminary Results
Maximal Area
Minimal Eccentricity
Special Result for Rectangles
Orthogonal Least Squares
Example
Tangency Chords and Conjugate Diameters Parallel to the Diagonals
Area Inequality
Midpoint Diagonal Quadrilaterals
Conjugate Diameters and Tangency Chords
Equal Conjugate Diameters and the Ellipse of Minimal Eccentricity
Example
Tangency Points as Midpoints of Sides of Q
Non-Trapezoids
Example
Trapezoids
Dynamics of Ellipses Inscribed in Quadrilaterals
Examples
Algorithms for Inscribed Ellipses
Transformations
Maximal Area and Minimal Eccentricity for Non-Parallelograms
Maximal Area and Minimal Eccentricity for Parallelograms
Dynamics
Ellipses Circumscribed about Quadrilaterals
Non-Parallelograms
Equation
Minimal Eccentricity
Minimal Area
Examples
Parallelograms
Equation
Minimal Eccentricity
Minimal Area
Example
Area Inequality
Inscribed versus Circumscribed
Bielliptic Quadrilaterals
Algorithms for Circumscribed Ellipses
Minimal Area and Minimal Eccentricity for Non-Parallelograms
Minimal Area and Minimal Eccentricity for Parallelograms
Related Research and Open Questions
Arc Length
Bielliptic Quadrilaterals
Other Families of Curves
Appendix
General Results on Ellipses
Coefficient formulas
Conjugate diameters
Proofs of Some Earlier Results
References
Index