1st Edition
Ellipses Inscribed in, and Circumscribed about, Quadrilaterals
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.
