Ellipses Inscribed In And Circumscribed About Quadrilaterals
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Ellipses Inscribed In And Circumscribed About Quadrilaterals
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Author : Alan Horwitz
language : en
Publisher: CRC Press
Release Date : 2024-05-29
Ellipses Inscribed In And Circumscribed About Quadrilaterals written by Alan Horwitz and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-05-29 with Mathematics categories.
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.
The Locus Of Centers Of Ellipses Inscribed In Quadrilaterals
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Author : Alan Horwitz
language : en
Publisher:
Release Date : 2018
The Locus Of Centers Of Ellipses Inscribed In Quadrilaterals written by Alan Horwitz and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018 with categories.
Let R be a four-sided convex polygon in the xy plane and let M1 and M2 be the midpoints of the diagonals of R. It is well-known that if E is an ellipse inscribed in R, then the center of E must lie on Z, the open line segment connecting M1 and M2. We prove the converse: If (h,k) lies in Z, then there is a unique ellipse with center (h,k) inscribed in R. This completely characterizes the locus of centers of ellipses inscribed in R. This result was proved by the author in (“Finding ellipses and hyperbolas tangent to two, three, or four given lines”, Southwest Journal of Pure and Applied Mathematics 1(2002), 6-32), but the approach given here is decidedly different and much shorter and more succinct. In addition, we are also able to prove that there is a unique ellipse of maximal area inscribed in R. The approach in this paper is to use a theorem of Marden relating the foci of an ellipse tangent to the lines thru the sides of a triangle and the zeros of a partial fraction expansion.
Finding Ellipses What Blaschke Products Poncelet S Theorem And The Numerical Range Know About Each Other
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Author : Ulrich Daepp
language : en
Publisher: American Mathematical Soc.
Release Date : 2018
Finding Ellipses What Blaschke Products Poncelet S Theorem And The Numerical Range Know About Each Other written by Ulrich Daepp and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018 with Conic sections categories.
Mathematicians delight in finding surprising connections between seemingly disparate areas of mathematics. Whole domains of modern mathematics have arisen from exploration of such connections--consider analytic number theory or algebraic topology. Finding Ellipses is a delight-filled romp across a three-way unexpected connection between complex analysis, linear algebra, and projective geometry. The book begins with Blaschke products, complex-analytic functions that are generalizations of disk automorphisms. In the analysis of Blaschke products, we encounter, in a quite natural way, an ellipse inside the unit disk. The story continues by introducing the reader to Poncelet's theorem--a beautiful result in projective geometry that ties together two conics and, in particular, two ellipses, one circumscribed by a polygon that is inscribed in the second. The Blaschke ellipse and the Poncelet ellipse turn out to be the same ellipse, and the connection is illuminated by considering the numerical range of a $2 \times 2$ matrix. The numerical range is a convex subset of the complex plane that contains information about the geometry of the transformation represented by a matrix. Through the numerical range of $n \times n$ matrices, we learn more about the interplay between Poncelet's theorem and Blaschke products. The story ranges widely over analysis, algebra, and geometry, and the exposition of the deep and surprising connections is lucid and compelling. Written for advanced undergraduates or beginning graduate students, this book would be the perfect vehicle for an invigorating and enlightening capstone exploration. The exercises and collection of extensive projects could be used as an embarkation point for a satisfying and rich research project. You are invited to read actively using the accompanying interactive website, which allows you to visualize the concepts in the book, experiment, and develop original conjectures.
Analytical Conics
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Author : Duncan M'Laren Young Sommerville
language : en
Publisher:
Release Date : 1924
Analytical Conics written by Duncan M'Laren Young Sommerville and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1924 with Conic sections categories.
Index To Mathematical Problems 1975 1979
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Author : Stanley Rabinowitz
language : en
Publisher: MathPro Press
Release Date : 1999
Index To Mathematical Problems 1975 1979 written by Stanley Rabinowitz and has been published by MathPro Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999 with Mathematics categories.
Oxford Cambridge And Dublin Messenger Of Mathematics
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Author :
language : en
Publisher:
Release Date : 1866
Oxford Cambridge And Dublin Messenger Of Mathematics written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1866 with categories.
Oxford Cambridge And Dublin Messenger Of Mathematics
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Author : William Allen Whitworth
language : en
Publisher:
Release Date : 1866
Oxford Cambridge And Dublin Messenger Of Mathematics written by William Allen Whitworth and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1866 with Mathematics categories.
The Oxford Cambridge And Dublin Messenger Of Mathematics
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Author :
language : en
Publisher:
Release Date : 1866
The Oxford Cambridge And Dublin Messenger Of Mathematics written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1866 with Mathematics categories.
The Messenger Of Mathematics
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Author :
language : en
Publisher:
Release Date : 1866
The Messenger Of Mathematics written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1866 with Mathematics categories.
Research Problems In Discrete Geometry
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Author : Peter Brass
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-01-27
Research Problems In Discrete Geometry written by Peter Brass and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-01-27 with Mathematics categories.
This book is the result of a 25-year-old project and comprises a collection of more than 500 attractive open problems in the field. The largely self-contained chapters provide a broad overview of discrete geometry, along with historical details and the most important partial results related to these problems. This book is intended as a source book for both professional mathematicians and graduate students who love beautiful mathematical questions, are willing to spend sleepless nights thinking about them, and who would like to get involved in mathematical research.