Finding Ellipses: What Blaschke Products, Poncelet’s Theorem, and the Numerical Range Know about Each Other is a mathematics book on "some surprising connections among complex analysis, geometry, and linear algebra",[1] and on the connected ways that ellipses can arise from other subjects of study in all three of these fields.
[3] Finding Ellipses studies a connection between Blaschke products, Poncelet's closure theorem, and the numerical range of matrices.
In the main case considered by the book, there are three distinct given points
These triples of preimages form triangles inscribed in the unit circle, and (it turns out) they all circumscribe an ellipse with foci at
The first part develops the mathematics of Blaschke products, Poncelet's closure theorem, and numerical ranges separately, before revealing the close connections between them.
The second part of the book generalizes these ideas to higher-order Blaschke products, larger matrices, and Poncelet-like results for the corresponding numerical ranges, which generalize ellipses.
[1] The third part consists of projects and exercises for students to develop this material beyond the exposition in the book.
[1] An online collection of web applets allow students to experiment with the constructions in the book.
[1][2] The first part of the book uses only standard undergraduate mathematics, but the second part is more demanding, and reviewer Bill Satzer writes that "even the best students might find themselves paging backward and forward in the book, feeling frustrated while trying to make connections".
[1] Despite that, Line Baribeau writes that it is "clear and engaging", and appealing in its use of modern topics.
[3] Yunus Zeytuncu is even more positive, calling it a "delight" that "realizes the dream" of bringing this combination of disciplines together into a neat package that is accessible to undergraduates.