With thanks to my mentor Foci of Algebraic Curves November 7, 2008 - PDF document

11/23/2008 With thanks to my mentor Foci of Algebraic Curves November 7, 2008 Outline of Talk What are Foci? Plücker Formulation 1

11/23/2008 Example 1: The k ‐ ellipse The 3 ‐ ellipse The innermost curve is convex 2

11/23/2008 Non ‐ smooth example A 3 ‐ ellipse with 8 (of nine) foci Example 2: Lemniscates Lemniscates can be anything E.T.: Phone Home! des cassiniennes alien ... par Alain Esculier 3

11/23/2008 Example 3: Siebeck’s Theorem Bôcher ‐ Grace Theorem Class Three 4

11/23/2008 Example 4: Non ‐ Euclidean polynomials n = 4 example The hyperbolic ellipse 5

11/23/2008 Example 5: Eigenvalues Kippenhann’s Theorem Example of an RZ curve The dual curve 6

11/23/2008 The dual of this RZ curve is not RZ curves and their duals convex, but its convex hull is a numerical range Example 6: Schwarz Reflection 7

11/23/2008 Application: Quadrature Domains Conclusion: Foci are interesting 8

11/23/2008 The End 9