Poncelet's porism in the finite real plane

Abstract

In this article, we continue exploring Poncelet's curves, leveraging their connection to the numerical ranges of special matrices. This connection was recently discovered by (H.-L. Gau and P.Y. Wu, Numerical range of S(ϕ), Lin. Multilin. Alg. 45 (1998), pp. 49–73) and (B. Mirman, Numerical ranges and Poncelet curves, Lin. Alg. Appl. 281 (1998), pp. 59–85). Main result: separated Poncelet's curves may be generated by alternating (in sign) functions and matrices of a special structure. Poncelet's polygons for separated Poncelet's curves have a typically saw-tooth shape.

Keywords:

• Blaschke product
• Steiner's problems
• Malyshev's theorem