Confocal Families of Hyperbolic Conics via Quadratic Differentials
We apply the theory of quadratic differentials, to present a classification of orthogonal pairs of foliations of the hyperbolic plane by hyperbolic conics. Light rays are represented by trajectories of meromorphic differentials, and mirrors are represented by trajectories of the quadratic differenti...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-05-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/12/6/507 |
| _version_ | 1827738601748692992 |
|---|---|
| author | Joel Langer David Singer |
| author_facet | Joel Langer David Singer |
| author_sort | Joel Langer |
| collection | DOAJ |
| description | We apply the theory of quadratic differentials, to present a classification of orthogonal pairs of foliations of the hyperbolic plane by hyperbolic conics. Light rays are represented by trajectories of meromorphic differentials, and mirrors are represented by trajectories of the quadratic differential that represents the geometric mean of two such differentials. Using the notion of a hyperbolic conic as a mirror, we classify the types of orthogonal pairs of foliations of the hyperbolic plane by confocal conics. Up to diffeomorphism, there are nine types: three of these types admit one parameter up to isometry; the remaining six types are unique up to isometry. The families include all possible hyperbolic conics. |
| first_indexed | 2024-03-11T02:46:40Z |
| format | Article |
| id | doaj.art-4af0c29f25e540c1ae8d16034fe5b335 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T02:46:40Z |
| publishDate | 2023-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-4af0c29f25e540c1ae8d16034fe5b3352023-11-18T09:16:02ZengMDPI AGAxioms2075-16802023-05-0112650710.3390/axioms12060507Confocal Families of Hyperbolic Conics via Quadratic DifferentialsJoel Langer0David Singer1Department of Mathematics, Applied Math and Statistics, Case Western Reserve University, Cleveland, OH 44106-7058, USADepartment of Mathematics, Applied Math and Statistics, Case Western Reserve University, Cleveland, OH 44106-7058, USAWe apply the theory of quadratic differentials, to present a classification of orthogonal pairs of foliations of the hyperbolic plane by hyperbolic conics. Light rays are represented by trajectories of meromorphic differentials, and mirrors are represented by trajectories of the quadratic differential that represents the geometric mean of two such differentials. Using the notion of a hyperbolic conic as a mirror, we classify the types of orthogonal pairs of foliations of the hyperbolic plane by confocal conics. Up to diffeomorphism, there are nine types: three of these types admit one parameter up to isometry; the remaining six types are unique up to isometry. The families include all possible hyperbolic conics.https://www.mdpi.com/2075-1680/12/6/507quadratic differentialhyperbolic conicPoincaré discconfocal families |
| spellingShingle | Joel Langer David Singer Confocal Families of Hyperbolic Conics via Quadratic Differentials Axioms quadratic differential hyperbolic conic Poincaré disc confocal families |
| title | Confocal Families of Hyperbolic Conics via Quadratic Differentials |
| title_full | Confocal Families of Hyperbolic Conics via Quadratic Differentials |
| title_fullStr | Confocal Families of Hyperbolic Conics via Quadratic Differentials |
| title_full_unstemmed | Confocal Families of Hyperbolic Conics via Quadratic Differentials |
| title_short | Confocal Families of Hyperbolic Conics via Quadratic Differentials |
| title_sort | confocal families of hyperbolic conics via quadratic differentials |
| topic | quadratic differential hyperbolic conic Poincaré disc confocal families |
| url | https://www.mdpi.com/2075-1680/12/6/507 |
| work_keys_str_mv | AT joellanger confocalfamiliesofhyperbolicconicsviaquadraticdifferentials AT davidsinger confocalfamiliesofhyperbolicconicsviaquadraticdifferentials |