Orthogonal Families of Bicircular Quartics, Quadratic Differentials, and Edwards Normal Form
Orthogonal families of bicircular quartics are naturally viewed as pairs of singular foliations of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi mathvariant="double-s...
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MDPI AG
2023-09-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/9/870 |
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| author | Joel C. Langer David A. Singer |
| author_facet | Joel C. Langer David A. Singer |
| author_sort | Joel C. Langer |
| collection | DOAJ |
| description | Orthogonal families of bicircular quartics are naturally viewed as pairs of singular foliations of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi mathvariant="double-struck">C</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula> by vertical and horizontal trajectories of a non-vanishing quadratic differential. Yet the identification of these trajectories with real quartics in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi><msup><mrow><mi mathvariant="normal">P</mi></mrow><mn>2</mn></msup></mrow></semantics></math></inline-formula> is subtle. Here, we give an efficient, geometric argument in the course of updating the classical theory of confocal families in the modern language of quadratic differentials and the Edwards normal form for elliptic curves. In particular, we define a <i>parameterized Edwards transformation,</i> providing explicit birational equivalence between each curve in a confocal family and a fixed curve in normal form. |
| first_indexed | 2024-03-10T23:02:14Z |
| format | Article |
| id | doaj.art-35d73093d7614b07ab86951c0d6d9e0b |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T23:02:14Z |
| publishDate | 2023-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-35d73093d7614b07ab86951c0d6d9e0b2023-11-19T09:32:51ZengMDPI AGAxioms2075-16802023-09-0112987010.3390/axioms12090870Orthogonal Families of Bicircular Quartics, Quadratic Differentials, and Edwards Normal FormJoel C. Langer0David A. 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, USAOrthogonal families of bicircular quartics are naturally viewed as pairs of singular foliations of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi mathvariant="double-struck">C</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula> by vertical and horizontal trajectories of a non-vanishing quadratic differential. Yet the identification of these trajectories with real quartics in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi><msup><mrow><mi mathvariant="normal">P</mi></mrow><mn>2</mn></msup></mrow></semantics></math></inline-formula> is subtle. Here, we give an efficient, geometric argument in the course of updating the classical theory of confocal families in the modern language of quadratic differentials and the Edwards normal form for elliptic curves. In particular, we define a <i>parameterized Edwards transformation,</i> providing explicit birational equivalence between each curve in a confocal family and a fixed curve in normal form.https://www.mdpi.com/2075-1680/12/9/870Edwards normal formquadratic differentialconfocal familybicircular quarticfocielliptic curve |
| spellingShingle | Joel C. Langer David A. Singer Orthogonal Families of Bicircular Quartics, Quadratic Differentials, and Edwards Normal Form Axioms Edwards normal form quadratic differential confocal family bicircular quartic foci elliptic curve |
| title | Orthogonal Families of Bicircular Quartics, Quadratic Differentials, and Edwards Normal Form |
| title_full | Orthogonal Families of Bicircular Quartics, Quadratic Differentials, and Edwards Normal Form |
| title_fullStr | Orthogonal Families of Bicircular Quartics, Quadratic Differentials, and Edwards Normal Form |
| title_full_unstemmed | Orthogonal Families of Bicircular Quartics, Quadratic Differentials, and Edwards Normal Form |
| title_short | Orthogonal Families of Bicircular Quartics, Quadratic Differentials, and Edwards Normal Form |
| title_sort | orthogonal families of bicircular quartics quadratic differentials and edwards normal form |
| topic | Edwards normal form quadratic differential confocal family bicircular quartic foci elliptic curve |
| url | https://www.mdpi.com/2075-1680/12/9/870 |
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