Solving the quartic by conics
Two conic sections C1{C}_{1} and C2{C}_{2} in the Euclidean plane that pass through two given points can generally have two further points of intersection. It is shown how these can be constructed using compass and ruler. The idea is to construct the degenerate conics in the pencil of the two conics...
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| Format: | Article |
| Language: | English |
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De Gruyter
2025-04-01
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| Series: | Open Mathematics |
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| Online Access: | https://doi.org/10.1515/math-2025-0132 |
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| author | Halbeisen Lorenz Hungerbühler Norbert |
| author_facet | Halbeisen Lorenz Hungerbühler Norbert |
| author_sort | Halbeisen Lorenz |
| collection | DOAJ |
| description | Two conic sections C1{C}_{1} and C2{C}_{2} in the Euclidean plane that pass through two given points can generally have two further points of intersection. It is shown how these can be constructed using compass and ruler. The idea is to construct the degenerate conics in the pencil of the two conics C1{C}_{1} and C2{C}_{2}. Their intersections are then the four intersection points of C1{C}_{1} and C2{C}_{2}. The same idea is then used to reduce a general quartic equation to a cubic equation and to solve it. This is performed by interpreting the solutions of the quartic as intersections of two complex conic sections. The degenerate complex conics in their pencil can then be found through a cubic equation. |
| format | Article |
| id | doaj-art-2c20daedcb5047bd80ce779a033fd9d7 |
| institution | OA Journals |
| issn | 2391-5455 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj-art-2c20daedcb5047bd80ce779a033fd9d72025-08-20T02:28:49ZengDe GruyterOpen Mathematics2391-54552025-04-0123115916610.1515/math-2025-0132Solving the quartic by conicsHalbeisen Lorenz0Hungerbühler Norbert1Department of Mathematics, ETH Zentrum, Rämistrasse 101, 8092 Zürich, SwitzerlandDepartment of Mathematics, ETH Zentrum, Rämistrasse 101, 8092 Zürich, SwitzerlandTwo conic sections C1{C}_{1} and C2{C}_{2} in the Euclidean plane that pass through two given points can generally have two further points of intersection. It is shown how these can be constructed using compass and ruler. The idea is to construct the degenerate conics in the pencil of the two conics C1{C}_{1} and C2{C}_{2}. Their intersections are then the four intersection points of C1{C}_{1} and C2{C}_{2}. The same idea is then used to reduce a general quartic equation to a cubic equation and to solve it. This is performed by interpreting the solutions of the quartic as intersections of two complex conic sections. The degenerate complex conics in their pencil can then be found through a cubic equation.https://doi.org/10.1515/math-2025-0132intersection of conicspencils of conicsquartic equationscubic equationsruler and compass constructions51m0551m1512d10 |
| spellingShingle | Halbeisen Lorenz Hungerbühler Norbert Solving the quartic by conics Open Mathematics intersection of conics pencils of conics quartic equations cubic equations ruler and compass constructions 51m05 51m15 12d10 |
| title | Solving the quartic by conics |
| title_full | Solving the quartic by conics |
| title_fullStr | Solving the quartic by conics |
| title_full_unstemmed | Solving the quartic by conics |
| title_short | Solving the quartic by conics |
| title_sort | solving the quartic by conics |
| topic | intersection of conics pencils of conics quartic equations cubic equations ruler and compass constructions 51m05 51m15 12d10 |
| url | https://doi.org/10.1515/math-2025-0132 |
| work_keys_str_mv | AT halbeisenlorenz solvingthequarticbyconics AT hungerbuhlernorbert solvingthequarticbyconics |