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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2025-04-01
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| Series: | Open Mathematics |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/math-2025-0132 |
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| Summary: | 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. |
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| ISSN: | 2391-5455 |