On Probabilities regarding Poncelet Polygons over Finite Fields
Access status:
Open Access
Type
ThesisThesis type
Masters by ResearchAuthor/s
Ragas, Ruzzel DizonAbstract
An n-sided polygon that is inscribed in a conic A and circumscribed about a conic B is called a Poncelet polygon, and we call the pair of conics (A,B) an n-Poncelet pair. In the projective plane over a finite field of characteristic not equal to 2, we study Poncelet polygons and ...
See moreAn n-sided polygon that is inscribed in a conic A and circumscribed about a conic B is called a Poncelet polygon, and we call the pair of conics (A,B) an n-Poncelet pair. In the projective plane over a finite field of characteristic not equal to 2, we study Poncelet polygons and n-Poncelet pairs, with emphasis on the cases n = 3 and n=4. In particular, we discuss the construction of Poncelet polygons and derive results regarding degenerate Poncelet polygons. Moreover, we provide in-depth results regarding the construction of Poncelet triangles. For our main result, we compute the probability of obtaining a 3-Poncelet pair or a 4-Poncelet pair when we randomly select a pair of distinct conics (A,B), with A smooth or singular and B smooth, in a fixed pencil of conics. We do this for all pencils, classified up to projective automorphism, with at least one smooth conic.
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See moreAn n-sided polygon that is inscribed in a conic A and circumscribed about a conic B is called a Poncelet polygon, and we call the pair of conics (A,B) an n-Poncelet pair. In the projective plane over a finite field of characteristic not equal to 2, we study Poncelet polygons and n-Poncelet pairs, with emphasis on the cases n = 3 and n=4. In particular, we discuss the construction of Poncelet polygons and derive results regarding degenerate Poncelet polygons. Moreover, we provide in-depth results regarding the construction of Poncelet triangles. For our main result, we compute the probability of obtaining a 3-Poncelet pair or a 4-Poncelet pair when we randomly select a pair of distinct conics (A,B), with A smooth or singular and B smooth, in a fixed pencil of conics. We do this for all pencils, classified up to projective automorphism, with at least one smooth conic.
See less
Date
2026Rights statement
The author retains copyright of this thesis. It may only be used for the purposes of research and study. It must not be used for any other purposes and may not be transmitted or shared with others without prior permission.Faculty/School
Faculty of Science, School of Mathematics and StatisticsAwarding institution
The University of SydneyShare