Bisector fields and pencils of conics, arXiv:2305.08324
Authors: Bruce Olberding, Elaine A. Walker
Abstract: Working over a field of characteristic other than 2, we show that the degenerations of the conics in a nontrivial pencil of affine conics form an arrangement of pairs of lines that we call a bisector field, a maximal set B of paired lines such that each line in B simultaneously bisects each pair in B. Conversely, every bisector field arises this way from a pencil of affine conics.
Submitted 14 May, 2023; originally announced May 2023
Authors: Bruce Olberding, Elaine A. Walker
Abstract: Working over a field of characteristic other than 2, we show that the degenerations of the conics in a nontrivial pencil of affine conics form an arrangement of pairs of lines that we call a bisector field, a maximal set B of paired lines such that each line in B simultaneously bisects each pair in B. Conversely, every bisector field arises this way from a pencil of affine conics.
Submitted 14 May, 2023; originally announced May 2023
Read the [pdf].
Bisector fields of quadrilaterals, arXiv:2305.11762
Authors: Bruce Olberding, Elaine A. Walker
Abstract: Working over a field of characteristic other than 2, we examine a relationship between quadrilaterals and the pencil of conics passing through their vertices. Asymptotically, such a pencil of conics is what we call a bisector field, a set 𝔹 of paired lines such that each line ℓ in 𝔹 simultaneously bisects each pair in 𝔹 in the sense that ℓ crosses the pairs of lines in 𝔹 in pairs of points that all share the same midpoint. We show that a quadrilateral induces a geometry on the affine plane via an inner product, under which we examine pencils of conics and pairs of bisectors of a quadrilateral. We show also how bisectors give a new interpretation of some classically studied features of quadrangles, such as the nine-point conic. Submitted 19 May, 2023; originally announced May 2023.
Authors: Bruce Olberding, Elaine A. Walker
Abstract: Working over a field of characteristic other than 2, we examine a relationship between quadrilaterals and the pencil of conics passing through their vertices. Asymptotically, such a pencil of conics is what we call a bisector field, a set 𝔹 of paired lines such that each line ℓ in 𝔹 simultaneously bisects each pair in 𝔹 in the sense that ℓ crosses the pairs of lines in 𝔹 in pairs of points that all share the same midpoint. We show that a quadrilateral induces a geometry on the affine plane via an inner product, under which we examine pencils of conics and pairs of bisectors of a quadrilateral. We show also how bisectors give a new interpretation of some classically studied features of quadrangles, such as the nine-point conic. Submitted 19 May, 2023; originally announced May 2023.