**Rectangles conformally inscribed in lines**, J. Geom. 113, 9 (2022)

**Authors: Bruce Olberding, Elaine A. Walker**

Abstract: A parallelogram is conformally inscribed in four lines in the plane if it is inscribed in a scaled copy of the configuration of four lines. We describe the geometry of the three-dimensional Euclidean space whose points are the parallelograms conformally inscribed in sequence in these four lines. In doing so, we describe the flow of inscribed rectangles by introducing a compact model of the rectangle inscription problem.

Submitted 2 August, 2021; originally announced August 2021.

[pdf, additional images)

**Paths of rectangles inscribed in lines over fields**, Beitr Algebra Geom (2022)

Authors: Bruce Olberding, Elaine A. Walker

Abstract: We study rectangles inscribed in lines in the plane by parametrizing these rectangles in two ways, one involving slope and the other aspect ratio. This produces two paths, one that finds rectangles with specified slope and the other rectangles with specified aspect ratio. We describe the geometry of these paths and its dependence on the choice of four lines. Our methods are algebraic and work over an arbitrary field. Submitted 16 December, 2020; v1 submitted 25 June, 2020; originally announced June 2020.

[pdf, additional images]

**The conic geometry of rectangles inscribed in lines,**Proc. Amer. Math. Soc. 149 (2021), 2625-2638

**Authors: Bruce Olberding, Elaine A. Walker**

Abstract: We develop a circle of ideas involving pairs of lines in the plane, intersections of hyperbolically rotated elliptical cones and the locus of the centers of rectangles inscribed in lines in the plane. Submitted 2 August, 2021; v1 submitted 15 August, 2019; originally announced August 2019.

[pdf, additional images]