THE CONIC GEOMETRY OF RECTANGLES INSCRIBED IN LINES, Proceedings of the American Math Society, to appear.
BRUCE OLBERDING AND 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. (pdf) (more images)


ASPECT RATIO AND SLOPE OF ALGEBRAIC RECTANGLES INSCRIBED IN LINES OVER FIELDS
BRUCE OLBERDING AND ELAINE A. WALKER
Abstract. Let k be a field. By an algebraic rectangle in k^2 we mean four points in k^2 subject to certain conditions that in the case where k is the field of real numbers yield four vertices of a rectangle. We study algebraic rectangles inscribed in lines in k^2 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. (pdf) (more images)