PATHS OF RECTANGLES INSCRIBED IN LINES OVER FIELDS
BRUCE OLBERDING AND ELAINE A. WALKER
Abstract. Let k be a field. By an algebraic rectangle in k2 we mean four points in k2 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 k2 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. (Read the pdf on the arXiv.)
The following video illustrates how the rectangle locus changes as one line in the configuration is rotated. The locus moves through degenerate and non-degenerate configurations. When the blue lines merge, it is the image of a hyperbola under an affine transformation.
The following video illustrates how the rectangle locus changes as one line is translated in the configuration. The diagonals of the underlying quadrilateral are shown as well. When these diagonals are perpendicular, the locus is a degenerate hyperbola.
The following video illustrates how the rectangle locus changes as two lines are translated in the configuration. At the end of the video the hyperbola flattens. This locus, when completely flattened, is the image of a hyperbola under an affine transformation.