An euclidean affine invariant of quadrilaterals

TL;DR

The paper introduces a new affine-invariant measure for quadrilaterals based on specific side and diagonal lengths, and applies it to derive a formula for the area of sectors in symmetric quadrics.

Contribution

It presents a novel affine-invariant for quadrilaterals and demonstrates its application to calculating areas in symmetric quadrics.

Findings

The invariant remains unchanged under affine transformations.

Derived a formula for sector areas in symmetric quadrics.

Established a link between the invariant and geometric properties of quadrilaterals.

Abstract

A certain real number, depending on two neighbouring sides of a quadrilateral and the diagonal meeting these two sides at their common point, is shown to be invariant under affinity. As an application we demonstrate a nice formula for the area of a finite sector at centre of a planar quadric with point symmetry.