A Model for the space of convex quadrilaterals

Abstract

This paper describes a 'model' for the space of convex quadrilaterals, that is, a set $\mathbb{Q}$ of convex quadrilaterals in the plane which gives a 'cross-section' of the equivalence classes of similar convex quadrilaterals, in the sense that every convex quadrilateral is similar to exactly one member of $\mathbb{Q}$. It is also has the property that quadrilaterals which have nearly the same vertices are close to each other in the model in the Hausdorff metric. We also parameterize our model with the 4-cell and use that to describe and investigate various classes of quadrilaterals. For example, we show that the trapezoids form a 3-dimensional closed subset of the model which separates the model into 3 disjoint open sets, each homeomorphic with a 4-cell. We used \textbf{SageMath} to compose the latex for this note, and to make the \textbf{Sage Cell Interacts} to explore the model, available at \textit{sagelets.org}.