AI 中文总结
本文为凸四边形空间构建了一个模型，每个凸四边形恰相似于其中一员，并用4-胞腔参数化，证明梯形构成3维闭子集且分割模型，附SageMath交互工具。
AI 中文摘要
本文描述了凸四边形空间的一个“模型”，即平面中凸四边形的一个集合Q，它给出了相似凸四边形等价类的“横截面”，其意义在于每个凸四边形都相似于Q中恰好一个成员。该模型还具有这样的性质：顶点几乎相同的四边形在模型中按Hausdorff度量彼此接近。我们还用4-胞腔对模型进行参数化，并利用该参数化描述和研究各类四边形。例如，我们证明梯形构成模型的一个3维闭子集，该子集将模型分割为3个不相交的开集，每个开集与4-胞腔同胚。我们使用SageMath编写本文的LaTeX，并制作了Sage Cell交互工具以探索该模型，可在sagelets.org获取。
英文摘要
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}.
Comments18 pages, 25 figures
