Every polygon $P$ can be companioned by a cap polygon $\hat P$ such that $P$ and $\hat P$ serve as two parts of the boundary surface of a polyhedron $V$. Pairs of vertices on $P$ and $\hat P$ are identified successively to become vertices of $V$. In this paper, we study the cap construction that asserts equal angular defects at these pairings. We exhibit a linear relation that arises from the cap construction algorithm, which in turn demonstrates an abundance of polygons that satisfy the closed cap condition, that is, those that can successfully undergo the cap construction process.

翻译：每个多边形$P$均可伴随一个帽多边形$\hat P$，使得$P$与$\hat P$作为多面体$V$边界曲面的两部分。$P$与$\hat P$上的顶点对被依次等同，成为$V$的顶点。本文研究帽构造中在这些配对处断言等角度亏缺的性质。我们揭示了由帽构造算法导出的线性关系，进而证明了满足闭合帽条件的多边形大量存在，即那些能够成功经历帽构造过程的多边形。