Closed cap condition under the cap construction algorithm

Abstract

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.

Figures (8)

• Figure 1: A possible cap of a square is glued to the square along their boundary to form the boundary surface of a tetrahedron.
• Figure 2: Angular defect at $v_k$.
• Figure 3: The cap construction algorithm applied on a square octagon, as described in Example \ref{['ex:antiprism']}
• Figure 4: Cap construction where $P$ results in $\hat{P}$ that is closed.
• Figure 5: Cap construction where $P$ results in $\hat{P}$ that is not closed.

Theorems & Definitions (27)

• Definition 1.1: Polygon
• Theorem 1.3: Alexandrov
• Example 1.4
• Remark 1.5
• Definition 1.6: Closed cap condition
• Theorem 1.7
• Theorem 1.8
• Proposition 2.1
• proof
• Proposition 2.2