Rigidity of bordered polyhedral surfaces

Abstract

This paper investigates the rigidity of bordered polyhedral surfaces. Using the variational principle, we show that bordered polyhedral surfaces are determined by boundary value and discrete curvatures on the interior edges. As a corollary, we reprove the classical result that two Euclidean cyclic polygons (or hyperbolic cyclic polygons) are congruent if the lengths of their sides are equal.

Appendices

Appendix A Some formulas in Trigonometry

This section is devoted to some results from trigonometry.

Proposition A.1

(Cosine law) Let $l_{1}$, $l_{2}$, $l_{3}$ be the edge lengths of a geometric triangle and $\theta _{1}$, $\theta _{2}$, $\theta _{3}$ be the opposite inner angles. Set $\{i,j,k\}=\{1,2,3\}$.

(a):

In Euclidean geometry, we have

$$\begin{aligned} \cos \theta _{i}=\frac{-l^2_{i}+l^2_{j}+l^2_{k}}{2l_{j}l_{k}},\quad 1=\frac{\cos \theta _{i}+\cos \theta _{j}\cos \theta _{k}}{\cos \theta _{j}\cos \theta _{k}}. \end{aligned}$$

(b):

In hyperbolic geometry, we have

$$\begin{aligned} \cos \theta _{i}=\frac{-\cosh l_{i}+\cosh l_{j}\cosh l_{k}}{\sinh l_{j}\sinh l_{k}},\quad \cosh l_{i}=\frac{\cos \theta _{i}+\cos \theta _{j}\cos \theta _{k}}{\sin \theta _{j}\sin \theta _{k}}. \end{aligned}$$

(c):

In spherical geometry, we have

$$\begin{aligned} \cos \theta _{i}=\frac{\cos l_{i}-\cos l_{j}\cos l_{k}}{\sin l_{j}\sin l_{k}},\quad \cos l_{i}=\frac{\cos \theta _{i}+\cos \theta _{j}\cos \theta _{k}}{\sin \theta _{j}\sin \theta _{k}}. \end{aligned}$$

Proposition A.2

(Tangent law) Let $l_{1}$, $l_{2}$, $l_{3}$ be three edges of a triangle and $\theta _{1}$, $\theta _{2}$, $\theta _{3}$ are the corresponding inner angles. Set

$$\begin{aligned} h_{i}=\cos \alpha _{i}\coth \frac{l_{i}}{2},\ h'_{i}=r_{i}\tan \frac{\theta _{i}}{2},\ s_{i}=\cos \alpha _{i}\cot \frac{l_{i}}{2},\ s'_{i}=\sinh r_{i}\tan \frac{\theta _{i}}{2}, \end{aligned}$$

where $i\in \{1,2,3\}$ and $\alpha _{i}=\frac{1}{2}(\theta _{i}-\theta _{j}-\theta _{k})$, $r_{i}=\frac{1}{2}(l_{j}+l_{k}-l_{i})$. Then $h_{i}$, $h'_{i}$ (resp. $s_{i}$, $s'_{i}$) are positive numbers independent of indices in hyperbolic (resp. spherical) geometry.

Appendix B Basic results about closed forms

In this section, we give a simple introduction to some results on differential forms and convex functions constructed by differential forms. One refers to [3, 21] for more background.

A differential 1-form $\omega =\sum ^{n}_{i=1}f_{i}(x)\textrm{d}x_{i}$ is said to be continuous in an open set $U\subset {\mathbb {R}}^{n}$ if each $f_{i}(x)$ is continuous on U. A continuous 1-form is called closed if $\int _{\partial \gamma }\omega =0$ for each piecewise $C^{1}$-smooth null homologous loop $\gamma$ in U.

Proposition B.1

Suppose X is an open set in ${\mathbb {R}}^{n}$ and $A\subset X$ is an open subset bounded by a smooth $(n-1)$-dimensional submanifold in X. If $\omega =\sum ^{n}_{i=1}f_{i}(x)\textrm{d}x_{i}$ is a continuous 1-form on X such that $\omega |_{A}$ and $\omega |_{X-{\overline{A}}}$ are closed where ${\overline{A}}$ is the closure of A in X, then $\omega$ is closed in X.

Proposition B.2

Suppose $X\subset {\mathbb {R}}^{n}$ is an open convex set and $A\subset X$ is an open subset of X bounded by a codimension-1 real analytic submanifold in X. If $\omega =\sum ^{n}_{i=1}f_{i}(x)\textrm{d}x_{i}$ is a continuous closed 1-form on X such that $F(x)=\int ^{x}_{a}\omega$ is locally convex in A and in $X-{\overline{A}}$, then F(x) is convex in X.