Advances in Discrete Differential Geometry pp 177195  Cite as
A Variational Principle for Cyclic Polygons with Prescribed Edge Lengths
Abstract
We provide a new proof of the elementary geometric theorem on the existence and uniqueness of cyclic polygons with prescribed side lengths. The proof is based on a variational principle involving the central angles of the polygon as variables. The uniqueness follows from the concavity of the target function. The existence proof relies on a fundamental inequality of information theory. We also provide proofs for the corresponding theorems of spherical and hyperbolic geometry (and, as a byproduct, in \(1+1\) spacetime). The spherical theorem is reduced to the Euclidean one. The proof of the hyperbolic theorem treats three cases separately: Only the case of polygons inscribed in compact circles can be reduced to the Euclidean theorem. For the other two cases, polygons inscribed in horocycles and hypercycles, we provide separate arguments. The hypercycle case also proves the theorem for “cyclic” polygons in \(1+1\) spacetime.
Keywords
Variational Principle Straight Line Segment Hyperbolic Geometry Affine Plane Short Geodesic1 Introduction
This article is concerned with cyclic polygons, i.e., convex polygons inscribed in a circle. We will provide a new proof of the following elementary theorem in Sect. 2.
Theorem 1.1
Our proof involves a variational principle with the central angles as variables. The variational principle has a geometric interpretation in terms of volume in 3dimensional hyperbolic space (see Remark 2.6). Another striking feature of our proof is the use of a fundamental inequality of information theory:
Theorem
The left hand side of inequality (2) is called the Kullback–Leibler divergence or information gain of q from p, also the relative entropy of p with respect to q. The inequality follows from the strict concavity of the logarithm function (see, e.g., Cover and Thomas [3]).
In Sects. 3 and 4 we provide proofs for nonEuclidean versions of Theorem 1.1. The spherical version requires an extra inequality:
Theorem 1.2
Inequality (3) is necessary because the perimeter of a circle in the unit sphere cannot be greater than \(2\pi \), and the perimeter of the inscribed polygon is a lower bound. We require strict inequality to exclude polygons that degenerate to great circles (with all interior angles equal to \(\pi \)).
In Sect. 3, we prove Theorem 1.2 by a straightforward reduction to Theorem 1.1: connecting the vertices of a spherical cyclic polygon by straight line segments in the ambient Euclidean \(\mathbb R^{3}\), one obtains a Euclidean cyclic polygon.

a hyperbolic circle if the curvature is greater than 1,

a horocycle if the curvature is equal to 1,

a hypercycle, i.e., a curve at constant distance from a geodesic if the curvature is strictly between 0 and 1.
Theorem 1.3
There exists a hyperbolic cyclic polygon with \(n\ge 3\) sides of lengths \(\ell _{1},\ldots ,\ell _n\in \mathbb R\) if and only if they satisfy the polygon inequalities (1), and this cyclic hyperbolic polygon is unique.
We prove this theorem in Sect. 4. The case of hyperbolic polygons inscribed in circles can be reduced to Theorem 1.1 by considering the hyperboloid model of the hyperbolic plane: Connecting the vertices of a hyperbolic polygon inscribed in a circle by straight line segments in the ambient \(\mathbb R^{2,1}\), one obtains a Euclidean cyclic polygon.
The cases of polygons inscribed in horocycles and hypercycles cannot be reduced to the Euclidean case because the intrinsic geometry of the affine plane of the polygon is not Euclidean: In the horocycle case, the scalar product is degenerate with a 1dimensional kernel. Hence, this case reduces to the case of degenerate polygons inscribed in a straight line. It is easy to deal with. In the hypercycle case, the scalar product is indefinite. This case reduces to polygons inscribed in hyperbolas in flat \(1+1\) spacetime. The variational principle of Sect. 2 can be adapted for this case (see Sect. 5), but the corresponding target function fails to be concave or convex. It may be possible to base a proof of existence and uniqueness on this variational principle, perhaps using a \(\min \)\(\max \)argument, but we do not pursue this route in this article. Instead, we deal with polygons inscribed in hypercycles using a straightforward analytic argument.
Some history, from ancient to recent. Theorems 1.1–1.3 belong to the circle of results connected with the classical isoperimetric problem. As the subject is ancient and the body of literature is vast, we can only attempt to provide a rough historical perspective and ask for leniency regarding any essential work that we fail to mention.
The early history of the relevant results about polygons is briefly discussed by Steinitz [13, Sect. 16]. Steinitz goes on to discuss analogous results for polyhedra, a topic into which we will not go. A more recent and comprehensive survey of proofs of the isoperimetric property of the circle was given by Blåsjö [2].
It was known to Pappus that the regular ngon had the largest area among ngons with the same perimeter, and that the area grew with the number of sides. This was used to argue for the isoperimetric property of the circle:
Theorem 1.4
(Isoperimetric Theorem) Among all closed planar curves with given length, only the circle encloses the largest area.
It is not clear who first stated the following theorem about polygons:
Theorem 1.5
(Secant Polygon) Among all ngons with given side lengths, only the one inscribed in a circle has the largest area.
This was proved by Moula [8], by L’Huilier [5] (who cites Moula), and by Steiner [12] (who cites L’Huilier). L’Huilier also proved the following theorem:
Theorem 1.6
(Tangent Polygon) Among all convex ngons with given angles, only the one circumscribed to a circle has the largest area when the perimeter is fixed and and smallest perimeter when the area is fixed.
Steiner also proves versions of Theorems 1.5 and 1.6 for spherical polygons. None of these authors deemed it necessary to prove the existence of a maximizer, an issue that became generally recognized only after Weierstrass [14]. For polygons, the existence of a maximizer follows by a standard compactness argument.
Neither Blaschke, nor Steiner, L’Huilier, or Moula provide an argument for the uniqueness of the maximizer in Theorem 1.5 or 1.6. It seems that even after Weierstrass, the fact that the sides determine a cyclic polygon uniquely was considered too obvious to deserve a proof.
Penner [9, Theorem 6.2] gives a complete proof of Theorem 1.1. He proceeds by showing that there is one and only one circumcircle radius that allows the construction of a Euclidean cyclic polygon with given sides (provided they satisfy the polygon inequalities).
2 Euclidean Polygons. Proof of Theorem 1.1
Proposition 2.1
(Variational Principle) A point \(\alpha \in D_n\) is a critical point of \(f_\ell \) restricted to \(D_{n}\) if and only if there exists an \(R\in \mathbb R\) satisfying equations (7).
Proof
Thus, to prove Theorem 1.1, we need to show that \(f_\ell \) has a critical point in \(D_n\) if and only if the polygon inequalities (1) are satisfied, and that this critical point is then unique. The following proposition and corollary deal with the uniqueness claim.
Proposition 2.2
The function \(f_\ell \) is strictly concave on \(D_n\).
Corollary 2.3
If \(f_\ell \) has a critical point in \(D_n\), it is the unique maximizer of \(f_{\ell }\) in the closure \( \bar{D}_n=\{\alpha \in \mathbb R_{\ge 0}^n \;\;\sum \alpha _k = 2\pi \}. \)
This proves the uniqueness claim of Theorem 1.1.
Proof
Since \(f_{\ell }\) attains its maximum on the compact set \(\bar{D}_{n}\), it remains to show that the maximum is attained in \(D_{n}\) if and only if the polygon inequalities (1) are satisfied. This is achieved by the following Propositions 2.4 and 2.5.
Proposition 2.4
If the function \(f_\ell \) attains its maximum on the simplex \(\bar{D}_{n}\) at a boundary point \(\alpha \in \partial \bar{D}_{n}\), then \(\alpha \) is a vertex.
Proof
Suppose \(\alpha \in \partial \bar{D}_{n}\) is not a vertex. We need to show that \(f_{\ell }\) does not attain its maximum at \(\alpha \). This follows from the fact that the derivative of \(f_{\ell }\) in a direction pointing towards \(D_{n}\) is \(+\infty \).
Proposition 2.5
Proof
This completes the proof of Theorem 1.1.
Remark 2.6
The function \(V_{n}\) has the following interpretation in terms of hyperbolic volume [6]. Consider a Euclidean cyclic ngon with central angles \(\alpha _{1},\ldots ,\alpha _{n}\). Imagine the Euclidean plane of the polygon to be the ideal boundary of hyperbolic 3space in the Poincaré upper halfspace model. Then the vertical planes through the edges of the polygon and the hemisphere above its circumcircle bound a hyperbolic pyramid with vertices at infinity. Its volume is \(\frac{1}{2}V_{n}(\alpha _{1},\ldots ,\alpha _{n})\). Together with Schläfli’s differential volume equation (rather, Milnor’s generalization that allows for ideal vertices [7]), this provides another way to prove Proposition 2.1.
3 Spherical Polygons. Proof of Theorem 1.2
Proposition 3.1
If the spherical lengths \(\ell \in \mathbb R_{>0}^{n}\) satisfy the inequalities (1) and (3), then the Euclidean lengths \(\bar{\ell }\) defined by (15) satisfy the inequalities (1) as well. By Theorem 1.1 there is then a unique Euclidean cyclic polygon \(P_{\bar{\ell }}\) with side lengths \(\bar{\ell }\).
Proposition 3.2
The circumradius \(\bar{R}\) of the polygon \(P_{\bar{\ell }}\) of Proposition 3.1 is strictly less than 1.
We will use the following estimate in the proof of Proposition 3.1:
Lemma 3.3
Proof
Remark 3.4
The statement of Lemma 3.3 can be strengthened. Equality holds in (16) if and only if at most one \(\beta _{k}\) is greater than zero. This is easy to see, but we do not need this stronger statement in the following proof.
Proof

\(\sum _{i\not =k}\ell _i \le \pi \). Inequality (18) simply follows from the polygon inequality \(\ell _{k}<\sum _{i\not =k}\ell _{i}\) and the monotonicity of the sine function on the closed interval \([0,\frac{\pi }{2}]\).
 \(\sum _{i\not =k}\ell _i \ge \pi \). Note that \(2\pi >\sum _i\ell _i\) implies \(2\pi \ell _{k}>\sum _{i\not =k}\ell _i\), and henceInequality (18) follows from \(\sin \frac{\ell _k}{2}=\sin (\pi \frac{\ell _k}{2})\) and the monotonicity of the sine function on the closed interval \([\frac{\pi }{2},\pi ]\).$$\begin{aligned} 2\pi>2\pi \ell _{k}>\sum _{i\not =k}\ell _{i}\ge \pi . \end{aligned}$$(19)
This completes the proof of (18) and hence the proof of Proposition 3.1. \(\square \)
Proof
First, suppose that \(\alpha _k \le \pi \) for all k. Since \( \sum _k \ell _k < 2 \pi = \sum _k \alpha _k\), there is some k such that \(\ell _k < \alpha _k\). Then \(\sin \frac{\ell _k}{2} < \sin \frac{\alpha _k}{2}\), and equation (20) implies that \(\bar{R} < 1\).
4 Hyperbolic Polygons. Proof of Theorem 1.3
The polygon inequalities (1) are clearly necessary for the existence of a hyperbolic cyclic polygon, because every side is a shortest geodesic. It remains to show that they are also sufficient, and that the polygon is unique, i.e., Proposition 4.2. First, we review some basic facts from hyperbolic geometry.
A curve of intersection of \(\mathbb H^{2}\) with an affine plane in \(\mathbb R^{2,1}\) that does not contain 0 is a hyperbolic circle, a horocycle, or a hypercycle, depending on whether the plane is spacelike, lightlike, or timelike.
Proposition 4.1
 (i)a circle then$$\begin{aligned} \bar{\ell }_{k}<\sum _{\begin{array}{c} i=1\\ i\not =k \end{array}}^{n}\bar{\ell }_{i} \quad \textit{for} \, \textit{all} \, k. \end{aligned}$$(24)
 (ii)a horocycle then$$\begin{aligned} \bar{\ell }_{k}=\sum _{\begin{array}{c} i=1\\ i\not =k \end{array}}^{n}\bar{\ell }_{i} \quad \textit{for} \, \textit{one} \, k. \end{aligned}$$(25)
 (iii)a hypercycle then$$\begin{aligned} \bar{\ell }_{k}>\sum _{\begin{array}{c} i=1\\ i\not =k \end{array}}^{n}\bar{\ell }_{i} \quad \textit{for}\, \textit{one} \, k. \end{aligned}$$(26)
Proof
(i) If \(P_{\ell }\) is inscribed in a circle, then the chordal polygon obtained by connecting the vertices of \(P_{\ell }\) by straight line segments in \(\mathbb R^{2,1}\) is a Euclidean polygon. Hence, its side lengths \(\bar{\ell }\) satisfy (24).
This completes the proof of Proposition 4.1. \(\square \)
Proposition 4.2
If \(\ell \in \mathbb R^n_{>0}\) satisfies the polygon inequalities (1), then there exists a unique hyperbolic cyclic polygon with these side lengths.
Proof
Suppose \(\ell \in \mathbb R^n_{>0}\) satisfies the polygon inequalities (1). Let \(\bar{\ell }\) be the corresponding chordal lengths (23). We will treat each case of Proposition 4.1 separately. In each case, we will tacitly use Proposition 4.1 and its proof. Our treatment of case (iii) is analogous to Penner’s proof [9] of Theorem 1.1 (his Theorem 6.2).
(i) If the chordal lengths \(\bar{\ell }\) satisfy condition (24), then the existence and uniqueness of a hyperbolic cyclic polygon with side lengths \(\ell \) follows from the existence and uniqueness of a Euclidean cyclic polygon with side lengths \(\bar{\ell }\), i.e., from Theorem 1.1 (see Fig. 4).
(ii) If the chordal lengths \(\bar{\ell }\) satisfy condition (25), then the corresponding hyperbolic cyclic polygon can be constructed by marking off the lengths \(\bar{\ell }_{i}\) for \(i\not =k\) along a horocycle (see Fig. 5). To see the uniqueness claim, note that all horocycles are congruent.
This concludes the proof of Proposition 4.2. \(\square \)
5 Concluding Remarks on 1 + 1 Spacetime
Theorem 5.1
Notes
Acknowledgments
This research was supported by DFG SFB/TRR 109 “Discretization in Geometry and Dynamics”.
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