1 Introduction

Deformation of quadrilaterals and addition on elliptic curves

Abstract.

The space of quadrilaterals with fixed side lengths is an elliptic curve. Darboux used this to prove a porism on foldings.

In this article, the space of oriented quadrilaterals is studied on the base of biquadratic equations between their angles. The space of non-oriented quadrilaterals is also an elliptic curve, doubly covered by the previous one, and is described by a biquadratic relation between diagonals. The spaces of non-oriented quadrilaterals with the side lengths and turn out to be isomorphic via identification of two quadrilaterals with the same diagonal lengths.

We prove a periodicity condition for foldings, similar to Cayley’s condition for the Poncelet porism.

Some applications to kinematics and geometry are presented.

Key words and phrases:
Folding of quadrilaterals; porism; elliptic curve; biquadratic equation
Supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no. 247029-SDModels

1. Introduction

1.1. The Darboux porism

Let be a planar quadrilateral. Denote by the image of under reflection in the line . We call the transformation

the -folding. It acts on the set of all quadrilaterals with . Define the -folding in a similar way, see Figure 1.

Figure 1. Composition of - and -foldings.

Alternate the - and the -foldings, so that the vertices and stay in their places, while and jump. If we are lucky, then after several iterations the points and return to their initial positions.

Definition 1.1.

A quadrilateral is called -periodic, if it is invariant under the -fold iteration of the composition of - and -foldings:

The quadrilateral from Figure 1 is -periodic, see Figure 2. Surprizingly enough, the periodicity depends only on the side lengths.

Figure 2. A -periodic quadrilateral. Side lengths are .
Theorem 1 (Darboux [9]).

If a quadrilateral is -periodic, then every quadrilateral with the same side lengths is -periodic.

The clue to the Darboux porism is the fact that the space of congruence classes of quadrilaterals with fixed side lengths is an elliptic curve, and the foldings and act on it as involutions. Hence is a translation, just as well as . If a translation has a fixed point (one quadrilateral is -periodic), then it is an identity (all quadrilaterals with the same side lengths are periodic).

Unlike the Poncelet porism [11], its relative, the Darboux porism is much less known and was rediscovered several times in the recent decades.

In the present article we take a closer look at the configuration space of quadrilaterals with side lengths . We study also the space of non-oriented quadrilaterals, which is an elliptic curve doubly covered by . There are some surprising algebraic identities that result in a natural isomorphism between the spaces and , where is a certain involution on . Geometrically this leads to “conjugate pairs” of quadrilaterals. Finally, we express -periodicity of a quadrilateral in terms of its side lengths.

1.2. Euler-Chasles correspondence and Jacobi elliptic functions

Introduce the variables

where and are adjacent angles of a quadrilateral. Then the space of congruence classes of quadrilaterals, respecting the orientation, with fixed side lengths becomes identified with the algebraic curve

(1)

Here coefficients depend on the side lengths. This is a special biquadratic equation in two variables. Biquadratic equations are also known under the name of Euler-Chasles correspondences [7]. In the past decades they attracted a lot of attention: as a basis for -maps in the theory of discrete integrable systems [12]; as a solution of the Yang-Baxter equation [4, 22]; as an approach to flexible polyhedra [26, 15, 18].

It is known that in a generic case the curve (1) can be parametrized as

where is a second-order elliptic function with simple poles and zeros. Since any pair of adjacent angles is related by a biquadratic equation, all angles turn out to be scaled shifts of the same elliptic function.

If are the side lengths of a quadrilateral, then the non-degeneracy condition ensuring that the configuration space is an elliptic curve is

for all choices of signs. Put differently, this means , where and are the lengths of the shortest and the longest side, and is the half-perimeter.

In kinematics, the inequality is knows as the Grashof condition. It means that the shortest side can make a full turn with respect to each of its neighbors; also it means that the real part of the configuration space has two components, that is the quadrilateral cannot be deformed into its mirror image. The following theorem describes a parametrization of and underlines the difference between Grashof and non-Grashof quadrilaterals.

Theorem 2.

If , then the complexified configuration space of quadrilaterals with side lengths is an elliptic curve .

  1. If , then the lattice is rectangular, and the cotangents of the halves of the exterior angles can be parametrized as

    Here , the amplitudes are real or purely imaginary, and the shifts satisfy

    The real part of the configuration space consists of two components corresponding to .

  2. If , then the lattice is rhombic, and the cotangents of the halves of the exterior angles can be parametrized as

    Here , each of the amplitudes is real or purely imaginary, and the shifts satisfy

    The real part of the configuration space has one component corresponding to .

The proof of Theorem 2 is contained in Sections 3.3.4 and 3.3.5. There one can also find exact values of the amplitudes and shifts. The numbers appearing in the formula for the Jacobi modulus are defined as follows.

(2)

The pair of opposite angles , in a quadrilateral is also subject to a relation of the form (1), but with a vanishing coefficient at . A scaling of variables brings the equation into the form

(3)

with a real . This curve can be parametrized as

where if and if . Equation (3) is closely related to the equation

suggested by Edwards [13] as a new normal form for elliptic curves. Edwards constructs a parametrization of and “from the scratch” using a version of theta-functions.

1.3. The space of non-oriented quadrilaterals

Let be the space of congruence classes, disregarding the orientation, of quadrilaterals with side lengths . Clearly, the map is a double cover, easy to describe in terms of the holomorphic parameter of Theorem 2. There is an unexpected natural isomorphism between the spaces and with as in (2): for every quadrilateral with the side lengths there is a quadrilateral with the same diagonal lengths and the side lengths .

Theorem 3.

If , then the complexified configuration space of non-oriented quadrilaterals with the side lengths is an elliptic curve with a rectangular lattice .

There is a natural isomorphism

that identifies two quadrilaterals with the same diagonal lengths.

If has a rectangular lattice, then has a rhombic lattice. In particular, the above isomorphism does not lift to the spaces of oriented quadrilaterals. The double covers

look as shown on Figure 3.

Theorem 3 is proved in Section 4.

Figure 3. Spaces and covering .

The isomorphism turns out to be equivalent to the Ivory theorem, see Figure 12. Theorem 3 holds also for spherical and hyperbolic quadrilaterals. This extends the Ivory theorem to the sphere and the hyperbolic plane, with ellipses and hyperbolas defined geodesically, see Theorem 7.

Computations that allow to express in a particularly nice way the Jacobi modulus, amplitudes, and shifts in Theorem 2, and those leading to Theorem 3 are based on a number of identities for dual quadruples and . These are collected below.

1.4. The periodicity condition

Benoit and Hulin [5] studied the periodicity condition by constructing a pair of circles whose Poncelet dynamics is equivalent to the folding dynamics of the quadrilateral.

The following theorem deals with the periodicity disregarding the orientation. Proposition 5.4 describes how the period lengths on the curves and are related.

Theorem 4.

A quadrilateral with the side lengths is -periodic disregarding the orientation if and only if the following condition is satisfied.

Here are the coefficients of the expansion

where

Note that due to the identities from Section 1.3, so that the quadrilaterals with the side lengths have the same folding period (disregarding the orientation) as the quadrilaterals with the side lengths . This is also obvious from the isomorphism via equal diagonal lengths.

1.5. Acknowledgments

Parts of this work were done during the author’s visits to IHP Paris and to the Penn State University. The author thanks both institutions for hospitality. Also he wishes to thank Arseniy Akopyan, Udo Hertrich-Jeromin, Boris Springborn, and Yuri Suris for useful discussions.

2. The space of oriented quadrilaterals in terms of their angles

2.1. Notation

A planar quadrilateral is for us an ordered quadruple of points in the euclidean plane such that

Two quadrilaterals and are called directly congruent if there exists an orientation-preserving isometry such that

In this article, we study the set of (direct) congruence classes of quadrilaterals with fixed side lengths. A mechanical interpretation of this is the configuration space of a four-bar linkage with the positions of two adjacent joints fixed and whose bars are allowed to cross each other.

For a quadruple of real numbers there exists a quadrilateral with side lengths if and only if the following inequalities hold:

(4a)
(4b)

where in the second line.

Definition 2.1.

For satisfying the conditions (4a) and (4b), denote by the set of direct congruence classes of quadrilaterals with

Remark 2.2.

We require a congruence to preserve the marking of the vertices (or, equivalently, the marking of the sides). This does matter only if the sequence is symmetric under the action of an element of the dihedral group on .

Denote by the angle between the sides marked by and . More exactly, are the turning angles for the velocity vector of a point that runs along the perimeter in the direction given by the cyclic order , see Figure 4, left.

Figure 4. Notations; finding a relation for opposite angles; finding a relation for adjacent angles.

2.2. Equations relating the angles of a quadrilateral

Lemma 2.3.

The cosines of opposite angles of a quadrilateral are subject to a linear dependence, with coefficients depending on the side lengths.

Proof.

By expressing the diagonal length on Figure 4 with the help of the cosine law first through and then through , we obtain

(5)

This result is classical. Bricard [6] mentions it as well-known. The following substitution also appears in [6].

(6)
Proposition 2.4.

The tangents of the opposite half-angles of a quadrilateral satisfy the following algebraic relation

(7)
Proof.

Substitute and in (5). A simple computation yields (7). ∎

With a bit more work one finds a relation between pairs of adjacent angles.

Proposition 2.5.

The tangents of the adjacent half-angles of a quadrilateral satisfy the equation

(8)
Proof.

Denote by the vector running along the -th side of the quadrilateral, see Figure 4, right. Expanding the scalar product on the left hand side of we obtain

The substitution and a tedious computation produce (8). ∎

2.3. Bihomogeneous equations and algebraic curves in

The substitution (6) identifies , which is the range of , with , which is the range of . Thus it is geometrically reasonable to consider equations (7) and (8) as equations on . The proper setting for this are bihomogeneous polynomials. That is, we introduce projective variables and and rewrite equation (8) as

(9)

This point of view doesn’t affect the affine part of the curve but may change the number of points at infinity (the infinity of is the union of two projective lines instead of one for ). Indeed, while the usual projectivization of (8) has two points and at infinity, the curve (9) has four:

For a generic choice of coefficients the curve (9) is non-singular, as opposed to the usual projectivization of (8).

2.4. A system of six equations

By Propositions 2.4 and 2.5, the angles of every quadrilateral satisfy a system of six equations: two of the form (7) and four of the form (8). In this section we show that, vice versa, under a certain genericity assumption every solution of the system corresponds to a quadrilateral.

Proposition 2.6.

Assume that the quadruple is not made of two pairs of equal adjacent numbers:

(10)

Then every solution of the system of six equations on the pairs of angles corresponds to a unique quadrilateral in .

Proof.

By reverting the argument in the proof of Proposition 2.5 one sees that for every solution of (8) there is a unique quadrilateral in with angles and . The same is true for the other three equations relating adjacent angles. For every solution of (7) there is also a quadrilateral with angles and . This quadrilateral can be non-unique only if the second and the fourth vertices coincide, for which and is needed.

Thus for every solution of the system of six equations there are six quadrilaterals with the property that in the angles and have the correct values. Our goal is to show that there is with all correct angles. Consider , , and . They all have the same . Assumption (10) implies that there are at most two quadrilaterals in with a given . Thus at least two of the three quadrilaterals must coincide. This yields a quadrilateral with three correct angles, one of which is . If the fourth angle is incorrect, then again, consider the three quadrilaterals where this angle is correct and obtain a quadrilateral with three correct angles, one of which is . As and have two angles in common, they coincide, and is the desired quadrilateral. ∎

2.5. Birational equivalence between (7) and (8)

We have just seen that the configuration space is an algebraic curve in given by six equations. It will often be convenient to restrict our attention to a single equation in two variables. The following statement allows us to do so.

Proposition 2.7.

The projection of to every coordinate plane is a birational equivalence.

The projection of to the coordinate plane is a birational equivalence unless and . (All indices are taken in modulo .)

Proof.

It suffices to consider the case . As noted before, the projection of to the -plane is injective. It suffices to show that for every the values of and are rational functions of and . By projecting the quadrilateral to its second side and to the line orthogonal to it, we obtain

(11)

It follows that

which is rational in and .

Similarly, if either or , then the projection of to the -plane is injective. To show that the inverse map is rational, rewrite (11) as

This is a system of linear equations on and with the determinant . The determinant vanishes only for and , which also implies . If it does not vanish, then by solving the system we can express and as rational functions of and . ∎

Now that the algebraic structure of became clear, we can define the complexified configuration space.

Definition 2.8.

The complexified configuration space of oriented quadrilaterals with edge lengths is the algebraic curve in defined by the six equations of the form (7) and (8).

Due to Proposition 2.7, is birationally equivalent to the curve (8) and is birationally equivalent to (7) unless and .

3. Parametrizations of the configuration spaces of oriented quadrilaterals

3.1. Classifying linkages by their degree of degeneracy

The shape of the configuration space turns out to depend on the number of solutions of the equation

(12)

It is easy to see that if (12) has at least two solutions, then consists of two pairs of equal numbers. This leads to the following classification of quadrilaterals.

Definition 3.1.

A quadrilateral with side lengths (in this cyclic order) is said to be

  • of elliptic type, if equation (12) has no solution;

  • of conic type, if the equation (12) has exactly one solution;

  • an isogram, if its opposite sides are equal: , ;

  • a deltoid, if two pairs of adjacent sides are equal: or ;

  • a rhombus, if all sides are equal.

Let us look at the last three simple cases before we proceed to the more interesting conic and elliptic types.

The rhombus

Equation (8) becomes , and (7) becomes (in the bihomogeneous form)

The other equations can be obtained by cyclically permuting the indices. The configuration space consists of three lines

The first two consist of “folded” configurations, when two opposite vertices are at the same point, and the edges rotate around this point; the third line corresponds to actual rhombi with .

Note that the condition (10) is violated, and the system has “fantom” solutions that don’t correspond to any quadrilateral, e. g. , .

The deltoid

Assume . Equation (8) becomes

Besides, in the bihomogeneous form the factor appears, so that the configuration space consists of two lines

Again, the first line corresponds to a folded deltoid, and the second expresses an angle at a “peak” of the deltoid through the angle at its base. The fantom solution , is also present in this case.

The isogram

Under assumption equation (8) becomes

which factorizes as

The configuration space consists of two lines: , consisting of parallelograms, and , consisting of antiparallelograms.

3.2. Conic quadrilaterals: parametrization by trigonometric functions

If equation (12) has exactly one solution, then two cases must be distinguished: either the two pairs of opposite sides add up to the same total length, or two pairs of adjacent sides do so.

In what follows we will be repeatedly using the fact that implies

Circumsribable quadrilaterals:

Proposition 3.2.

Assume that is the unique solution of the equation (12) and that . Then the configuration space is isomorphic to the one-point compactification of , and a bijection can be established by the parametrization

where

with , , obtained by cyclically permuting the indices, and

The square roots of negative numbers are assumed to take value in .

Proof.

Due to equations (7) and (8) become

(13a)
(13b)

Equation (13a) rewrites as

with

so that the coordinates can be parametrized as

The substitution of and in (13b) under the assumption results in

which has the form

(14)

with and as stated in the theorem. On the other hand, and in (14) can be parametrized as and . This leads to

The relation between each pair of opposite angles imply that the above parametrization is completed by

for some choices of the signs. To determine the signs, one may analyze the equations between and and between and in a similar way. Alternatively, one looks at special configurations of quadrilaterals.

Edge lengths satisfy

Equation (7) relating and takes the same form as in the case of curcumscribable quadrilaterals:

so that we have again a parametrization

But for the pair we have

or, equivalently,