Iterating evolutes of spatial curves and of spatial

arXiv:1611.08836v2 [math.DG] 29 Nov 2016

Iterating evolutes of spatial curves and of spatial polygons Dmitry Fuchs∗

1

Serge Tabachnikov†

Introduction

The evolute of a smooth plane curve is the locus of its centers of curvature or, equivalently, the envelope of the family of its normal lines. The construction of the evolute can be iterated. In our recent paper [1] we studied such iterations; we also investigated discrete versions of this problem where smooth curves are replaced by polygons. In the present paper we study three-dimensional analogs of this problem, the iterations of evolutes of closed piecewise smooth and of polygonal curves in R3 . Our investigation consists of two parts, concerning curves and polygons, respectively. In the first part, we consider curves with non-vanishing curvature and non-vanishing torsion (the former is a general position property, whereas the latter is not). Our curves may have cusps, such as the curve (t2 , t3 , t4 ) at the origin. Each smooth piece of the curve is oriented, and the orientations agree at cusps (defining a coorientation of the curve). The number of cusps is necessarily even. The tangent Gauss image of such a curve, called its tangent indicatrix, is an immersed closed curve on the unit sphere. The osculating sphere of a spatial curve passes through a quadruple of infinitesimally close points of the curve. The evolute of a curve is the locus of the centers of its osculating spheres (this evolute is also called the evolute of the 2nd kind in [3] and the focal curve in [10]). Equivalently, the evolute ∗

Department of Mathematics, University of California, Davis, CA 95616; [email protected] † Department of Mathematics, Pennsylvania State University, University Park, PA 16802; [email protected]

1

is the enveloping curve of the family of the normal planes of a spatial curve, and it is called the edge of regression of the polar developable in [8]. See [4, 10] for a study of evolutes of spatial curves. The space of curves in R3 fibers over the space of spherical curves, their tangent indicatrices. The space of curves with a fixed tangent indicatrix is an infinite-dimensional vector space. One of our theorems is that the tangent indicatrix of the evolute of a spatial curve is spherically dual to the indicatrix of the original curve, and the evolute map of the fibers over these spherical curves is linear (see Theorem 1). The second evolute of a spatial curve has the same, up to a central symmetry, tangent indicatrix as the original curve, and one wonders whether there exist curves homothetic to their second evolutes. In the plane, the curves with this property are the classical hypocycloids, see [1], Corollary 2.8. We construct a family of spatial curves that are homothetic to their second evolutes. The tangent indicatrix of these curves are circles on the unit sphere. These curves may be regarded as spatial analogs of hypocycloids, and indeed, they look not unlike the classical hypocycloids, see Figure 1. We do not know whether there are other spatial curves homothetic to their second evolutes. For an n-gon in R3 , we define the evolute as the n-gon whose vertices are the centers of the spheres through the quadruples of its consecutive vertices (we assume that n ≥ 5). (There exists an alternative definition of an evolute of an n-gon as an n-gon whose vertices are the centers of spheres tangent to the quadruples of consecutive sides; in this article we almost never consider this option.) For an n-gon, the tangent indicatrix becomes a spherical polygon or, equivalently, a cyclically ordered sequence of n unit vectors. For polygons with fixed directions of sides, the sides of their evolutes also have fixed directions, and the sides of the second evolute are parallel to the sides of the original polygon. The space of n-gons with given directions of sides, considered modulo parallel translations, is an (n − 3)-dimensional vector space, with the signed side lengths as coordinates. We show that the evolute map is a linear transformation between two such vector spaces, and the second evolute map is a linear self-map of such a vector space. It follows that if the maximum module eigenvalue of the second evolute transformation is real, then the whole sequence of multiple evolutes of a polygon, considered up to parallel translations and homotheties, is asymptotically 2

2-periodic (the limit coeficient of the homothety equals the maximum module eigenvalue; in particular, it may be negative). In Figures 2 and 3, we show an example of such behavior for heptagons. Why do we need heptagons? For a spatial pentagon, a theorem by E. Tsukerman [9] states that its second evolute is always homothetic to it (actually, Thukerman proves this for (n + 2)-gons in Rn ). For a spatial hexagon, we prove in this paper that its third evolute is always homothetic to its first evolute. We conjecture that a similar fact holds for (n + 3)-gons in Rn ; for n = 2, this was conjectured by B. Gr¨ unbaum in [6, 7] and proved in our paper [1]. Speaking of higher dimensions, we expect that the results of this paper can be extended to arbitrarily high dimensions and to the classical geometries of constant curvature. We hope to investigate these questions in the near future. Acknowledgment. The second author was supported by NSF grant DMS1510055.

2 2.1

Piecewise smooth curves Spherical curves

Let us recall some facts about spherical curves; see, e.g., [2]. A coorientation of a smooth curve on the unit sphere is a choice of a unit normal vector field. Let γ be a cooriented closed spherical curve. The dual curve γ is obtained by moving each point of γ distance π/2 in the direction of the coorientation. For example, the Southward cooriented Arctic Circle is dual to the Tropic of Capricorn, and the Northward cooriented Arctic Circle is dual to the Tropic of Cancer. The dual curve is the locus of centers of great circles tangent to the original curve. The second dual curve is antipodal to the original one. An inflection of a spherical curve is a point where the geodesic curvature vanishes, that is, where the curve is abnormally well approximated by a great circle. The spherical duality interchanges inflection points with cusps. We shall be concerned with locally convex, i.e., inflection free, smooth curves. This class of curves is invariant under the duality. 3

2.2

Curves in space and their tangent indicatrices

Let Γ(x) be an arc length parameterized curve in R3 with non-vanishing curvature k and non-vanishing torsion τ . Let (T, N, B) be the Frenet frame, and let γ(t) be the arc length parameterized tangent indicatrix. Set ρ = 1/k, the radius of curvature. Lemma 2.1 One has:

dt dx = ρ, = k. dt dx The tangent indicatrix γ is smooth and inflection free. Its spherically dual curve γ is given by the vector B. Proof. The Frenet equations read: Tx = kN, Nx = τ B − kT, Bx = −τ N. One has: 1 = |Tt | = |Tx |xt = kxt , hence xt = ρ. Since γ = T and k 6= 0, the velocity of γ does not vanish, so γ is smooth. The vector B is orthogonal to T , hence tangent to the sphere, and normal to N , therefore it given the tangent indicatrix a coorrientation and describes the dual curve. Since τ 6= 0, the Frenet equations imply that this dual curve is smooth as well. 2 Remark 2.2 The geodesic curvature vector of the tangent indicatrix is T + Ttt . A straightforward calculation using the Frenet formulas yields ρτ for the geodesic curvature of this spherical curve. This confirms that the tangent indicatrix of a spatial curve free from inflection and flattening points is locally convex, i.e., free from inflections. Let γ(t) be a closed smooth locally convex arc length parameterized spherical curve. Consider the piecewise smooth spatial curves Γ having γ as the tangent indicatrix, modulo parallel translations. The space of such curves is identified with the space of periodic functions ρ(t), satisfying the linear relation Z ρ(t)γ(t) dt = 0, 4

saying that Γ is a closed. The zeros of the function ρ correspond to the cusps of the respective space curve Γ. Denote the space of such curves Γ by Cγ . Let Γ be the evolute of Γ and y be its arc length parameter. The formula for the evolute, originally due to Monge, can be found in, e.g., [8, 10] and [4]: ρx B. τ

Γ = Γ + ρN + Furthermore, Γx = ΦB, where Φ = ρτ +

ρ  x

τ

(1) x

see [10] and [4] again. Therefore dx 1 dy = Φ, = . dx dy Φ

(2)

Remark 2.3 At a cusp (at2 , bt3 , ct4 ), both the curvature k and the torsion τ are infinite. However, the quantity ρx /τ has a finite limit, and the center of the osculating sphere is located on the binormal, the vertical axis, at distance a2 /(2c) from the origin. Likewise, the tangent indicatrix is smooth and inflection free at a point corresponding to a cusp of a spatial curve. Since the velocity vector of the evolute Γ is proportional to the binormal of Γ, we have Corollary 2.4 The tangent indicatrix of Γ is the curve γ spherically dual to the curve γ, the tangent indicatrix of Γ. Remark 2.5 Equation (1) implies that ρ  x Φ = ρτ + =0 τ x is a necessary and sufficient condition for a spatial curve to lie on a sphere. This is a classical result, see, e.g., [8]. Let ρ and τ be the radius of curvature and the torsion of Γ. Lemma 2.6 One has

Φ 1 ρ=− , τ = . τ Φ 5

(3)

Proof. Let (T , N , B) be the Frenet frame along Γ. Then T = B and, using the Frenet formulas, dT 1 dB τ = = − N. dy Φ dx Φ τ Hence N = N and k = − . This implies the first equation of the lemma. Φ Also B = T × N = B × N = −T . Next, 1 dB = − N, dy Φ therefore τ =

k , implying the second equation. 2 Φ

Let s be the arc length parameter on γ. The next result describes the operation of taking the evolute as a linear operator Cγ → Cγ . Theorem 1 One has: ρ = −ρ − ρss . Proof. We have

dy ds = ρ, = k. ds dy

Combining (2), (3), and (4), yields d 1 d =− . ds τ dx Using (1) and (3), one obtains ρ = −ρ −

1  ρx  = −ρ − ρss , τ τ x

as claimed. 2

6

(4)

2.3

Spatial hypocycloids

Consider an arc length parameterized circle of latitude     √ t t , y = r sin , z = 1 − r2 , x = r cos r r

(5)

with 0 ≤ t ≤ 2πr. Let ρ(t) be the respective 2πr-periodic function. Then     Z 2πr Z 2πr Z 2πr t t sin ρ(t)dt = 0. ρ(t) dt = ρ(t) dt = cos r r 0 0 0 t Set α = ; then r Z 2π Z cos α ρ(α) dα = 0

2π

Z sin α ρ(α) dα =

0

2π

ρ(α) dα = 0, 0

that is, the Fourier expansion of ρ(α) is free from the constant term and from the first harmonics. We mention, in passing, that this implies that the function ρ(α) has at least four zeros on the interval [0, 2π); see, e.g., [5]. Next, consider the arc length parameter s on the dual circle of latitude. One has √ 1 − r2 √ = 1 − r2 α, s= r hence 1 ρ = −ρ − ρss = −ρ − ραα . (6) 1 − r2 Consider now the case when the function ρ is a pure harmonic: ρ(α) = cos kα with integral k ≥ 2. Then, by formula (6), ρ is proportional to ρ. In this case, the parametric equations of the curve Γ(t) (with the indicatrix being the circle of latitude (5)) are   sin(k − 1)t sin(k + 1)t x=r + , k−1 k+1   (7) √ cos(k − 1)t cos(k + 1)t sin kt 2 y=r − , , z =2 1−r . k−1 k+1 k It follows from formula (6) (and can be confirmed by a direct√computation) that the evolute Γ is obtained from Γ by switching r ←→ 1 − r2 and a r2 + k2 − 1 . homothety with the coefficient √ r 1 − r2 7

Corollary 2.7 The second evolute Γ(t) is obtained from Γ(t) by a homothety r2 (1 − r2 ) + k 2 (k 2 − 1) with the coefficient . r2 (1 − r2 ) This makes the curve Γ(t) similar to the classical hypocycloid.

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Figure 1: Spatial hypocycloids: k = 3 and k =

5 2

Let us provide some geometric information regarding these “hypocycloids.” The next proposition is proved by a straightforward calculation. Proposition 2.8 The curve Γ(t) with the parametric equations (7) is contained in the hyperbo oid k2 1 k2 − 1 2 2 (x + y ) − z2 = 2 2 2 4r 4(1 − r ) k −1 √ 2 1 − r2 between the p anes z = ± . It has 2k cusps at the points k √   2i − 1 2i − 1 1 − r2 2kr cos , 2kr sin , 2(−1) , i = 1, 2, . . . , 2k. 2k 2k k The number k may be rational, k = p/q > 1, (p, q) = 1. In this case the indicatrix will be the circle (5), traversed q times, the curve Γ will be contained in the same hyperboloid and will have 2p cusps. See Figure 1. 8

3

Polygons

We consider a closed n-gon P = (P1 , P3 , . . . , P2n−1 ) in R3 ; we assume that n ≥ 5 and that no 3 consecutive sides of P are parallel to the same plane.

3.1

Evolutes: definitions

There are two natural notions of the evolute of the polygon P; following the terminology of [1], we call them P-evolute and A-evolute. The P-evolute of P is the n-gon Q = (Q2 , Q4 , . . . , Q2n ) where Qi is the center of the sphere through the vertices Pi−3 , Pi−1 , Pi+1 , Pi+3 of P (we consider the subscripts as defined modulo 2n). The A-evolute of P is the n-gon R = (R1 , R3 , . . . , R2n−1 ) where Ri is the center of the sphere tangent to the (extensions of the) sides Pi−4 Pi−2 , Pi−2 Pi , Pi Pi+2 , Pi+2 Pi+4 of P. In other words, Qi is the point of intersection of the perpendicular bisector planes of the sides Pi−3 Pi−1 , Pi−1 Pi+1 , Pi+1 Pi+3 of P, and Ri is the point of intersection of the bisectorial planes of the angles Pi−4 Pi−2 Pi , Pi−2 Pi Pi+2 , Pi Pi+2 Pi+4 of P. Since every angle has two different bissectorial planes, a polygon has, in general, 2n different A-evolutes. Proposition 3.1 The sides of the second P-evoliute of a polygon P are parallel to the sides of the polygon P. Proof. Let S = (S1 , S3 , . . . , S2n−1 ) be the P-evolute of Q. The side Pi−1 Pi+1 of P and the side Si−1 Si+1 of S are both perpendicular to the sides Qi−2 Qi and Qi Qi+2 of Q; hence they are parallel to each other. 2

3.2

Involutes: existence and uniqueness

If an n-gon Q is the P-evolute of an n-gon P, then we call P a P-involute of Q. If an n-gon Q is one of the 2n A-evolutes of an n-gon P, then we call P an A-involute of Q. 9

Lemma 3.2 Let σ : R3 → R3 be a generic1 isometry. If σ preserves orientation, then it has no fixed points; if σ reverses orientation, then it has a unique fixed point; in all cases, σ has a unique invariant line. Proof. If σ preserves orientation, then it is a composition of rotation around a line and a parallel translation in the direction of this line, that is, a screw motion. This line is invariant with respect to σ; if the translation is not the identity, then σ has no fixed points. If σ reverses the orientation, then there exists a direction such that every line of this direction is mapped by σ to some other line of this direction, and this map between parallel lines reverses orientation. In the quotient of R3 by these lines, σ acts as an orientation preserving map. Generically, this map is a rotation of a plane about a point, so precisely one line is invariant with respect to σ, and hence σ has a fixed point on this line. 2 Proposition 3.3 If n is even, then a generic n-gon has no P-involutes. If n is odd, then a generic n-gon has a unique P-involute. For any n ≥ 5, a generic n-gon has a unique A-involute. Proof. Let Q = (Q2 , Q4 , . . . Q2n ) be an n-gon satisfying the conditions formulated at the beginning of this section. Let si , i = 2, 4, . . . , 2n, be the reflection of R3 is the plane containing the points Qi−2 , Qi , Qi+2 , and let σ = s2n ◦ · · · ◦ s4 ◦ s2 . Obviously, for a generic n-gon, σ is a generic isometry, preserving or reversing orientation, depending on the parity of n. The polygon Q is a P-evolute of a polygon P = (P1 , P3 , . . . , P2n−1 ) if and only if, for every i = 2, 4, . . . , 2n, the point Qi belongs to the perpendicuar bisector planes of the segments Pi−3 Pi−1 , Pi−1 Pi+1 , Pi+1 Pi+3 . In other words, for every i, the points Qi−2 , Qi , Qi+2 lie in the perpendicular bisector plane of Pi−1 Pi+1 , which means, in turn, that Pi+1 = si (Pi−1 ). Thus, σ(P1 ) = P2n+1 = P1 . If n is even, then σ preserves orientation, and by Lemma 3.2, it has no fixed points, so P cannot exist. If n is odd, then σ has a fixed point. We take this point for P1 and put P3 = s2 (P1 ), P5 = s4 (P3 ), P7 = s6 (P5 ), and so on; we obtain an n-gon P, whose P-evolute is Q. The proof for A-involutes is similar, the only difference is that we begin not with a fixed point, but with an invariant line of σ. We succesively apply to 1

The word generic means that the statement holds within a dense open set of isometries.

10

this line the reflections s2 , s4 , s6 , . . .. We obtain n lines which form an n-gon whose A-evolute is Q. (Notice that if we reflect a line in a plane not parallel to this line, then the line and its reflection intersect at the intersection point of the line with the plane.) 2 In the rest of the paper, we will consider neither A-evolutes nor Ainvolutes. So, we will refer to P-evolutes and P-involutes simply as to evolutes and involutes. Remark 3.4 It is interesting to study iterations of the involute transformations as well; we do not do it in this paper. In dimension two, such a study was made in [1].

3.3

Spherical polygons and polygons with fixed directions of sides

A spherical n-gon is a cyclically ordered collection of n points of the unit sphere S 2 ; notation: v = (v2 , v4 , . . . , v2n ) or u = (u1 , u3 , . . . , u2n−1 ). We assume that the polygons are generic: every consecutive pair of vectors is not coplanar. Put ci = vi−1 · vi+1 , ei = |vi−1 × vi+1 |, fi = vi−2 · vi+2 , di = det(vi−2 , vi , vi+2 ), where all indices are understood cyclically. Note the identities c2i + e2i = 1, d2i = 1 + 2fi ci−1 ci+1 − fi2 − c2i−1 − c2i+1 (the latter is the Gram determinant). The signature of a polygon v is the cyclic sequence s of signs si of the determinants di . For three unit vectors a, b, c, we write c ∼ a × b for c=

a×b . |a × b|

For a spherical n-gon v = (v2 , v4 , . . . , v2n ), let u = v∗ be the dual n-gon u = (u1 , u3 , . . . , u2n−1 ) given by ui ∼ vi1 × vi+1 . The result of applying duality twice depends on the signature. Lemma 3.5 One has (v∗ )∗ = (s2 v2 , s4 v4 , . . . , s2n v2n ). 11

Proof. Let w = u∗ . It suffices to consider three consecutive vertices v−2 , v0 , v2 . Then u−1 ∼ v−2 × v0 , u1 ∼ v0 × v2 , and w0 ∼ (v−2 × v0 ) × (v0 × v2 ) = d0 v0 , as claimed. 2 What is the signature of the dual polygon? Lemma 3.6 The signature of v∗ is as follows: (s2 s4 , s4 s6 , s6 s8 , ...). Proof. The result follows from the triple product identities for cross product: det(vi−2 × vi , vi × vi+2 , vi+2 × vi+4 ) = (vi−2 × vi ) · ((vi × vi+2 ) × (vi+2 × vi+4 )) = di+2 (vi−2 × vi ) · vi+2 = di+2 di , as needed. 2 Fix a spherical n-gon v, and let Pv be the space of n-gons in R3 whose sides are parallel to the respective vectors vi . If P = (P1 , P3 , . . . , P2n−1 ) is such a polygon, define the real numbers xi (signed side lengths) by Pi+1 − Pi−1 = xi vi . The vector x = (x2 , . . . , x2n ) uniquely determines the polygon up to parallel translation. P The coordinates xi satisfy the three linear relations xi vi = 0, saying that the respective polygon closes up. Thus Pv is a vector space of dimension n − 3. Notice that the space Pv stays the same if we replace some vectors vi be the opposite vectors (although some coordinates xi change signs).

3.4

The evolute transformation

In this section we show that the evolute of a polygon P ∈ Pv belongs to Pu , where u = v∗ , and that the evolute map E : Pv → Pu is a linear transformation. Lemma 3.7 Let P ∈ Pv and Q = E(P). Then Q ∈ Pu , where u = v∗ . 12

Proof. Both points Q2i Q2i+2 lie in the planes perpendicular the vectors v2i and v2i+2 . Therefore the line Q2i Q2i+2 is perpendicular to these vectors, and hence parallel to their cross-product. 2 Let a, b, c, d be four points in R3 , and let z be their circumcenter. Lemma 3.8 The point z is determined by the system of linear equations z · (b − a) =

c 2 − b2 d 2 − c2 b 2 − a2 , z · (c − b) = , z · (d − c) = . 2 2 2

Proof. The circumcenter z satisfies the equations |z − a|2 = |z − b|2 = |z − c|2 = |z − d|2 , that are equivalent to the above system of linear equations. 2 Let P = (P1 , P3 , . . . , P2n−1 ) ∈ Pv , and let Q = (Q2 , Q4 , . . . , Q2n ) = E(P) ∈ Pu . Define the real numbers y2i+1 by Q2i+2 − Q2i = y2i+1 u2i+1 . We obtain a vector y = (y1 , y3 , . . . , y2n−1 ), and the evolute transformation E : Pv → Pu acts by x 7→ y. Theorem 2 The vector y is obtained from x by a linear transformation depending on v. Proof. The claim is local, so consider five consecutive vertices of P, and let the consecutive vectors of the sides be x0 v0 , x2 v2 , x4 v4 , and x6 v6 . Assume that the middle vertex is the origin. Then the five vertices are − x0 v0 − x2 v2 , −x2 v2 , 0, x4 v4 , x4 v4 + x6 v6 .

(8)

The two relevant vertices of the evolute are Q2 and Q4 , and the relevant unit vector is u3 . Apply Lemma 3.8 to the first four points in (8) to obtain: Q2 · v0 = −

x2 x4 x0 + 2x2 (v0 · v2 ) , Q2 · v2 = − , Q2 · v4 = . 2 2 2 13

This is a linear system on Q2 whose right hand side depends linearly on x. Hence Q2 is a linear function of x0 , x2 , x4 . Likewise, Q4 is a linear function of x2 , x4 , x6 : Q4 · v2 = −

x4 x6 + 2x4 (v4 · v6 ) x2 , Q4 · v4 = , Q4 · v6 = . 2 2 2

Hence Q4 −Q2 is a linear function of x0 , x2 , x4 , x6 , and so is y3 = (Q4 −Q2 )·u3 . 2 In the next section, we shall give an explicit formula for the evolute transformation.

3.5

Rank of the evolute transformation and an explicit formula

The following proposition describes the rank of the evolute transformation. Proposition 3.9 Generically, the evolute transformation E : Pv → Pu has 1-dimensional kernel if n is even, and has full rank if n is odd. Proof. If n is odd, then the map E : Pv → Pu has a full rank, because it is onto: a generic odd-gon has an involute (Proposition 3.3). Let us prove that if n is even, then dim Ker E ≥ 1 and, generically, this dimension is 1. A polygon belongs to the kernel of the evolute map if and only if it is inscribed into a sphere. We will show that, for every v, Pv contains a polygon inscribed into the unit sphere S 2 and, generically, this polygon is unique up to the reflection in the center. Take a point P1 ∈ S 2 and reflect it, successively, in planes passing through 0 and perpendicular to the vectors v2 , v4 , . . . , v2n ; we obtain points P3 , P5 , . . . , P2n+1 ∈ S 2 . If P2n+1 = P1 , then P = (P1 , P3 , . . . , P2n−1 ) ∈ Pv is a polygon inscribed in S 2 , and all polygons from Pv inscribed in S 2 are obtained by this construction. The transformation P1 7→ P2n+1 is an orientation preserving isometry of S 2 . It is either a rotation about some axis, in which (generic) case it has two antipodal fixed points, or the identity. Proposition follows. 2 Next we derive explicit formulas for the evolution map E : Pv → Pu . In the next proposition, the symbols c, d, e, f have the same meaning as described in Section 3.3. We compute y3 as a linear function of x0 , x2 , x4 and x6 ; 14

the values of other yi are obtained from this formula by cyclic permutations of the indices. Proposition 3.10 One has     c3 f2 + c1 − 2c1 c23 f4 − c3 c5 1 − c23 + x2 + 2e3 y3 = x0 d2   d4 2  .  d2 1 − c3 f2 − c1 c3 c3 f4 + c5 − 2c5 c23 + + x6 , +x4 d2 d4 d4 Proof. Let 

   1 2c1 0 −1 0 0 1 1 0 , B =  0 1 0 , A=  0 1 2 2 0 0 −1 0 2c5 1 and let M2 and M4 be the matrices with the columns v0 , v2 , v4 and v2 , v4 , v6 , respectively. Let ξ2 = (x0 , x2 , x4 )T , ξ4 = (x2 , x4 , x6 )T . Then, by formulas in the proof of Theorem 2, M2T (Q2 ) = −A(ξ2 ), M4T (Q4 ) = B(ξ4 ); hence, y3 = (M4T )−1 B(ξ4 )·u3 +(M2T )−1 A(ξ2 )·u3 = ξ4 ·B T (M4−1 (u3 )+ξ2 ·AT M2−1 (u3 ). One has

1 (v2 × v4 , v4 × v0 .v0 × v2 )T , d2 and likewise for M4−1 . One computes the vectors M4−1 (u3 ) and M2−1 (u3 ), using the fact that u3 = (v2 × v4 )/e3 , and the identity M2−1 =

(a × b) · (c × d) = (a · c)(b · d) − (a · d)(b · c). The result is as follows: 1 (c3 c5 − f4 , f4 c3 − c5 , 1 − c23 )T , e 3 d4 1 M2−1 (u3 ) = (1 − c23 , c3 f2 − c1 , c1 c3 − f2 )T . e 3 d2

M4−1 (u3 ) =

It remains to multiply these vectors on the left by the matrices B T and AT and to take dot product with the vectors ξ4 and ξ2 . This yields the result. 2 15

3.6

Iterations of the evolute transformation

We start with small values of n. The case n = 4 is trivial: the evolute degenerates to a point. Pentagons. The following statement was proved by E. Tsukerman [9]. For completeness, we reproduce his proof here. Theorem 3 For a generic pentagon P in space, its second evolute E 2 (P) is homothetic to P. Proof. Let P = (P1 , P3 , P5 , P7 , P9 ), E(P) = (Q2 , Q4 , Q6 , Q8 , Q10 ), and E 2 (P) = (R1 , R3 , R5 , R7 , R9 ). We claim that the planes (Pi Pj Pk ) and (Ri Rj Rk ) are parallel for all triples (i, j, k). This claim implies that the tetrahedra (Pi Pj Pk Pm ) and (Ri Rj Rk Rm ) are homothetic for every quadruple (i, j, k, m) since their respective faces are parallel. Since these tetrahedra share elements, the centres and the coefficients of the homotheties coincide for all quadruples (i, j, k, m). To prove the claim, let us show that the planes (P1 P3 P5 ) and (R1 R3 R5 ) are perpendicular to the line (Q2 Q4 ); the other triples of indices are treated in the same way. Indeed, the points Q2 and Q4 lie in the perpendicular bisector plane of the segment P1 P3 and in the perpendicular bisector plane of the segment P3 P5 , hence (Q2 Q4 ) is perpendicular to the plane (P1 P3 P5 ). Likewise, the points R1 , R3 , and R5 lie in the perpendicular bisector plane of the segment Q2 Q4 . Hence (Q2 Q4 ) is perpendicular to the plane (R1 R3 R5 ). 2 Actually, Tsukerman’s paper contains the same result for all (n + 2)-gons in Rn , and the proof goes along the same lines. Tsukerman’s paper also contains analogs of this result in the spherical and hyperbolic geometries. Hexagons. Here the situation becomes the same as with pentagons after the first application of the evolute transformation. Theorem 4 For a generic hexagon P in space, its third evolute E 3 (P) is homothetic to its first evolute E(P).

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Proof. Let P = (P1 , P3 , P5 , P7 , P9 , P11 ), E(P) = (Q2 , Q4 , Q6 , Q8 , Q10 , Q12 ). A hexagon P with fixed generic directions of the sides, modulo parallel translations, is uniquely characterized by the vector of its long diagonal (P1 P7 ). Likewise, the evolute Q is characterized by its long diagonal Q4 Q10 . The evolute transformation E : P1 P7 7→ Q4 Q10 is linear. We claim that the vector Q4 Q10 is perpendicular to P1 P7 . Iindeed, Q4 lies in the perpendicular bisector plane of the segment P1 P7 , and so does Q10 . Thus the map E is skew-symmetric, and hence Tr(E) = 0. This implies that, on the image of E, the map E 2 is a scalar map. 2 Conjecture For n ≥ 4, the third evolute of an (n + 3)-gon in Rn is homothetic to its first evolute. For n = 2, this is Proposition 4.15 of [1]. General case, n ≥ 7. The second evolute transformation acts separately on even numbered and odd numbered evolutes, see Figures 2, 3 below. Let v be a spherical polygon and u its dual. Proposition 3.11 The second evolute maps Pv → Pv and Pu → Pu have the same eigenvalues. Proof. One has two first evolute maps E1 : Pv → Pu and E2 : Pu → Pv . Hence one also has two second evolute maps E2 ◦ E1 : Pv → Pv and E1 ◦ E2 : Pu → Pu , and these two compositions share the eigenvalues. 2 The asymptotic behavior of the iterated evolutes E 2k (P) of a generic n-gon P depends on the maximum module eigenvalue of the linear transformation E 2. If this eigenvalue is real, then the sequence of iterated evolutes E 2k (P), after appropriate parallel translations and rescalings, is asymptotically 2periodic (with or without flips at every step, depending on the sign of this real eigenvalue). If the maximum value eigenvalue is not real, then this sequence looks chaotic. The sequences of evolutes of heptagons are shown in Figures 2, 3.

17

•........A

′′

..... .. ... . ... .. .. ... ... .... ... ..... . . .. . . .. .... .. ... .. ..... .. ... .. .. ... ... . ... ... .... . ... ... .. . . . ... ... .. .. .. .. ... . ... . . ... F ..... . ′ . . . ... . ..•... . .. . ... F . . ... . .. ... .. . . ... ′ ′ .• . . . . ′ . ... .. .... ... . . . G . . . . . . . . . . .. .... ... B ...................................C . ... . . . . . . • ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . •• . • . . . . .. .... . • ... ... ... ... E . . . . . . ′ . .. .... . . . ... ... ... ... ... . . F ′′ D ... .. .. ... ... ′′... .• ... ... ... . C . . .. ...... . ′′ . . . C . . . D ... . . . . . . . . . . . . D..........................................................................................•............................................................•.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...............................• ........ ... ... ..................................................................... .. ′′ . . • • . . . . . . ... . . .. E B .. .. • ... B ′′ ...... ... .... .. .. .. .. E .. ... ... .. .... G′′•.. .. .. ... ... . . ...... ... ... .. .. .. .. ... ... •.. G .. .. ... ... . . . . ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. .... ..... ..... ...... . .•.. ′ ... .. ... ... .... .. ... . A . .... .. .. . .. ... ... ... . ... . ... .. ... .. ... .. .. ... ... . ... . ... ... . .. . ... ... . . ... .. ... ... . ... ... .. ... ... ... .. .. . . . . . ... . . . . ... ... . . . . . . . . . . . . . . . .................................................................. . . . . . . . . .............................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . ... ... ... . . . .. . .. . . . ... ... . ... . . . . . . . . . . . . . ... ... ... ... ... . .. ... .. .. .. ... ... ... ... .. ... ... .. .. . ... ... ... .. ... ... .. ............................................................................ .................................................................................. . .. ... . ... .. . . . . . ... ... ... .. .. ... ... .. ... ... ... ... .. ... .. ... ... ... .. ... .. . . . . . . . . . . ... ... ... . ... .. ... ... ... .. .. ... ... ... ... .. .. ... .. ... ... ... . .. ... . . . ... ... .. ... .. .. ... .. ... ... ... ... ... .... ... .. ... .. ... ... ... .. ... .. ... ... . . . ..... ..... ..... ..... .. ..

•..... A

Figure 2: A spatial heptagon ABCDEF G and the sequence of its even numbered evolutes (likewise denoted sides are parallel to each other). The maximum module value of the second evolute transformation is real and negative. For this reason, the evolute is flipping at every step.

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... ..................... ................... ... ............................. ...... ......................................... ...... ..................................... ...... ................................... . . . . ............... ...................... . . ............... ..................... . . . . . . . . . . . . . ....................................... ..... . . . . . . . . . . .......... . ............ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............... ... ... ... .. .. ....... . . . . . . . . ............... . . . . . . . . . . ................................................................. ... .. .......... ... .. . . . . . . . ... ... ................... ..................... . . ... ... . . ... .. ... .. ................................ ..................... ............................................................ ... .. ... .. ... ... ..................... ............... ..................... ............... ... ... . . . . . . . . . . . . . . . . . . . . ... .. . . . . . . . . . . . .................................... ................................... . . ... ... . . . . . . ......................... .......................... ... .. ... ... ... .. ....... ....... . ... ... . . . . ... .. ... .. ... .. ... .. ... .. ... .. ... ... ... ... ...... ... .. . ... .. ..... ... .. ... .. ..... ... .. ... ... .. ... ... ..... . . ..... ..... ..... .... ... .... . ... . ... .. ...... . .................... ...... ...... .................... ...... . . . . ........................................... . . ...... . . . . . . . . . . . . . . . .................................... .... ..... ................................... . ...... . ..................................... . . . . . . . . ... ... . . . . . . . . . . . . . . . . . . ............... .................... ............... ..................... ... .. .. ... . .. ................................. .... ... .. ........................................ . ....... . . . . . . . . . . . . . . . . . . . . . ................ .................. . . ................ . . . . . . . . . . . . . . . . . .... ............. ....... . . . .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. ................................. ................. .................................. ................. ... .. ..................................................................... ... .. .................................... .................................... .................................... ... ... .................................... . . ................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................................... .................................. .................................. ... ... ... .. ... ............... .............. .............. ... ... ... ... ... ..... ... .. . . ..... ..... .... ..... .. .... .

Figure 3: The sequence of odd numbered evolutes of the same heptagon.

References [1] M. Arnold, D. Fuchs, I. Izmestiev, S. Tabachnikov, E. Tsukerman. Iterating evolutes and involutes. Preprint arXiv:1510:07742v2. [2] V. Arnold. The geometry of spherical curves and quaternion algebra. Russian Math. Surveys 50 (1995), 1–68. [3] W. Blaschke, K. Leichtweiß. Elementare Differentialgeometrie. SpringerVerlag, Berlin-New York, 1973. [4] D. Fuchs. Evolutes and involutes of spatial curves. Amer. Math. Monthly 120 (2013), 217–231. [5] D. Fuchs, S. Tabachnikov. Mathematical omnibus. Thirty lectures on classic mathematics. Amer. Math. Soc., Providence, RI, 2007. [6] B. Gr¨ unbaum. Quadrangles, pentagons, and computers. Geombinatorics 3 (1993), no. 1, 4–9. [7] B. Gr¨ unbaum. Quadrangles, pentagons, and computers, revisited. Geombinatorics 4 (1994), no. 1, 11–16. 19

[8] D. Struik. Lectures on Classical Differential Geometry. Addison-Wesley Press, Inc., Cambridge, Mass., 1950. [9] E. Tsukerman. The perpendicular bisector construction in n-dimensional Euclidean and non-Euclidean geometries. Preprint arXiv:1203.6429. [10] R. Uribe-Vargas. On vertices, focal curvatures and differential geometry of space curves. Bull. Braz. Math. Soc. 36 (2005), 285–307.

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