Iterating evolutes and involutes

Work in progress with M. Arnold, D. Fuchs, I. Izmestiev, and E. Tsukerman

Integrability in Mechanics and Geometry: Theory and Computations

June 2015

1 2 Evolutes and involutes

Evolute: the envelope of the normals, the locus of the centers of curvature; free from inflections and has zero algebraic length. Involute: string construction; come in 1-parameter families. 3 Hedgehogs

Wave fronts without inflection and total rotation 2π, given by their support function p(α).

...... π ...... α + ...... 2 ...... p!(α)...... O ...... γ ...... •...... p(α) ...... α

The evolute map: p(α) 7→ p0(α − π/2), invertible on functions with zero average. 4 Steiner point: the center of mass of the curve with the density equal to the curvature: 1 Z 2π ! p(α) (cos α, sin α) dα π 0 Choosing the origin there “kills” the first harmonics of p(α).

Theorem: the iterated involutes converge to the Steiner point, the limiting shape being a hypocycloid, generically, of order two.

...... • ...... • ...... • ...... 5 Corollary: if a hedgehog is similar to its evolute then it is a hypocycloid.

......

The converse is a classical fact, due to C. Huygens (1678).

6 The 76th iterated evolute of the curve with p = exp(sin α):

Proposition: unless the support function is a trigonometric poly- nomial, the number of cusps of the iterated evolutes increases without bound (a consequence of a Polya-Wiener theorem).

7 Other rotation numbers

......

Cardioid (3/2 full turns) and nephroid (2 full turns).

8 Cycloid

γ(α + 2π) = γ(α) − (2π, 0), p(α) = −α cos α + f(α)

Theorem: under this periodicity condition, iterated involutes converge to the cycloid, which is the only such curve that coin- cides with its evolute, up to a parallel translation.

9 Discretization: P -evolute and A-evolute

...... P ...... P ...... Q ...... Q ......

The angle evolute depends on orientations of the sides.

10 Existence and uniqueness of involutes

Let S be the composition of reflections in the sides of Q. For P -involute, one wants a fixed point, and for A-involute, a fixed line. n odd: generically, S is a glide reflection (a unique fixed line and no fixed points), but may be a reflection; n even: generically, S is a rotation (a unique fixed point and no fixed lines), but may be a parallel translation or the identity.

Generically “good” cases: odd n, for A-involute, and even n, for P -involute.

11 Formulas for evolutes

Space of lines (a cylinder):

12 * * Li+1/2 Li L i Li+1

Li

A-evolute: α + α π p − p α∗ = i i+1 + , p∗ = i+1 i . i+1/2 2 2 i+1/2 αi+1−αi 2 sin 2

∗ π P -evolute: αi = αi + 2, sin sin + sin( ) ∗ pi+1 θi−1/2 − pi−1 θi+1/2 pi θi+1/2 − θi−1/2 pi = . 2 sin θi−1/2 sin θi+1/2 where θi+1/2 = αi+1 − αi is the turning angle.

13 Discrete hedgehogs

Oriented lines L1,...,Ln with the turning angle from Li to Li+1 in (0, π), and the total turning equal to 2π, an analog of inflection- free curves. Side lengths are signed.

Important subclass: equiangular hedgehogs. 14 Discrete Fourier decomposition and discrete hypocycloids

The hedgehogs 2πk 2πmk 2πk 2πmk Cm(n) = , cos ,Sm(n) = , sin , k = 1, . . . , n, n n n n are analogs of pure harmonics (0 ≤ m ≤ n/2).

1 6 2 1 3 2

6 3 4 5 5 4

Discrete hypocycloid: equiangular and tangent to a hypocycloid. 15 Proposition: let L1,...,Ln be the consecutive sides of a discrete hypocycloid tangent to a hypocycloid of order n+1 or n−1. Then all these lines are concurrent at a point on the circle inscribed into the hypocycloid.

Case n = 3: the astroid is the envelope of the chords of a circle whose endpoints move in the opposite directions, one three times faster than the other (Cremona).

16 Theorem: Let Q be an n-sided discrete hypocycloid, tangent to a hypocycloid H of order m. Then the A-evolute of Q is tangent to the evolute of H, scaled by sin(πm/n)/m sin(π/n). Likewise for its P -evolute, with the scaling factor sin(2πm/n)/m sin(2π/n). The second evolute of Q is similar to Q.

17 Define the Steiner point of an equiangular hedgehog (αi, pi), i = 1, . . . , n (not the standard definition): n 2 X pi (cos αi, sin αi). n i=1

Theorem: an equiangular hedgehog possesses an A-involute if P and only if pk = 0. The Steiner point of an equiangular hedge- hog and of its A-evolute coincide, and the iterated A-involutes converge to the Steiner point. Likewise, for the P -evolutes and involutes.

18 Dynamics of the shapes of A-evolutes

For n-sided equiangular hedgehogs, the highest discrete hypocy- cloid dominates. If n is even, it is Cn/2, and if n is odd, it is aC(n−1)/2 + bS(n−1)/2, whose shape evolves with period 2.

And in general,

Theorem: the iterated A-evolutes of an n-sided hedgehog con- verge, in shape, to a discrete hypocycloid, generically, of the highest order: n/2 for even n, and (n − 1)/2 for odd n.

19 Dynamics of the angles of A-involutes, n odd

Recall the composition of reflections S in the sides of a polygon. For evolution of angles, assume that the lines, (L1,...,Ln), pass through the origin. Then S is a reflection is some line. Its orbit under the consecutive reflections is the new cyclically ordered ∗ ∗ collection of lines (L1,...,Ln).

∗ ∗ Theorem: The n-torus map (L1,...,Ln) 7→ (L1,...,Ln) is er- godic.

20 Case n = 3: pedal triangles

Well studied (J. Kingston, J. Synge, P. Lax, P. Ungar, J. Alexan- der, A. Manning): the iterated pedal triangles converge to a point, but their shapes behave chaotically; the map is ergodic and is isomorphic to the shift on four symbols.

21 A different kind of angle bisector evolute ...... Oriented the sides...... by... the cyclic order...... of the vertices...... = 4: the iterated evolutes of...... a convex quadrilateral converge.... . n ...... to a square, and those of a non-convex quadrilateral degenerate...... to a segment...... = 5...... :...... observed a...... strange...... eventual...... 4-periodicity,.. with rotation...... n ...... ◦ ...... through 108... . Conjecturally,...... it’s. a local attractor......

......

...... 22 ......

...... Iterating perpendicular bisector evolutes

∗ Recall the linear map Aα : p 7→ p : sin sin + sin( ) ∗ pi+1 θi−1/2 − pi−1 θi+1/2 pi θi+1/2 − θi−1/2 pi = , 2 sin θi−1/2 sin θi+1/2 and let

C = (cos α1, cos α2,..., cos αn),S = (sin α1, sin α2,..., sin αn).

Theorem: (i) Aα(C) = −S, Aα(S) = C; (ii) Let α be generic. Then, for even n, the map Aα is non- degenerate, and for odd n, it has a 1-dimensional kernel; (iii) If n is even and α1 − α2 + α3 + ... − αn = 0 mod π (parallel to an inscribed n-gon), then Aα has a 2-dimensional kernel.

23 Theorem: The spectrum of Aα is symmetric with respect to the origin.

Proof: M−1A>M = −A, where the matrix M computes the side lengths of a polygon from its support numbers: ` = M(p).

∗ Equiangular case: pi = pi+1 − pi−1, and the eigenvalues are 2i sin(2πj/n), j = 0, 1, . . . , n − 1. For n = 5, the dominating eigenvalue looks like this (the space is 2-dimensional):

24 ......

...... Evolution...... of...... shapes...... The dominating eigenvalues may be real or complex (what are

...... the...... probabilities?)...... In the...... former case,...... the limiting...... shape...... is... 2- ...... periodic:......

......

A complex example (also n = 6), close to period 9:

25 ......

...... 26 Curious examples: a self-evolute and a pair of mutual evolutes

27 Small values of n n = 4: T 2(P ) is homothetic to P (Bennett, Gr¨unbaum,King, Langr, Shephard). n = 5: T 3(P ) is homothetic to T (P ) (conjectured by Gr¨unbaum, now a theorem).

28 3 n = 6: If α1 − α2 + α3 + ... − α6 = 0 mod π, then T (P ) is homothetic to T (P ), and if the opposite sides of P are parallel, then T 3(P ) = T (P ).

29 Thank you!

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