Straightening Soliton Curves
Straightening Soliton Curves Joel Langer Abstract. A canonical straightening process is described for soliton curves associated with the localized induction hierarchy. Following computer animated examples, the present topic is placed in the context of a larger theme: the soliton class is a natural setting for representation of diverse topological and geometrical behavior of curves and their motions.
The constructions we describe here S∞pertain to the soliton class, Γ, of unit speed curves γ in Euclidean 3-space: Γ = n=0 Γn , where Γ1 = {lines}, Γ2 = {helices}, Γ3 = {elastic rods}, 4 ⊃ {buckled rings under pressure}, . . . For concise definiPΓ ∞ tion, consider X = n=0 λn Xn , a formal series of vectorfields along γ(s) satisfying X0 = − ∂γ ∂s = −T and JXn = ∂Xn−1 , n = 1, 2, . . . , i.e., (1)
JX = λ∂X
∂ Here, J = T × (cross product with unit tangent), and ∂ = ∂s = ∇T (covariant derivative). Let Γn = {γ : 0 = Xn }; as will be seen, Γn is defined by an nth -order ODE for T = γs , depending on n constants. Since J 2 = −Id on normal vectorfields, Xn = fn T − J∂Xn−1 , where ∂fn = ∂hT, Xn i = h∂T, Xn i + hT, ∂Xn i = h−JX1 , Xn i determines fn up to a constant of integration. Alternatively, (1) implies λ∂hX, Xi = 2hJX, Xi = 0, so for some constants Cn , ∞ X (2) hX, Xi = p(λ) = 1 + Cn λn n=1
Pn−1 The λ -term of (2) gives fn without antidifferentiation: 2fn = −Cn + k=1 hXk , Xn−k i, n = 2, 3, . . . .PThe normalization p(λ) = 1 (all Cn are zero) yields a special solution ∞ to (1), Y = n=0 λn Yn , whose terms are generated inductively: 3 Y0 = −γs , Y1 = γs × γss , Y2 = hγss , γss iγs + γsss , 2 3 Y3 = hγs × γss , γsss iγs − γs × γssss − hγss , γss iγs × γss , 2 n
n−1
(3)
Yn = (
1X hYk , Yn−k i) T − J∂Yn−1 2 k=1
Mathematics Subject Classification. 35Q51, 53A04. Key words and phrases. solitons, curves, localized induction hierarchy.
1
2
Joel Langer
P∞ n The general solution to (1), X = n=0 λ Xn , may be written in terms of Y , A0 = 1, and “integration constants” A1 , A2 , . . . : ∞ n ∞ ∞ X X X X p An−k Yk = ( Ai λi )( λj Yj ) = p(λ) Y (4) X= λn n=0
k=0
i=0
j=0
The above formulas are presented in more detail in [1], where background and related topics are also discussed. The localized induction hierarchy of soliton equations, γt = Yn , is considered from a Hamiltonian viewpoint in [2], along with the connection to the better known nonlinear Schr¨ odinger hierarchy. Here we focus on interpretation of Equation 2. Note that Y may be regarded as a (formal) differential operator, depending on a parameter λ, acting on spherical curves (tangent indicatrices) T ; as such, Y is unambiguously defined by (3). On the other phand, Equation 4 shows that Y may be variously represented in the form Y = X/ p(λ), in terms of arbitrary constants A1 , A2 , . . . . We now use this fact to address the issue of convergence of Y = Y [γs ], evaluated on soliton curves γ: Pn Theorem 1. Let γ ∈ Γn ; specifically, let 0 = Xn = k=0 An−k Yk along γ, and assume γ does not belong to Γn−1 . Then X as in (1) may be assumed to terminate, p(λ) = hX, Xi is a non-vanishing polynomial, and Y converges to a homotopy of spherical curves, (5)
n−1 r X X p 1 Ar−k Yk , Tλ = X/ p(λ) = p λr p(λ) r=0 k=0
deforming T0 = −T to a point, as λ → ±∞. P∞ P∞ Pr Proof. Let X = r=0 λr Xr = r=0 λr k=0 Ar−k Yk be a solution to (1), where A1 , A2 , . . . , An are such that Xn = 0 along γ. Then JXn = ∂Xn−1 implies Xn−1 = Pn−1 Pm k=0 An−1−k Yk is constant. For m ≥ n, let Am+1 = k=0 Am−k hT, Yk+1 i. One verifies by induction that, for m ≥ n, Am+1 is a constant such that Xm+1 = Pm+1 Pn−1 = 0. Thus, X is terminating: X = r=0 λr Xr . Further, vanishk=0 Am+1−k Yk P 2n−2 ing of p(λ) = 1 + r=1 Cr λr for some real λ would imply γ ∈ Γn−1 . The values of Tλ at λ = ±∞ may now be read off: T±∞ = (±1)(n−1) Xn−1 /kXn−1 k. Since Y is local, Tλ is periodic in s when T is. Corollary 2. Antidifferentiation of (5) yields a quasiperiod-preserving regular homotopy of unit speed curves in E 3 , deforming γ0 = −γ to a straight line, as λ → ±∞; in fact, γλ has curvature-normal vector, κλ Nλ = ∂Y = JY λ (λ 6= 0), whose length is at most 1/|λ|. Assuming κλ non-vanishing, Fλ = {Tλ , Nλ , Bλ } gives a homotopy in SO(3), deforming the Frenet frame of γ to a one-parameter subgroup (reparametrized and translated). Example 1 For γ ∈ Γ2 , γλ is a homotopy of helices. The loop (one period of) Fλ is nontrivial in π1 (SO(3)); the (quasi)writhe of γ is converted into twist as λ → ±∞, and {γλ , Fλ } becomes a (uniform) spinning line. Fλ lifts to a minimal torus in SU (2) ∼ torus - with conformal parameters θ, φ given by θ = cs, = S 3 - the Clifford √ τ −λc2 2 cot φ = κ , and c = κ + τ 2 (κ and τ being the constant curvature and torsion √ of γ). These conclusions follow from Y = (Y0 + λ(A1 Y0 + Y1 ))/ 1 − 2λτ + λ2 c2 = hsin φ cos θ, sin φ sin θ, cos φi (Cartesian coordinates in the latter expression with zaxis parallel to the constant vector X1 = A1 Y0 + Y1 ).
Straightening Soliton Curves
3
Example 2 Γ3 consists of (centerlines of) elastic rods. Such γ satisfy a variational principle involving bending, twisting, and extension energies, with Euler equation equivalent to 0 = X3 = A3 Y0 + A2 Y1 + A1 Y2 + Y3 . (In the planar case, the next equation, 0 = X4 , simply adds a hydrostatic pressure term to bending and extension terms, yielding a standard model for cross sections of cylindrical pipes buckling under pressure.) The equation 0 = X3 has first integrals hX2 , X2 i = j 2 and hX1 , X2 i = h, which may be expressed as a pair of ODE’s for the curvature κ and torsion τ of γ. One thus obtains κ and τ in terms of elliptic functions: 2 2 2 κ2 (s) = κ20 (1 − α β 2 sn (κ0 s/2β, α)), κ (s)(2τ (s) + A1 ) = c; here, 0 ≤ α ≤ β ≤ 1, and α is the modulus of the Jacobi elliptic sine, sn(u, α). The (congruence classes of) curves in Γ3 form a four-dimensional family, parametrized by α, β, A1 , A2 , (the remaining constants being determined by these). One may express cylindrical coordinates along a soliton curve γ ∈ Γ by quadrature in terms of its curvature and torsion (details for the case of elastic rods may be found in [3]). E.g., in the special case of inflectional (non-inflectional) planar elas1 tica, β = α (β = 1), we may write r = κ/j, θ = θ0 , and zs = 2j (κ2 − 2A2 ). Further, p A3 = A1 = 0, and Y = (Y0 + λY1 + λ2 X2 )/ p(λ), where X2 = A2 Y0 + Y2 = j kˆ is a constant vector pointing in the z-direction. Integration of Y yields the homotopy γλ explicitly, and the Frenet frame Fλ may be obtained by differentiation. Straightening homotopies for inflectional and non-inflectional cases are shown in Figure 1 and Figure 2, respectively. Mathematica was used to compute binormal ribbons γ(s, t; λ) = γλ + tBλ (s), with −1 ≤ t ≤ 1, = 0.1, s ranging over two periods of the curvature function κλ , and λ = tan(0.222m), m = 0, . . . , 7. See [4] for an animated version of Figure 1, but showing twice as much of the curve, and λ = tan(.0468m), m = −33, . . . , 33. Cartesian coordinates were chosen with γ0 = −γ lying in the x1 , x2 -plane. In the inflectional case - we omit details for the (similar) non-inflectional case - the coordinates of γλ are: √ x1 = [16α3 E(φ, α) − 2α(4α2 + λ2 )u]/ q, √ x2 = −16α4 cn/ q, √ x3 = 8α2 λ arcsin(α sn)/ q, q
= 16α4 p(λ) = 16α4 + 8α2 (2α2 − 1)λ2 + λ4 .
Here, u = s/2α, and we use shorthand for the amplitude √ φ = am(u, α), elliptic functions cn = cn(u, α), sn = sn(u, α), dn = dn(u, α) = 1 − α2 sn2 , and E is the elliptic integral of the second kind. The normalization κ0 = 1 and the formulas 2 1 A2 = 2α4α−1 2 , j = 4α2 were applied. We note that the curvature-normal vector of γλ is given by κλ Nλ = ∂Y = √ (−8α3 sn cn dn, 4α2 cn(1 − 2α2 sn2 ), −2αλsn dn)/ q, and the curvature of γλ is √ 2α 2 2 2 2 2 κλ = √ q 4α cn + λ sn dn , which is non-vanishing for 0 < |λ| < ∞. The binormal Bλ is computed as Bλ = Y × ∂Y /κλ . As λ → 0, the binormal ribbon develops a kink, corresponding to the singularity of the Frenet frame of the original inflectional elastica (exceptional among γ ∈ Γ3 ). A similar phenomenon - apparent in the computer animations - occurs as λ → ±∞, for both classes of planar elastica; one may regard τ±∞ as a sum of Dirac delta functions, one for each vertex of the original elastica. (In this respect, the helix example better represents generic behavior for non-planar γ ∈ Γ3 .)
4
Joel Langer
Figure 1. Straightening an inflectional planar elastica; β = α = 0.93, λ = tan(0.222m), m = 0, . . . , 7.
Straightening Soliton Curves
5
Figure 2. Straightening a non-inflectional planar elastica; β = 1, α = 0.95, λ = tan(0.222m), m = 0, . . . , 7. The above constructions and examples are meant to contribute to a larger program: to explore the natural role of soliton curves in geometry, topology, and graphics. Low order examples in Γ are geometrically interesting in themselves even physically meaningful - and have surprising connections to other topics in geometry, too numerous to mention here. Stepping up the soliton hierarchy, the
6
Joel Langer
curves γ ∈ Γn exhibit increasing complexity, but are always selected by geometric variational principles and permit a variety of interesting deformations and constructions; for example, the powerful techniques of soliton theory, including the inverse scattering method and B¨acklund transformations, enable one to construct elements of Γ and their related Hamiltonian evolutions, γt = Yn . In particular, a major boost to the program was provided by the recent work of Calini and Ivey ([5],[6]), who used B¨acklund transformations to produce exotic knots of constant torsion from elastic rod torus knots. This work leads to an appealing idea: one expects all knot types to be represented in Γ (already by constant torsion elements of Γ). In fact, roughly speaking, Γ should be dense among smooth curves. Here we have exploited some of the structure of Γ for a rather different, specialised purpose: to embed any γ ∈ Γ in a parametrized family of regular curves γλ with γ0 = γ and γ∞ a straight line - behavior more like a gradient flow. The construction is direct enough to be given a self-contained account here, and has the computationally appealing feature that the deformation itself has essentially R polynomial dependence on λ; once the antiderivatives Yr ds are computed for a given curve γ, smooth animation of γλ requires minimal additional computation. We have illustrated the straightening process with explicit formulas, in the case of a planar elastica - this was relatively easy to do; for higher order examples, one would focus more on qualitative behavior, and expect to rely more on numerical computations. With this in mind, we conclude by listing some properties which ought to be characteristic of graphical output generated by the straightening process, and suggest some of the topological content of such animations: • No kinks develop in the curves themselves, which have unit-speed, and cur1 vature decreasing as |λ| . • Frenet ribbon/tube surface representations of (γλ , Fλ ) exhibit interesting interconversions of twist and writhe - as in the Calugareanu-White Theorem even though γλ is not closed. • Quasiperiodicity is preserved (periodicity necessarily lost). • Quasiperiodic planar curves with nonzero rotation index break symmetry and evolve in R3 (think “quasiperiodic” Whitney-Graustein Theorem).
References [1] J. Langer, Recursion in curve geometry, New York J. Math. 5 (1999) 25-51. [2] J. Langer and R. Perline, Poisson geometry of the filament equation, J. Nonlinear Sci. 1 (1991) 71-93. [3] T. Ivey and D. Singer, Knot types and homotopies of elastic rods, London Math. Soc. Proceedings, to appear. [4] J. Langer, A ribbon animation, http://www.cwru.edu/artsci/math/langer/gifs/str.gif [5] A. Calini and T. Ivey, B¨ acklund transformations and knots of constant torsion, J. Knot Theory and its Ramifications, Vol. 7, No. 6 (1998) 719-746. [6] A. Calini and T. Ivey, Topology of constant torsion curves evolving under the sine-Gordon equation, Phys. Lett. A 254(1999) 235-243. Dept. of Mathematics, Case Western Reserve University, Cleveland, OH 44106 [email protected]
The constructions we describe here S∞pertain to the soliton class, Γ, of unit speed curves γ in Euclidean 3-space: Γ = n=0 Γn , where Γ1 = {lines}, Γ2 = {helices}, Γ3 = {elastic rods}, 4 ⊃ {buckled rings under pressure}, . . . For concise definiPΓ ∞ tion, consider X = n=0 λn Xn , a formal series of vectorfields along γ(s) satisfying X0 = − ∂γ ∂s = −T and JXn = ∂Xn−1 , n = 1, 2, . . . , i.e., (1)
JX = λ∂X
∂ Here, J = T × (cross product with unit tangent), and ∂ = ∂s = ∇T (covariant derivative). Let Γn = {γ : 0 = Xn }; as will be seen, Γn is defined by an nth -order ODE for T = γs , depending on n constants. Since J 2 = −Id on normal vectorfields, Xn = fn T − J∂Xn−1 , where ∂fn = ∂hT, Xn i = h∂T, Xn i + hT, ∂Xn i = h−JX1 , Xn i determines fn up to a constant of integration. Alternatively, (1) implies λ∂hX, Xi = 2hJX, Xi = 0, so for some constants Cn , ∞ X (2) hX, Xi = p(λ) = 1 + Cn λn n=1
Pn−1 The λ -term of (2) gives fn without antidifferentiation: 2fn = −Cn + k=1 hXk , Xn−k i, n = 2, 3, . . . .PThe normalization p(λ) = 1 (all Cn are zero) yields a special solution ∞ to (1), Y = n=0 λn Yn , whose terms are generated inductively: 3 Y0 = −γs , Y1 = γs × γss , Y2 = hγss , γss iγs + γsss , 2 3 Y3 = hγs × γss , γsss iγs − γs × γssss − hγss , γss iγs × γss , 2 n
n−1
(3)
Yn = (
1X hYk , Yn−k i) T − J∂Yn−1 2 k=1
Mathematics Subject Classification. 35Q51, 53A04. Key words and phrases. solitons, curves, localized induction hierarchy.
1
2
Joel Langer
P∞ n The general solution to (1), X = n=0 λ Xn , may be written in terms of Y , A0 = 1, and “integration constants” A1 , A2 , . . . : ∞ n ∞ ∞ X X X X p An−k Yk = ( Ai λi )( λj Yj ) = p(λ) Y (4) X= λn n=0
k=0
i=0
j=0
The above formulas are presented in more detail in [1], where background and related topics are also discussed. The localized induction hierarchy of soliton equations, γt = Yn , is considered from a Hamiltonian viewpoint in [2], along with the connection to the better known nonlinear Schr¨ odinger hierarchy. Here we focus on interpretation of Equation 2. Note that Y may be regarded as a (formal) differential operator, depending on a parameter λ, acting on spherical curves (tangent indicatrices) T ; as such, Y is unambiguously defined by (3). On the other phand, Equation 4 shows that Y may be variously represented in the form Y = X/ p(λ), in terms of arbitrary constants A1 , A2 , . . . . We now use this fact to address the issue of convergence of Y = Y [γs ], evaluated on soliton curves γ: Pn Theorem 1. Let γ ∈ Γn ; specifically, let 0 = Xn = k=0 An−k Yk along γ, and assume γ does not belong to Γn−1 . Then X as in (1) may be assumed to terminate, p(λ) = hX, Xi is a non-vanishing polynomial, and Y converges to a homotopy of spherical curves, (5)
n−1 r X X p 1 Ar−k Yk , Tλ = X/ p(λ) = p λr p(λ) r=0 k=0
deforming T0 = −T to a point, as λ → ±∞. P∞ P∞ Pr Proof. Let X = r=0 λr Xr = r=0 λr k=0 Ar−k Yk be a solution to (1), where A1 , A2 , . . . , An are such that Xn = 0 along γ. Then JXn = ∂Xn−1 implies Xn−1 = Pn−1 Pm k=0 An−1−k Yk is constant. For m ≥ n, let Am+1 = k=0 Am−k hT, Yk+1 i. One verifies by induction that, for m ≥ n, Am+1 is a constant such that Xm+1 = Pm+1 Pn−1 = 0. Thus, X is terminating: X = r=0 λr Xr . Further, vanishk=0 Am+1−k Yk P 2n−2 ing of p(λ) = 1 + r=1 Cr λr for some real λ would imply γ ∈ Γn−1 . The values of Tλ at λ = ±∞ may now be read off: T±∞ = (±1)(n−1) Xn−1 /kXn−1 k. Since Y is local, Tλ is periodic in s when T is. Corollary 2. Antidifferentiation of (5) yields a quasiperiod-preserving regular homotopy of unit speed curves in E 3 , deforming γ0 = −γ to a straight line, as λ → ±∞; in fact, γλ has curvature-normal vector, κλ Nλ = ∂Y = JY λ (λ 6= 0), whose length is at most 1/|λ|. Assuming κλ non-vanishing, Fλ = {Tλ , Nλ , Bλ } gives a homotopy in SO(3), deforming the Frenet frame of γ to a one-parameter subgroup (reparametrized and translated). Example 1 For γ ∈ Γ2 , γλ is a homotopy of helices. The loop (one period of) Fλ is nontrivial in π1 (SO(3)); the (quasi)writhe of γ is converted into twist as λ → ±∞, and {γλ , Fλ } becomes a (uniform) spinning line. Fλ lifts to a minimal torus in SU (2) ∼ torus - with conformal parameters θ, φ given by θ = cs, = S 3 - the Clifford √ τ −λc2 2 cot φ = κ , and c = κ + τ 2 (κ and τ being the constant curvature and torsion √ of γ). These conclusions follow from Y = (Y0 + λ(A1 Y0 + Y1 ))/ 1 − 2λτ + λ2 c2 = hsin φ cos θ, sin φ sin θ, cos φi (Cartesian coordinates in the latter expression with zaxis parallel to the constant vector X1 = A1 Y0 + Y1 ).
Straightening Soliton Curves
3
Example 2 Γ3 consists of (centerlines of) elastic rods. Such γ satisfy a variational principle involving bending, twisting, and extension energies, with Euler equation equivalent to 0 = X3 = A3 Y0 + A2 Y1 + A1 Y2 + Y3 . (In the planar case, the next equation, 0 = X4 , simply adds a hydrostatic pressure term to bending and extension terms, yielding a standard model for cross sections of cylindrical pipes buckling under pressure.) The equation 0 = X3 has first integrals hX2 , X2 i = j 2 and hX1 , X2 i = h, which may be expressed as a pair of ODE’s for the curvature κ and torsion τ of γ. One thus obtains κ and τ in terms of elliptic functions: 2 2 2 κ2 (s) = κ20 (1 − α β 2 sn (κ0 s/2β, α)), κ (s)(2τ (s) + A1 ) = c; here, 0 ≤ α ≤ β ≤ 1, and α is the modulus of the Jacobi elliptic sine, sn(u, α). The (congruence classes of) curves in Γ3 form a four-dimensional family, parametrized by α, β, A1 , A2 , (the remaining constants being determined by these). One may express cylindrical coordinates along a soliton curve γ ∈ Γ by quadrature in terms of its curvature and torsion (details for the case of elastic rods may be found in [3]). E.g., in the special case of inflectional (non-inflectional) planar elas1 tica, β = α (β = 1), we may write r = κ/j, θ = θ0 , and zs = 2j (κ2 − 2A2 ). Further, p A3 = A1 = 0, and Y = (Y0 + λY1 + λ2 X2 )/ p(λ), where X2 = A2 Y0 + Y2 = j kˆ is a constant vector pointing in the z-direction. Integration of Y yields the homotopy γλ explicitly, and the Frenet frame Fλ may be obtained by differentiation. Straightening homotopies for inflectional and non-inflectional cases are shown in Figure 1 and Figure 2, respectively. Mathematica was used to compute binormal ribbons γ(s, t; λ) = γλ + tBλ (s), with −1 ≤ t ≤ 1, = 0.1, s ranging over two periods of the curvature function κλ , and λ = tan(0.222m), m = 0, . . . , 7. See [4] for an animated version of Figure 1, but showing twice as much of the curve, and λ = tan(.0468m), m = −33, . . . , 33. Cartesian coordinates were chosen with γ0 = −γ lying in the x1 , x2 -plane. In the inflectional case - we omit details for the (similar) non-inflectional case - the coordinates of γλ are: √ x1 = [16α3 E(φ, α) − 2α(4α2 + λ2 )u]/ q, √ x2 = −16α4 cn/ q, √ x3 = 8α2 λ arcsin(α sn)/ q, q
= 16α4 p(λ) = 16α4 + 8α2 (2α2 − 1)λ2 + λ4 .
Here, u = s/2α, and we use shorthand for the amplitude √ φ = am(u, α), elliptic functions cn = cn(u, α), sn = sn(u, α), dn = dn(u, α) = 1 − α2 sn2 , and E is the elliptic integral of the second kind. The normalization κ0 = 1 and the formulas 2 1 A2 = 2α4α−1 2 , j = 4α2 were applied. We note that the curvature-normal vector of γλ is given by κλ Nλ = ∂Y = √ (−8α3 sn cn dn, 4α2 cn(1 − 2α2 sn2 ), −2αλsn dn)/ q, and the curvature of γλ is √ 2α 2 2 2 2 2 κλ = √ q 4α cn + λ sn dn , which is non-vanishing for 0 < |λ| < ∞. The binormal Bλ is computed as Bλ = Y × ∂Y /κλ . As λ → 0, the binormal ribbon develops a kink, corresponding to the singularity of the Frenet frame of the original inflectional elastica (exceptional among γ ∈ Γ3 ). A similar phenomenon - apparent in the computer animations - occurs as λ → ±∞, for both classes of planar elastica; one may regard τ±∞ as a sum of Dirac delta functions, one for each vertex of the original elastica. (In this respect, the helix example better represents generic behavior for non-planar γ ∈ Γ3 .)
4
Joel Langer
Figure 1. Straightening an inflectional planar elastica; β = α = 0.93, λ = tan(0.222m), m = 0, . . . , 7.
Straightening Soliton Curves
5
Figure 2. Straightening a non-inflectional planar elastica; β = 1, α = 0.95, λ = tan(0.222m), m = 0, . . . , 7. The above constructions and examples are meant to contribute to a larger program: to explore the natural role of soliton curves in geometry, topology, and graphics. Low order examples in Γ are geometrically interesting in themselves even physically meaningful - and have surprising connections to other topics in geometry, too numerous to mention here. Stepping up the soliton hierarchy, the
6
Joel Langer
curves γ ∈ Γn exhibit increasing complexity, but are always selected by geometric variational principles and permit a variety of interesting deformations and constructions; for example, the powerful techniques of soliton theory, including the inverse scattering method and B¨acklund transformations, enable one to construct elements of Γ and their related Hamiltonian evolutions, γt = Yn . In particular, a major boost to the program was provided by the recent work of Calini and Ivey ([5],[6]), who used B¨acklund transformations to produce exotic knots of constant torsion from elastic rod torus knots. This work leads to an appealing idea: one expects all knot types to be represented in Γ (already by constant torsion elements of Γ). In fact, roughly speaking, Γ should be dense among smooth curves. Here we have exploited some of the structure of Γ for a rather different, specialised purpose: to embed any γ ∈ Γ in a parametrized family of regular curves γλ with γ0 = γ and γ∞ a straight line - behavior more like a gradient flow. The construction is direct enough to be given a self-contained account here, and has the computationally appealing feature that the deformation itself has essentially R polynomial dependence on λ; once the antiderivatives Yr ds are computed for a given curve γ, smooth animation of γλ requires minimal additional computation. We have illustrated the straightening process with explicit formulas, in the case of a planar elastica - this was relatively easy to do; for higher order examples, one would focus more on qualitative behavior, and expect to rely more on numerical computations. With this in mind, we conclude by listing some properties which ought to be characteristic of graphical output generated by the straightening process, and suggest some of the topological content of such animations: • No kinks develop in the curves themselves, which have unit-speed, and cur1 vature decreasing as |λ| . • Frenet ribbon/tube surface representations of (γλ , Fλ ) exhibit interesting interconversions of twist and writhe - as in the Calugareanu-White Theorem even though γλ is not closed. • Quasiperiodicity is preserved (periodicity necessarily lost). • Quasiperiodic planar curves with nonzero rotation index break symmetry and evolve in R3 (think “quasiperiodic” Whitney-Graustein Theorem).
References [1] J. Langer, Recursion in curve geometry, New York J. Math. 5 (1999) 25-51. [2] J. Langer and R. Perline, Poisson geometry of the filament equation, J. Nonlinear Sci. 1 (1991) 71-93. [3] T. Ivey and D. Singer, Knot types and homotopies of elastic rods, London Math. Soc. Proceedings, to appear. [4] J. Langer, A ribbon animation, http://www.cwru.edu/artsci/math/langer/gifs/str.gif [5] A. Calini and T. Ivey, B¨ acklund transformations and knots of constant torsion, J. Knot Theory and its Ramifications, Vol. 7, No. 6 (1998) 719-746. [6] A. Calini and T. Ivey, Topology of constant torsion curves evolving under the sine-Gordon equation, Phys. Lett. A 254(1999) 235-243. Dept. of Mathematics, Case Western Reserve University, Cleveland, OH 44106 [email protected]
