Plane curves with curvature depending on distance to a line

Differential Geometry and its Applications 44 (2016) 77–97

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Differential Geometry and its Applications www.elsevier.com/locate/difgeo

Plane curves with curvature depending on distance to a line Ildefonso Castro a,∗ , Ildefonso Castro-Infantes b a b

Departamento de Matemáticas, Universidad de Jaén, 23071 Jaén, Spain Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain

a r t i c l e

i n f o

Article history: Received 22 April 2015 Received in revised form 2 October 2015 Available online 18 November 2015 Communicated by E. Garcia Rio MSC: primary 53A04 secondary 74H05

a b s t r a c t Motivated by a problem posed by David A. Singer in 1999 and by the classical Euler elastic curves, we study the plane curves whose curvature is expressed in terms of the (signed) distance to a line. In this way, we provide new characterizations of some well known curves, like the catenary or the grim-reaper. We also find out several interesting families of plane curves (including closed and embedded ones) whose intrinsic equations are expressed in terms of elementary functions or Jacobi elliptic functions and we are able to get arc length parametrizations of them and they are depicted graphically. © 2015 Elsevier B.V. All rights reserved.

Keywords: Plane curves Curvature Distance Catenary Grim-reaper

1. Introduction The fundamental existence and uniqueness theorem in the theory of plane curves states that a curve is uniquely determined up to rigid motion by its curvature given as a function of its arc-length. However, in most cases such curves are impossible to find explicitly in practice, due to the difficulty in solving the quadratures appearing in the integration process. In [9], David A. Singer considered a different sort of problem: Can a plane curve be determined if its curvature is given in terms of its position? Having the above comment in mind, it is expectable that if the curvature κ is given by a function of its position, i.e. κ = κ(x, y), the situation is even more complicated. The general form of this problem translates into solving a nonlinear differential equation: * Corresponding author. E-mail addresses: [email protected] (I. Castro), [email protected] (I. Castro-Infantes). http://dx.doi.org/10.1016/j.difgeo.2015.11.002 0926-2245/© 2015 Elsevier B.V. All rights reserved.

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Fig. 1. Euler elastic curves.

x (t)y  (t) − y  (t)x (t) = κ(x(t), y(t)) (x (t)2 + y  (t)2 )3/2 Probably the most interesting solved problem in this setting corresponds to the Euler elastic curves. Classically an elastica is the solution to a variational problem proposed by Daniel Bernoulli to Leonhard Euler in 1744, that of minimizing the bending energy of a thin inextensible wire. From a mathematical point of view, the modelling for this problem consists of minimizing the integral of the squared curvature for curves of a fixed length satisfying given first order boundary data (see [3,10]). These curves satisfy a remarkable property: the curvature is proportional to one of the coordinate functions, say κ(x, y) = 2λy (see Section 3). Among the Euler elasticae, there is a unique closed curve with the shape of a figure-eight (see Fig. 1) and the expression of the curvature in terms of the arc parameter involves elliptic Jacobi  functions. Singer in [9] started to study the posed problem by considering the condition κ(x, y) = x2 + y 2 and tried to find analytic representations for these curves. First he proved (see Theorem 3.1 in [9]) that the problem of determining a curve whose curvature is κ(r), where r is the distance from the origin, is solvable by quadratures when rκ(r) is a continuous function. The proof is based in giving to such curvature an interpretation of a central potential in the plane and finding the trajectories by the standard methods in classical mechanics. See also [11] for a group-theoretical approach of the same result viewing the Frenet equations as a fictitious dynamical system. But the simple case κ(r) = r, where elliptic integrals appear, illustrated that the fact that the corresponding differential equation is integrable by quadratures does not mean that it is easy to perform the integrations. In [9], only the very pleasant special case of the classical Bernoulli lemniscate, r2 = 3 cos 2θ in polar coordinates, was solved explicitly, where the corresponding elliptic integral becomes elementary. Some other families of plane curves whose curvature depends on distance from the origin were considered later. In a chronological order, we emphasize the following: Bernoulli’s lemniscate, κ = σ r, σ > 0 (see [9] and [4]); Levy’s elasticae, κ = a r2 + c, a, c ∈ R, a = 0 (see [2]); generalized Sturm spirals, κ = σ/r, σ > 0 (see [5,7,8]) and deformations of Cassinian ovals: κ = λ/r3 + 3μ r, λ, μ ∈ R (see [6]). In this article, inspired by Euler elastic curves, we propose to study the plane curves whose curvature depends on distance to a line instead from a point. After a rigid motion, we can suppose this line as the x-axis of the reference system for the curve and so we afford the case κ = κ(y) of the aforementioned Singer’s problem, including the very important family of Euler elastic curves corresponding to the linear function κ(y) = 2λy.

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Of course that if one writes locally the plane curve as the graph (f (y), y), it is obvious that κ = κ(y) and it is required to solve the nonlinear differential equation f  (y) + κ(y)(1 + f  (y)2 ) 2 = 0 3

that it is not difficult to prove that can be solved by quadratures taking  f (y) = −



(F (y) + c)dy

 , where 1 − (F (y) + c)2

κ(y)dy =: F (y) + c, c ∈ R

(1.1)

But we aim to study the problem from a global point of view in the following sense: for a unit speed parametrization α(s) = (x(s), y(s)) of a regular plane curve, we prescribe the curvature with the geometric extrinsic condition κ = κ(y) and want to determine the analytic representation of α(s) and so its intrinsic equation κ = κ(s). In the same spirit of Theorem 3.1 in [9], we show in Theorem 2.1 that the problem of determining a plane curve whose curvature is κ(y), y being the (signed) distance to the x-axis, is solvable by quadratures considering the unit speed curve (x(s), y(s)), where y(s) is the inverse of s = s(y) that is given by the integration of  



dy 1 − (F (y) + c)2

= s, where

κ(y)dy =: F (y) + c, c constant

(1.2)

and x(s) is expressed in terms of y(s) and c by    x(s) = − c s + F (y(s))ds

(1.3)

Of course these last two formulas (1.2) and (1.3) are compatible with (1.1) and, in addition, for a given value of c such a curve is uniquely determined up to translations in the x-direction. The constant c appearing in (1.2) parameterizes the one-parameter family of plane curves satisfying κ = κ(y). We show in Section 2 that  it is required that −1 < F (y) + c < 1 and, since c is just the integration constant corresponding to κ(y)dy, every chosen antiderivative of κ(y) determines a different curve (see Remark 2.1). In the remarkable family  of the Euler elasticae ( κ(y)dy = λy 2 + c), this constant c determines the tension σ of the elastic curve (see Section 3) since σ = −4λc. The important case of the free elasticae, when the constraint on arc length is removed (see [3]), corresponds to c = 0 and these are among the classical Euclidean curves studied by Radon. In general, when one carries out the computations of (1.2) and (1.3), one finds in most cases the difficulties described in Remark 2.2. However, after considering translations, reflections and dilations, we are successful (at least considering a particular antiderivative of κ(y) corresponding to the free case c = 0) in the following families (studied in Sections 4–9) described—according to Singer’s problem—by the geometric conditions: κ(y) = λ/y 2 , κ(y) = λ cos y, κ(y) = λ cosh y, κ(y) = λ/ey , κ(y) = λ/ cos2 y and κ(y) = λ/ cosh2 y. Without loss of generality, we can consider λ > 0 once assumed λ non-null. In this way, on the one hand we provide new characterizations of some well known curves, like the catenary (see Corollaries 4.1 and 9.2) or the grim-reaper (see Corollaries 5.2 and 7.2). On the other hand, we also find out several interesting families of plane curves whose intrinsic equations are expressed in terms of elementary functions or Jacobi elliptic functions (like cosine amplitude cn(·, p) and delta amplitude dn(·, p) of modulus p ∈ (0, 1)) and we are able to get arc length parametrizations of them. Concretely, we obtain the following intrinsic equations of explicit plane curves studied along the paper: κ(s) = λ cn(s, λ), s ∈ R, 0 < λ < 1 κ(s) = λ dn(λs, 1/λ), s ∈ R, λ > 1

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κ(s) = λ/ dn



1+

λ2

1 s, √ 1 + λ2

 , s ∈ R, λ > 0

1 − c2 √ , s ∈ R, −1 < c < 1 cosh( 1 − c2 s) − c   c2 − 1 π π √ κ(s) = , s ∈ −√ ,√ , c>1 c + sin( c2 − 1s) c2 − 1 c2 − 1   λ(1 + λ2 ) π −π √ , λ>0 κ(s) = ,s∈ √ ,√ λ2 + cos2 ( 1 + λ2 s) 1 + λ2 1 + λ2 κ(s) =

λ(1 − λ2 ) √ , s ∈ R, 0 < λ < 1 cosh2 ( 1 − λ2 s) − λ2   −π π λ(λ2 − 1) √ ,s∈ √ ,√ , λ>1 κ(s) = λ2 − cos2 ( λ2 − 1s) λ2 − 1 λ2 − 1 κ(s) =

Moreover, they include four one-parameter families of closed embedded curves (see Subsection 5.3, Section 6, Section 8 and Subsection 9.3). 2. Plane curves such that κ = κ(y) Let α : I ⊂ R → R2 ≡ C be a regular plane curve parameterized by arc-length; that is, |α(s)| ˙ = 1, ∀s ∈ I, where I is some interval in R. Here ˙ means derivation with respect to s. We denote by ·, · the usual inner product in R2 . Let T := α˙ be the unit tangent vector to the curve α and let N := iα˙ be the vector orthogonal to T such that the frame (T, N ) is positively oriented. The Frenet equations of α are given by T˙ (s) = κ(s)N (s),

N˙ (s) = −κ(s)T (s)

(2.1)

where κ = κ(s) = det(α(s), ˙ α ¨ (s)) is the (signed) curvature of α. If ϑ denotes the slope of the tangent vector T = (cos ϑ, sin ϑ), we can rewrite (2.1) as ˙ ϑ(s) = κ(s),

x(s) ˙ = cos ϑ(s),

y(s) ˙ = sin ϑ(s)

(2.2)

Thus once κ has been determined, the curve can be found by three quadratures. We are interested in the geometric condition that the curvature of α depends on the distance to a line. If e ∈ R2 is a unit length vector, then α, e is the signed distance to the orthogonal line to e passing through the origin. Without restriction, we consider e := (0, 1) and write α = (x, y). Hence we are devoted to study the condition κ = κ(y) since y = α, e represents the signed distance to the x-axis. We first show that we have certain control on the normal component of the vector e := (0, 1) when κ = κ(y). Lemma 2.1. Let α = (x, y) be a plane curve with unit normal vector N . If κ = κ(y), then there exists c ∈ R such that N, e + F (y) + c = 0,

dF = κ(y) dy

(2.3)

Proof. Without restriction we can suppose that α is arc-length parameterized. We use the second Frenet equation of (2.1) and the hypothesis κ = κ(y) to deduce that N, e ˙ = −κ T, e = −κ(y)y˙ So we have that N, e + F (y) is constant, what proves the result. 2

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We make use of Lemma 2.1 to determine by quadratures a plane curve such that κ = κ(y). Writing α = (x, y) in Cartesian coordinates, it is clear that α˙ = (x, ˙ y) ˙ and N = (−y, ˙ x), ˙ with x˙ 2 + y˙ 2 = 1 since α is unit-speed. Joint with (2.3), we get that, up to a translation of α, its first component is given by    x(s) = − c s + F (y(s))ds

(2.4)

and its second component is recovered from y˙ 2 = 1 − (F (y) + c)2

(2.5)

The possible constant solutions of (2.5) correspond to horizontal lines. Excluding these, we can always separate variables in this ordinary differential equation. Up to a change of sign and a translation in the parameter s, we have that 

dy  =s 1 − (F (y) + c)2

(2.6)

where −1 < F (y) + c < 1. As a summary, we have proved the following result in the spirit of Theorem 3.1 in [9]. Theorem 2.1. Let κ = κ(y) be a non-null continuous function. Then the problem of determining a curve whose curvature is κ(y), y being the signed distance to the x-axis, is solvable locally by quadratures considering the unit speed curve (x(s), y(s)), where y(s) is given by the integration of (2.6) with  κ(y)dy = F (y) + c, c constant

(2.7)

and x(s) is expressed in terms of y(s) and c by (2.4). For any given c, such a curve is uniquely determined up to translations in the x-direction. Remark 2.1. The constant c appearing in (2.7) parameterizes the one-parameter family of plane curves satisfying κ = κ(y). Using (2.6) it is required that −1 < F (y) + c < 1. Looking at Lemma 2.1 or (2.7),  c is related to the integration constant corresponding to κ(y)dy. So every chosen antiderivative of κ(y) determines a different curve with prescribed curvature κ(y). We will call the primitive curvature and denote by K(y) the election for the antiderivative of κ(y) in the uniqueness results of Sections 4–9. As an illustrative example of the use of Theorem 2.1, we show  the case that κ is a non-null constant,  say κ ≡ k0 > 0. Then we take F (y) = k0 y and s = dy/ 1 − (k0 y + c)2 = arcsin(k0 y + c)/k0 . So y(s) = (sin(k0 s) −c)/k0 . This gives that x(s) = cos(k0 s)/k0 and finally, up to translations in the x-direction, α(s) = 1/k0 (cos(k0 s), sin(k0 s) − c) that correspond obviously to arbitrary circles of radius 1/k0 . Remark 2.2. The main three difficulties one can find carrying out the strategy described in Theorem 2.1 to determine a plane curve whose curvature depends on the signed distance to a line (κ = κ(y)) are the following: (1) The integration of (2.6): Even in the case that F (y) were a polynomial, the above integral (2.6) is not necessarily elementary. For example, when F (y) is a quadratic polynomial, (2.6) can be solved using Jacobi elliptic functions (see e.g. [1]). This is equivalent to κ(y) be linear, so that κ(y) = 2λy + μ, λ, μ ∈ R. We will study such curves in Section 3.

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(2) The integration performed in (2.6) gives us s = s(y); it is not always possible to obtain explicitly y = y(s), what is necessary to determine the analytic representation of the curve. (3) Even knowing explicitly y = y(s), the integration (2.4) to get x(s) may be impossible to compute using elementary or known functions. Nevertheless, along the paper we will study six different situations where we are—at least partially— successful with the procedure described in Theorem 2.1 and we will recover some very well known curves and find out new families of plane curves characterized by this kind of geometric property. 3. Elastic curves revisited A unit speed curve α is said to be an elastica under tension σ (see [3]) if it satisfies the differential equation 2¨ κ + κ3 − σκ = 0

(3.1)

for some value of σ ∈ R. If σ = 0, then α is called a free elastica. Multiplying (3.1) by 2κ˙ and integration allow us to introduce the energy E ∈ R of an elastica: 1 σ E := κ˙ 2 + κ4 − κ2 4 2

(3.2)

In this section we will study those plane curves satisfying κ(y) = 2λy + μ, λ > 0, μ ∈ R

(3.3)

Up to a translation in the y-direction, it is equivalent to consider κ(y) = 2λy, λ > 0

(3.4)

The trivial solution y ≡ 0 to (3.1) corresponds to the x-axis. Following Theorem 2.1, we can take F (y) = λy 2 and so the differential equation for y = y(s) is given by y˙ 2 = 1 − (λy 2 + c)2 = (1 + c + λy 2 )(1 − c − λy 2 )

(3.5)

for some constant c. In the following result we show that we are studying precisely elastic curves. Proposition 3.1. Let α be a plane curve whose curvature satisfies (3.4). Then α is an elastica under tension σ = −4λc and energy E = 4λ2 (1 − c2 ). Proof. Without restriction we can consider α parameterized by arc length. In order α to be an elastica, we must check that κ given by (3.4) satisfies (3.1). We have that κ˙ 2 = 4λ2 y˙ 2 and using (3.5) and substituting κ y = 2λ it is an exercise to finish the proof. 2 It is clear that (3.5) implies that −1 ≤ λy 2 +c ≤ 1, and so we deduce that c < 1. If c = 1 necessarily y ≡ 0. Following the strategy described in Theorem 2.1, we can control the plane curves (x(s), y(s)) satisfying (3.4) by means of  s=



dy  1 + c + λy 2 1 − c − λy 2

(3.6)

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Fig. 2. Free elastica: κ(y) = 2λy and K(y) = λy 2 (wavelike).

Fig. 3. Elastica with κ(y) = 2λy and K(y) = λy 2 − 1 (borderline).

and  x(s) = − cs + λ



 y(s) ds 2

(3.7)

with c < 1. The integrations of (3.6) and (3.7) involve Jacobi functions and elliptic integrals. √ elliptic √ According to [10], the maximum curvature is given by k0 = 2 λ 1 − c and we must distinguish three cases: (1) the wavelike case, when c > −1; the curvature is expressed in terms of the Jacobi elliptic cosine amplitude of modulus p:  κ(s) = k0 cn

 1−c k0 s , p , p2 = ,s∈R 2p 2

(2) the borderline case, when c = −1; the curvature is given by κ(s) = k0 sech(k0 s/2), s ∈ R (3) the orbitlike case, when c < −1; the curvature is expressed in terms of the Jacobi elliptic delta amplitude of modulus p:  κ(s) = k0 dn

 k0 s 2 , p , p2 = ,s∈R 2 1−c

As an illustration, we show the pictures of three representative examples of elastic curves (see Proposition 3.1): a free elastica, i.e. c = 0 (see Fig. 2) that is wavelike; an elastica of null energy c = −1 (see Fig. 3) that is borderline; and an elastica with c = −2 that is orbitlike (see Fig. 4). 4. Curves such that κ(y) = λ/y 2 In this section we are devoted to study those plane curves whose curvature satisfies

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Fig. 4. Elastica with κ(y) = 2λy and K(y) = λy 2 − 2 (orbitlike).

κ(y) =

λ , λ>0 y2

(4.1)

Since y must not vanish, we can also consider y > 0 without loss of generality. Following the ideas of Theorem 2.1, we can take F (y) = −λ/y and the differential equation for y = y(s) is given by y˙ 2 = 1 − (c − λ/y)2 = (1 + c − λ/y)(1 − c + λ/y)

(4.2)

for some constant c. We observe that (4.2) implies that −1 ≤ c − λ/y ≤ 1 and so we deduce that c > −1, taking into account we are considering λ, y > 0. Hence we can control the plane curves (x(s), y(s)) satisfying (4.1) by means of  s=



1+c−

dy 

λ y

 1−c+

λ y

=



ydy  (1 + c)y − λ (1 − c)y + λ

(4.3)

and  x(s) = −cs + λ

ds y(s)

(4.4)

with c > −1. We can always perform the integration given in (4.3), but only if c = 0 and c = 1 it is possible to get y explicitly as a function of s, being this absolutely necessary to obtain x(s) through (4.4). Therefore we only consider the aforementioned cases (see Remark 2.2). In addition, up to dilations, we can consider only λ = 1. In fact, if α is a plane curve satisfying κα = λ/y 2 , λ > 0, then κρα = λρ/(ρy)2 , for any ρ > 0. Taking ρ = 1/λ it is enough to consider the value λ = 1 in (4.1). 4.1. Case λ = 1 and c = 0 In this case, we have that y ≥ 1. Using (4.3) we easily obtain  s=



ydy y2 − 1

=



y 2 − 1 ⇒ y(s) =



s2 + 1

and from (4.4) we get  x(s) = In conclusion, we arrive at

√

 ds = log(s + s2 + 1) s2 + 1

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Fig. 5. The catenary y = cosh x: κ(y) = 1/y 2 and K(y) = −1/y.

(x(s), y(s)) = (log(s +

  s2 + 1), s2 + 1), s ∈ R

(4.5)

whose intrinsic equation, using (4.1) is given by κ(s) =

1 ,s∈R s2 + 1

In fact, this curve is nothing but the catenary y = cosh x (see Fig. 5) and we have proved this simple uniqueness result. Corollary 4.1. The catenary y = cosh x, x ∈ R—see (4.5)—is the only plane curve (up to translations in the x-direction) with curvature κ(y) = 1/y 2 and primitive curvature K(y) = −1/y. 4.2. Case λ = 1 and c = 1 In this case, we have that y ≥ 1/2. Using (4.3) we have that  s=

√

√ (y + 1) 2y − 1 ydy ⇒ 9s2 = (y + 1)2 (2y − 1) = 3 2y − 1

and this implies that 1 y(s) = 2

  1 (1 + 18s2 + 6s 1 + 9s2 ) 3 +



1 √ 1 − 1 2 (1 + 18s + 6s 1 + 9s2 ) 3

On the other hand, using (4.4), we know that x(s) = −s + x(s) = −s +



(4.6)

ds/y(s). From (4.6) we get

p(s) , q(s)

  1 p(s) = −1 + (1 + 18s2 + 6s 1 + 9s2 ) 3     × 1944s6 + s 1 + 9s2 + 324s4 (1 + 2s 1 + 9s2 + 12s2 (1 + 6s 1 + 9s2 )) ,  2 q(s) = s(1 + 18s2 + 6s 1 + 9s2 ) 3    × 1 + 324s4 + 9s 1 + 9s2 + 9s2 (5 + 12s 1 + 9s2 )

(4.7)

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Fig. 6. The curve with κ(y) = 1/y 2 and K(y) = 1 − 1/y.

Using (1.1), the implicit equation of this curve is given by cubic 9x2 = (y − 2)2 (2y − 1). In conclusion, we have proved the following characterization of the curve with intrinsic equation κ(s) = 

4 √ 1 (1 + 18s2 + 6s 1 + 9s2 ) 3 +

2 1

√ 1 (1+18s2 +6s 1+9s2 ) 3

−1

s ∈ R, plotted in Fig. 6. Corollary 4.2. The curve (x(s), y(s)), s ∈ R, given by (4.7) and (4.6) is the only plane curve (up to translations in the x-direction) with curvature κ = 1/y 2 and primitive curvature K(y) = 1 − 1/y. 5. Plane curves such that κ(y) = λ cos y The purpose of this section is the study of the plane curves satisfying κ(y) = λ cos y, λ > 0

(5.1)

The trivial solutions of (5.1) are the horizontal lines y ≡ (2m + 1)π/2, m ∈ Z. In order to study the non-trivial solutions we proceed according to Theorem 2.1 and, since we can take F (y) = λ sin y in this case, we get the following o.d.e. for y = y(s): y˙ 2 = 1 − (λ sin y + c)2

(5.2)

for some constant c. From (5.2) we have that −1 ≤ λ sin y + c ≤ 1 and this implies that −(1 + λ) ≤ c ≤ 1 + λ. Thus we can control the plane curves (x(s), y(s)) satisfying (5.1) by means of  s=

dy  , 1 − (λ sin y + c)2

 x(s) = −cs − λ

sin y(s)ds

with c ∈ (−1 − λ, 1 + λ). In general, the integration of the above first integral does not allow to get explicitly y = y(s) since it is involved elliptic integrals and trigonometric functions of y in a very complicated way (see Remark 2.2). We then only afford the case c = 0. It is remarkable that in this case κ satisfies the differential equation of similar type to (3.2): κ˙ 2 + κ4 + (1 − 2λ2 )κ2 = λ2 (λ2 − 1) Even in this case, we must distinguished subcases according to the values of λ which determine the sign of the second member of the above equation.

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Fig. 7. The curves with κ(y) = λ cos y and K(y) = λ sin y, 0 < λ < 1.

5.1. Case c = 0, 0 < λ < 1 Using the elliptic integral of first class F (·, λ) and the elementary Jacobi elliptic functions amplitude am(·, λ), sine amplitude sn(·, λ), cosine amplitude cn(·, λ) and delta amplitude dn(·, λ) of modulus λ (see e.g. [1]), we get that  s=



dy 1 − λ2 sin2 y

= F (y, λ) ⇒ y(s) = am(s, λ)

and so  x(s) = −λ

 sin y(s)ds = −λ

sn(s, λ)ds = − log (dn(s, λ) − λ cn(s, λ))

In conclusion, we arrive at (x(s), y(s)) = (− log (dn(s, λ) − λ cn(s, λ)) , am(s, λ)) whose intrinsic equation, using (5.1) is given by κ(s) = λ cn(s, λ), s ∈ R (see Fig. 7). 5.2. Case c = 0, λ = 1 In this case we only need elementary functions:  s=

dy = 2 arctanh(tan y/2) ⇒ y(s) = 2 arctan(tanh s/2) cos y

(5.3)

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Fig. 8. The grim-reaper x = log cos y: κ(y) = cos y and K(y) = sin y.

and so  x(s) = −

 sin y(s) ds = −

tanh s ds = − log cosh s

In conclusion, we arrive at (x(s), y(s)) = (− log cosh s, 2 arctan(tanh s/2))

(5.4)

whose intrinsic equation, using (5.1), is given by κ(s) = sech s, s ∈ R We have obtained the arc-length parametrization of the graph (log cos y, y), y ∈ (−π/2, π/2), known in the literature as the grim-reaper curve (see Fig. 8). It is the only translator for the curve shortening flow. 5.3. Case c = 0, λ > 1 In a similar way that in Subsection 5.1, using formula 162.01 of [1], we finally arrive at the periodic curves    1 1 1 1 1 sn(λs, ) − log dn(λs, ) − cn(λs, ) , arcsin λ λ λ λ λ

 (x(s), y(s)) =



(5.5)

whose intrinsic equation, using (5.1), is given by κ(s) = λ dn(λs, 1/λ), s ∈ R (see Fig. 9). As a summary, we conclude this section with the following uniqueness results. Corollary 5.1. Let α(s) = (x(s), y(s)) be a plane curve whose curvature satisfies κ(y) = λ cos y, λ > 0, with primitive curvature K(y) = λ sin y. Then, up to translations in the x-direction, α is given by (5.3) if λ ∈ (0, 1), by (5.4) if λ = 1 and by (5.5) if λ > 1. Corollary 5.2. The grim-reaper x = log cos y, −π/2 < y < π/2—see (5.4)—is the only plane curve (up to translations in the x-direction) with curvature κ(y) = cos y and primitive curvature K(y) = sin y.

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Fig. 9. The curves with κ = λ cos y and K(y) = λ sin y, λ > 1.

6. Plane curves such that κ(y) = λ cosh y In this section we claim to study those plane curves satisfying κ(y) = λ cosh y, λ > 0

(6.1)

Following Theorem 2.1, since we can take F (y) = λ sinh y, we arrive at the o.d.e. for y = y(s): y˙ 2 = 1 − (λ sinh y + c)2

(6.2)

for some real constant c. So we can determine the plane curves (x(s), y(s)) satisfying (6.1) by means of  s=



dy 1 − (λ sinh y + c)2

 ,

x(s) = −cs − λ

sinh y(s)ds

with c ∈ R. When c = 0, the integration of the above first integral does not allow to get explicitly y = y(s) because elliptic integrals and hyperbolic functions of y appear in non easy way (see Remark 2.2). We then only afford the case c = 0, where in we remark that κ satisfies the differential equation of similar type to (3.2): κ˙ 2 + κ4 − (1 + 2λ2 )κ2 = −λ2 (λ2 + 1) Using formula 295.20 in [1] and the reciprocal nd(·, p) of the elementary Jacobi delta amplitude dn(·, p) of modulus p, we have that  s=

dy 1 1  nd−1 (cosh y, p), p2 = =√ 2 2 1 + λ2 1+λ 1 + λ2 − λ2 cosh y

and so   y(s) = arccosh nd( 1 + λ2 s, p) , p2 =

1 1 + λ2

(6.3)

90

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Fig. 10. The curves with κ(y) = λ cosh y and K(y) = λ sinh y, λ > 0.

On the other hand, sinh y(s) = gives us

√ sn( 1+λ2 s,p) √ √ 2 1+λ dn( 1+λ2 s,p)

x(s) = − arctan

and, using formula 318.01 of [1], a direct computation

 √ λ(1 + cn( 1 + λ2 s, p)) 1 √ , p2 = 1 + λ2 cn( 1 + λ2 s, p) − λ2

(6.4)

Using (6.1), the intrinsic equation of the curve (x(s), y(s)) given by (6.4) and (6.3) is given by  κ(s) = λ nd( 1 + λ2 s, p), p2 =

1 , s∈R 1 + λ2

(see Fig. 10). As a summary we have proved the following uniqueness result. Corollary 6.1. Given λ > 0, the curve α(s) = (x(s), y(s)), s ∈ R, given by (6.4) and (6.3), is the only plane curve (up to translations in the x-direction) with curvature κ(y) = λ cosh y and primitive curvature K(y) = λ sinh y. 7. Plane curves such that κ(y) = λ/ey We determine in this section the plane curves whose curvature satisfies the geometric condition κ(y) = λe−y , λ > 0. Up to a translation in the y-direction, it is enough to consider κ(y) = e−y

(7.1)

Changing y by −y we would obtain a congruent family of curves satisfying κ(y) = ey . We start to apply Theorem 2.1 taking F (y) = −e−y in order to get the differential equation for y = y(s) given by y˙ 2 = 1 − (c − e−y )2

(7.2)

for some constant c, necessarily c > −1 using (7.2). In this way, we can determine the plane curves (x(s), y(s)) satisfying (7.1) by means of

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Fig. 11. The curves with κ(y) = e−y and K(y) = c − e−y , −1 < c < 1. In the middle, the grim-reaper y = − log sin x corresponding to c = 0.



dy  1 − (c − e−y )2

s=

(7.3)

and  x(s) = −cs +

e−y(s) ds

(7.4)

with c > −1. To perform the integration in (7.3) we make the change of variable u = ey obtaining  

s=

du (1 −

c2 )u2

+ 2cu − 1

(7.5)

what implies to distinguish cases according to the values of the parameter c > −1. 7.1. Case −1 < c < 1 Now (7.5) gives   √ log c + u − c2 u + 1 − c2 (1 − c2 )u2 + 2cu − 1 √ s= , u = ey 1 − c2 and using (7.4) we get finally

(x(s), y(s)) =

2 arctan

√ 1−c2 s

e

−c √ 1 − c2



− cs, log

 √ cosh( 1 − c2 s) − c 1 − c2

(7.6)

whose intrinsic equation, using (7.1), is given by κ(s) =

1 − c2 √ ,s∈R cosh( 1 − c2 s) − c

(see Fig. 11). We remark that the case c = 0 corresponds to the grim-reaper (x(s), y(s)) = (2 arctan es , log cosh s)

(7.7)

and we have obtained in (7.7) the arc-length parametrization of the graph (x, − log sin x), x ∈ (0, π). 7.2. Case c = 1 In this case, (7.5) reduces to s = plane curve

√

2u − 1 and so u = ey = (s2 + 1)/2; putting this in (7.4) we get the

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Fig. 12. The curve with κ(y) = e−y and K(y) = 1 − e−y .

 (x(s), y(s)) =

s2 + 1 2 arctan s − s, log 2

 (7.8)

whose intrinsic equation, using (7.1), is given by κ(s) =

2 ,s∈R s2 + 1

(see Fig. 12). 7.3. Case c > 1 In this last case, from (7.5) we have that s = −√

1 arcsin((1 − c2 )u + c), u = ey c2 − 1

and using (7.4) we get finally

(x(s), y(s)) =

2 arctan



 √ √ c + sin( c2 − 1s) c tan( c2 − 1s/2) + 1 √ − cs, log c2 − 1 c2 − 1

(7.9)

whose intrinsic equation, using (7.1), is given by κ(s) =

c2 − 1 √ ,s∈ c + sin( c2 − 1s)

 −√

π π ,√ 2 2 c −1 c −1



(see Fig. 13). In conclusion, we finish this section with the following uniqueness results. Corollary 7.1. Let α(s) = (x(s), y(s)) be a plane curve whose curvature satisfies κ(y) = e−y . Then its primitive curvature is written as K(y) = c − e−y , with c greater than -1 and α is given, up to translations in the x-direction, by (7.6) if −1 < c < 1, by (7.8) if c = 1, and by (7.9) if c > 1. Corollary 7.2. The grim-reaper y = − log sin x, 0 < x < π—see (7.7)—is the only plane curve (up to translations in the x-direction) with curvature κ(y) = e−y and primitive curvature K(y) = −e−y . 8. Plane curves such that κ(y) = λ/ cos2 y We study in this section the plane curves whose curvature satisfies the geometric condition: κ(y) = λ/ cos2 y, λ > 0

(8.1)

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Fig. 13. The curves with κ(y) = e−y and K(y) = c − e−y , c > 1.

We assume −π/2 < y < π/2 and apply Theorem 2.1 taking F (y) = λ tan y, and thus we get the differential equation for y = y(s) given by y˙ 2 = 1 − (λ tan y + c)2

(8.2)

for some constant c. In this way, we can determine the plane curves (x(s), y(s)) satisfying (8.1) by means of  dy  s= (8.3) 1 − (λ tan y + c)2 and  x(s) = −cs − λ

tan y(s)ds

(8.4)

with c ∈ R. To perform the integrations (8.3) and (8.4) we have the problems described in Remark 2.2 except at the case c = 0. In this case, we make the change of variable u = sin y to compute the integration in (8.3) obtaining   du 1  s= =√ arcsin( 1 + λ2 u) 1 + λ2 1 − (1 + λ2 )u2 and then  y(s) = arcsin



√

 1 sin( 1 + λ2 s) 1 + λ2

(8.5)

Using (8.4) we get      λ 2 2 2 2 x(s) = √ log cos( 1 + λ s) + λ + cos ( 1 + λ s) 1 + λ2 whose intrinsic equation, using (8.1), is given by κ(s) =

λ(1 + λ2 ) √ ,s∈ λ2 + cos2 ( 1 + λ2 s)

 √

π −π ,√ 1 + λ2 1 + λ2

(see Fig. 14). As a summary we have proved the following uniqueness result.



(8.6)

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Fig. 14. The curves with κ(y) = λ/ cos2 y and K(y) = λ tan y, λ > 0. −π √ π Corollary 8.1. Given λ > 0, the curve α(s) = (x(s), y(s)), s ∈ ( √1+λ ), given by (8.6) and (8.5), is 2, 1+λ2 the only plane curve (up to translations in the x-direction) with curvature κ(y) = λ/ cos2 y and primitive curvature K(y) = λ tan y.

9. Plane curves such that κ(y) = λ/ cosh2 y The aim of this section is the study of the plane curves whose curvature satisfies the geometric condition: κ(y) = λ/ cosh2 y, λ > 0

(9.1)

Applying Theorem 2.1 taking F (y) = λ tanh y we obtain the differential equation for y = y(s) given by y˙ 2 = 1 − (λ tanh y + c)2

(9.2)

for some constant c that must satisfy −(1 + λ) < c < 1 + λ by (9.2). In order to determine the plane curves (x(s), y(s)) satisfying (9.1), we must compute the integrations  s=

dy  1 − (λ tanh y + c)2

(9.3)

and  x(s) = −cs − λ

tanh y(s)ds

(9.4)

with c ∈ (−(1 + λ), 1 + λ). But again only in the case c = 0 we are successful to perform the integrations (9.3) and (9.4) due to the difficulties mentioned in Remark 2.2. In this case, we make the change of variable u = sinh y to compute the integration in (9.3) obtaining  s=

du  1 + (1 − λ2 )u2

and it is necessary to distinguish cases according to the values of λ.

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Fig. 15. The curves with κ(y) = λ/ cosh2 y and K(y) = λ tanh y, 0 < λ < 1.

9.1. Case c = 0, 0 < λ < 1 In this case  s=

 du 1  =√ arcsinh( 1 − λ2 u) 1 − λ2 1 + (1 − λ2 )u2

and then  y(s) = arcsinh

√

  1 sinh( 1 − λ2 s) 1 − λ2

(9.5)

Using (9.4) we get      λ 2 2 2 2 x(s) = − √ log cosh( 1 − λ s) + cosh ( 1 − λ s) − λ 1 − λ2

(9.6)

whose intrinsic equation, using (9.1), is given by κ(s) =

λ(1 − λ2 ) √ ,s∈R cosh2 ( 1 − λ2 s) − λ2

(see Fig. 15). 9.2. Case c = 0, λ = 1 √ Now s = sinh y and thus y = arcsinh s. Using (9.4) we get x(s) = − 1 + s2 . Hence we arrive at the plane curve  (x(s), y(s)) = (arcsinh s, − 1 + s2 )

(9.7)

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Fig. 16. The catenary x = − cosh y: κ(y) = 1/ cosh2 y and K(y) = tanh y.

whose intrinsic equation, using (9.1), is given by κ(s) =

1 1 + s2

So we get the catenary as the graph x = − cosh y (see Fig. 16). 9.3. Case c = 0, λ > 1 In this case  s=



du 1 + (1 −

λ2 )u2

=√



1 λ2

−1

arcsin(

λ2 − 1 u)

and then  y(s) = arcsinh

√

1 λ2 − 1

  sin( λ2 − 1s)

(9.8)

Using (9.4) we get x(s) = √

λ λ2 − 1

arcsin

 √ cos( λ2 − 1s) λ

(9.9)

whose intrinsic equation, using (9.1), is given by κ(s) =

λ(λ2 − 1) √ ,s∈ 2 λ − cos2 ( λ2 − 1s)

 √

π −π ,√ λ2 − 1 λ2 − 1



(see Fig. 17). We finish this section with the following uniqueness results. Corollary 9.1. Let α(s) = (x(s), y(s)) be a plane curve whose curvature satisfies κ(y) = λ/ cosh2 y, λ > 0, with primitive curvature K(y) = λ tanh y. Then, up to translations in the x-direction, α is given by (9.6) and (9.5) if λ ∈ (0, 1), by (9.7) if λ = 1 and by (9.9) and (9.8) if λ > 1. Corollary 9.2. The catenary x = − cosh y, y ∈ R—see (9.7)—is the only plane curve (up to translations in the x-direction) with curvature κ(y) = 1/ cosh2 y and primitive curvature K(y) = tanh y.

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Fig. 17. The curves with κ(y) = λ/ cosh2 y and K(y) = λ tanh y, λ > 1.

Acknowledgements Research of the first named author partially supported by a MEC-FEDER grant MTM2014-52368-P. The authors would like to thank to the referees for their many valuable comments and suggestions. References [1] P.F. Byrd, M.D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Springer Verlag, 1954. [2] P.A. Djondjorov, V.M. Vassilev, I.M. Mladenov, Plane curves associated with integrable dynamical systems of the Frenet– Serret type, in: Proc. of the 9th International Workshop on Complex Structures, Integrability and Vector Fields, World Scientific, Singapore, 2009, pp. 57–63. [3] J. Langer, D. Singer, The total squared curvature of closed curves, J. Differ. Geom. 20 (1984) 1–22. [4] I.M. Mladenov, M.T. Hadzhilazova, P.A. Djondjorov, V.M. Vassilev, On the plane curves whose curvature depends on the distance from the origin, in: XXIX Workshop on Geometric Methods in Physics, in: AIP Conf. Proc., vol. 1307, 2010, pp. 112–118. [5] I.M. Mladenov, M.T. Hadzhilazova, P.A. Djondjorov, V.M. Vassilev, On the generalized Sturmian spirals, C. R. Acad. Bulgare Sci. 64 (2011) 633–640. [6] I.M. Mladenov, M.T. Hadzhilazova, P.A. Djondjorov, V.M. Vassilev, On some deformations of the Cassinian oval, in: International Workshop on Complex Structures, Integrability and Vector Fields, in: AIP Conf. Proc., vol. 1340, 2011, pp. 81–89. [7] P.I. Marinov, M.T. Hadzhilazova, I.M. Mladenov, Elastic Sturmian spirals, C. R. Acad. Bulgare Sci. 67 (2014) 167–172. [8] I.M. Mladenov, P.I. Marinov, M.T. Hadzhilazova, Elastic spirals, in: Workshop on Application of Mathematics in Technical and Natural Sciences, in: AIP Conf. Proc., vol. 1629, 2014, pp. 437–443. [9] D. Singer, Curves whose curvature depends on distance from the origin, Am. Math. Mon. 106 (1999) 835–841. [10] D. Singer, Lectures on elastic curves and rods, in: Curvature and Variational Modeling in Physics and Biophysics, in: AIP Conf. Proc., vol. 1002, 2008, pp. 3–32. [11] V. Vassilev, P. Djondjorov, I. Mladenov, Integrable dynamical systems of the Frenet–Serret type, in: Proc. of the 9th International Workshop on Complex Structures, Integrability and Vector Fields, World Scientific, Singapore, 2009, pp. 234–244.