Euler - Savary’s Formula on Minkowski Geometry

T. Ikawa

Dedicated to the Memory of Grigorios TSAGAS (1935-2003), President of Balkan Society of Geometers (1997-2003)

Abstract

We consider a base curve, a rolling curve and a roulette on Minkowski plane and give the relation between the curvatures of these three curves. This formula is a generalization of the Euler - Savary’s formula of Euclidean plane.

Mathematics Subject Classifications: 53A35, 53B30 Key words: base curve, curvature, Euler - Savary’s formula, rolling curve, roulette.

1 Introduction

On the Euclidean plane E^{2}, we consider two curves c_{B} and c_{R}. Let P be a point relative to c_{R}. When c_{R} rolles without splitting along c_{B}, the locus of the point P makes a curve, say c_{L}. On this set of curves, c_{B}, c_{R} c_{L} are called the base curve, rolling curve and roulette, respectively. For example, if c_{B} is a straight line, c_{R} is a quadratic curve and P is a focus of c_{R}, then c_{L} is the Delaunay curve that are used to study surfaces of revolution with the constant mean curvature.

Since this ”rolling situation” makes up three curves, it is natural to ask questions: what is the relation between the curvatures of these curves, when given two curves, can we find the third one? Many geometers studied these questions and generalized the situation [3]. Today the relation of the curvatures is called as the Euler - Savary’s formula.

However, the ”rolling situation” on the Minkowski geometry is not studied yet. Only the Delaunay curve is considered to study surfaces of revolution with the constant mean curvature [1]. The purpose of this paper is to give answers to the abovementioned general questions on the Minkowski geometry. After the preliminaries of section 2, in section 3, we consider the associated curve that is the key concept to study the roulette, for, the roulette is one of associated curves of the base curve. Section 4 is devoted to give the Euler - Savary’s formula on the Minkowski plane. In the final section, we determine the third curve from other two.

∗ Balkan Journal of Geometry and Its Applications, Vol.8, No.2, 2003, pp. 31-36.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2003.

2 Preliminaries

Let L^{2} be the Minkowski plane with metric g = (+, −). A vector X of L^{2} is said to be spacelike if g(X, X) > 0 or X = 0, timelike if g(X, X) < 0 and null if g(X, X) = 0 and X ≠ 0.

A curve c is a smooth mapping c : I → L^{2} from an open interval I into L^{2}. Let t be a parameter of c. By c(t) = (x(t), y(t)), we denote the orthogonal coordinate

representation of c(t). The vector field \frac{dc}{dt} = (\frac{dx}{dt}, \frac{dy}{dt} ) =: X is called the tangent

vector field of the curve c(t). If the tangent vector field X of c(t) is a spacelike, timelike, or null, then the curve c(t) is called spacelike, timelike, or null, respectively.

In the rest of this paper, we mostly consider non-null curves. When the tangent vector field X is non-null, we can have the arc length parameter s and have the Frenet formula

\frac{dX}{ds} = kY, \frac{dY}{ds} = kX, (2.1)

where k is the curvature of c(s) (cf. [2]). The vector field Y is called the normal vector field of the curve c(s). Remark that we have the same representation of the Frenet formula regardless of whether the curve is spacelike or timelike.

If ϕ(s) is the slope angle of the curve, then we have \frac{dϕ}{ds} = k.

3 Associated curve

In this section, we give general formulas of the associated curve. Let c(s) be a non-null curve with the arc length parameter s, and {X, Y } the Frenet frame of c(s).

If we put c_{A} = c(s) + u_{1}(s)X + u_{2}(s)Y, (3.1)

then c_{A}(s) generally makes a curve. This curve is called the associated curve of c(s).

Remark that {u_{1}(s), u_{2}(s)} is a relative coordinate of c_{A}(s) with respect to {c(s), X, Y }.

If we put

\frac{dc_{A}}{ds} = \frac{δu_{1}}{ds} X + \frac{δu_{2}}{ds} Y,

then, since

\frac{dc_{A}}{ds} = \frac{dc}{ds} + \frac{du_{1}}{ds} X +u_{1} \frac{dX}{ds} + \frac{du_{2}}{ds} Y +u_{2} \frac{dY}{ds} = (1 + \frac{du_{1}}{ds} + ku_{2}) X +(ku_{1} + \frac{du_{2}}{ds} Y ) ,

by virtue of (2.1), we have

\begin{matrix}\frac{δu_{1}}{ds} = \frac{du_{1}}{ds} + ku_{2} + 1, \\ \frac{δu_{2}}{ds} = \frac{du_{2}}{ds} + ku_{1}. \end{matrix} (3.2)

Let s_{A} be the arc length parameter of c_{A}. Then, from

\frac{dc_{A}}{ds} = \frac{dc_{A}}{ds_{A}} \frac{ds_{A}}{ds} = v_{1}X + v_{2}Y,

v_{1} := \frac{du_{1}}{ds} + ku_{2} + 1, v_{2} := \frac{du_{2}}{ds} + ku_{1},

the Frenet frame {Z, W } of c_{A} has following equations;

\begin{matrix}\frac{dZ}{ds_{A}} = k_{A}W, \\ \frac{dW}{ds_{A}} = k_{A}Z, \end{matrix} (3.3)

where k_{A} is the curvature of c_{A}.

Let θ (resp. ω) be the slope angle of c (resp. c_{A}). Then

k_{A} = \frac{dω}{ds_{A}} = \frac{dω}{ds_{A}} \frac{ds}{ds_{A}} = (k + \frac{dϕ}{ds}) \frac{1}{\sqrt{|v_{1}^{2} − v_{2}^{2}|}}, (3.4)

where ϕ = ω − θ.

If c_{A} is space-like, then we can put

cosh ϕ = \frac{v_{1}}{\sqrt{v_{1}^{2} − v_{2}^{2}}} ,

sinh ϕ = \frac{v_{2}}{\sqrt{v_{1}^{2} − v_{2}^{2}}} .

Since

\frac{dϕ}{ds} = \frac{d}{ds} (cosh^{−1} \frac{v_{1}}{\sqrt{v_{1}^{2} − v_{2}^{2}}} ) ,

(3.4) reduces to

k_{A} = (k + \frac{v_{1}v′_{2} − v′_{1}v_{2}}{v_{1}^{2} − v_{2}^{2}} ) \frac{1}{\sqrt{v_{1}^{2} − v_{2}^{2}}} ,

where dash represents the derivative with respect to s.

If c_{A} is time-like, since sinh ϕ = \frac{v_{1}}{\sqrt{v_{2}^{2} − v_{1}^{2}}} , we have

k_{A} = (k + \frac{v′_{1}v_{2} − v_{1}v′_{2}}{v_{2}^{2} − v_{1}^{2}} ) \frac{1}{\sqrt{v_{2}^{2} − v_{1}^{2}}} ,

4 Euler - Savary’s formula

In this section, we consider the roulette and give the Euler - Savary’s formula.

Let c_{B} (resp. c_{R} ) be the base (resp. rolling) curve and k_{B} (resp. k_{R}) the curvature of c_{B} (resp. c_{R}). Let P be a point relative to c_{R}. By c_{L}, we denote the roulette of the locus of P.

We can consider that c_{L} is an associated curve of c_{B}, then the relative coordinate {x, y} of c_{L} with respect to c_{B} satisfies

\begin{matrix}\frac{δx}{ds_{B}} = \frac{dx}{ds_{B}} + k_{B}y + 1, \\ \frac{δy}{ds_{B}} = \frac{dy}{ds_{B}} + k_{B}x, \end{matrix} (4.1)

by virtue of (3.2).

Since c_{R} rolles without splitting along c_{B}, at each point of contact, we can consider {x, y} is a relative coordinate of c_{L} with respect to c_{R} for a suitable parameter s_{R}. In this case, the associated curve is reduced to a point P. Hence it follows that

\begin{matrix}\frac{δx}{ds_{R}} = \frac{dx}{ds_{R}} + k_{R}y + 1 = 0, \\ \frac{δx}{ds_{R}} = \frac{dx}{ds_{R}} + k_{R}y = 0. \end{matrix} (4.2)

Substituting these equations into (4.1), we have

\frac{δx}{ds_{B}} = (k_{B} − k_{R})y, \frac{δy}{ds_{B}} = (k_{B} − k_{R})x, (4.3)

so

\frac{δx}{δy} = \frac{x}{y}. (4.4)

Proposition 4.1 Let c_{R} rolles without splitting along c_{B} from the starting time t = 0. Then at each time t = t_{0} of this motion, the normal at the point c_{L}(t_{0}) passes through the point of contact c_{B}(t_{0}) = c_{R}(t_{0}).

Suppose that c_{L} is spacelike. Then, from (4.3),

0 < (\frac{δx}{ds_{B}} )^{2} − (\frac{δy}{ds_{B}} )^{2} = (k_{B} − k_{R})^{2}(y^{2} − x^{2}). (4.5)

Hence we can put

x = sinh ϕ, y = cosh ϕ.

Differentiating these equations, we have

\frac{dx}{ds_{R}} = \frac{dr}{ds_{R}} sinh ϕ + r cosh ϕ\frac{dϕ}{ds_{R}} = −1 − k_{R}r cosh ϕ,

\frac{dy}{ds_{R}} = \frac{dr}{ds_{R}} cosh ϕ + r sinh ϕ\frac{dϕ}{ds_{R}} = −k_{R}r sinh ϕ,

by virtue of (4.2). From these equations, it follows that

r \frac{dϕ}{ds_{R}} = − cosh ϕ − k_{r}r.

Therefore, substituting this equation into (3.4), we have

rk_{L} = ±1 − \frac{cosh ϕ}{r|k_{B} − k_{R}|}.

If c_{L} is timelike, by similar calculation, we have

rk_{L} = ±1 + \frac{sinh ϕ}{r|k_{B} − k_{R}|}.

We can easily see that the case c_{L} is null makes a contradiction.

Theorem 4.1 On the Minkowski plane L^{2}, suppose that a curve c_{R} rolles without splitting along a curve c_{B}. Let c_{L} be a locus of a point P that is relative to c_{R}. Let Q be a point on c_{L} and R a point of contact of c_{B} and c_{R} corresponds to Q relative to the rolling relation. By (r, ϕ), we denote a polar coordinate of Q with respect to the origin R and the base line c′_{B}|_{R}. Then curvatures k_{B}, k_{R} and k_{L} of c_{B}, c_{R} and c_{L}, respectively, satisfies

rk_{L} = ±1 − \frac{cosh ϕ}{r|k_{B} − k_{R}|} (when c_{L} is space like),

rk_{L} = ±1 + \frac{sinh ϕ}{r|k_{B} − k_{R}|} (when c_{L} is time like).

5 Determining the curve

Since the roulette is a locus of a point, it is determined by the base curve and the rolling curve. In theis section, we consider the converse problem.

First suppose that a base curve c_{B} and a roulette c_{L} is given.

Let (x(s_{B}), y(s_{B})) be the orthogonal coordinates of the base curve c_{B} with the arc length parameter s_{B}. For a point Q of c_{B}, draw the normal to the roulette c_{L}. Let R be the foot of this normal with the orthogonal coordinate (f(s_{B}), g(s_{B})). Then the length of QR is

QR = \sqrt{|(f(s_{B}) − x(s_{B}))^{2} − (g(s_{B}) − y(s_{B}))^{2}|}. (5.1)

If we consider (5.1) on the rolling curve c_{R}, this equation represents the length of the point P relative to c_{R} and a point of c_{R}. Hence the orthogonal coordinate (u(s_{B}), v(s_{B})) of c_{R} is given by the equations

u(s_{B})^{2} − v(s_{B})^{2} = (f(s_{B}) − x(s_{B}))^{2} − (g(s_{B}) − y(s_{B}))^{2},

(\frac{du}{ds_{B}} )^{2} − (\frac{dv}{ds_{B}} )^{2} = ±1,

the sign of ±1 depends on spacelike or timelike of c_{R}.

Next suppose that a rolling curve c_{R} and a roulette c_{L} is given.

Let (x(s_{L}), y(s_{L})) be the orthogonal coordinate of c_{L} with arclength parameter s_{L}. Suppose that the polar coordinate r(s_{R}) of c_{R} is given by the arc length parameter s_{R} of c_{R}.

Since the normal of c_{L} is (\frac{dy}{ds_{L}} , \frac{dx}{ds_{L}} ), a point (u, v) of the base curve c_{B} is given

by

\begin{matrix}u = x(s_{L}) ± r(s_{R})\frac{dy}{ds_{L}} , \\ v = y(s_{L}) ± r(s_{R})\frac{dx}{ds_{L}} , \end{matrix} (5.2)

Then, from

\frac{du}{ds_{R}} = \frac{dx}{ds_{L}} \frac{ds_{L}}{ds_{R}} ± \frac{dr}{ds_{R}} \frac{dy}{ds_{L}} ± r \frac{d}{s_{L}} (\frac{dy}{ds_{L}} ) \frac{ds_{L}}{ds_{R}} ,

\frac{dv}{ds_{R}} = \frac{dy}{ds_{L}} \frac{ds_{L}}{ds_{R}} ± \frac{dr}{ds_{R}} \frac{dx}{ds_{L}} ± r \frac{d}{s_{L}} (\frac{dx}{ds_{L}} ) \frac{ds_{L}}{ds_{R}} ,

we have

\frac{du}{ds_{R}} = \frac{dx}{ds_{L}} (1 ± rk_{L}) \frac{ds_{L}}{ds_{R}} ± \frac{dr}{ds_{R}} \frac{dy}{ds_{L}} ,

\frac{dv}{ds_{R}} = \frac{dy}{ds_{L}} (1 ± rk_{L}) \frac{ds_{L}}{ds_{R}} ± \frac{dr}{ds_{R}} \frac{dx}{ds_{L}} ,

where k_{L} is the curvature of c_{L}.

Since s_{R} is also the arc length of c_{B}, it follows that

(\frac{du}{ds_{R}} )^{2} − (\frac{dv}{ds_{R}} )^{2} = (\frac{ds_{L}}{ds_{R}} )^{2} (1 ± rk_{L})^{2} − \frac{dr}{ds_{R}}^{2} = ±1,

where the sign of ±1 depends on spacelike or timelike of c_{B}. From this differential equation, we can solve s_{L} = s_{L}(s_{R}). Substituting this equation into (5.2), we can have the orthogonal coordinate of c_{B}.

The solvability of these differential equations is easily checked. For example, we have solutions like that : c_{B} is x-axis, c_{R} is quadratic curve and c_{L} is ”Delaunay curve” (cf. [1]).

References

[1] J. Hano, and K. Nomizu, Surfaces of revolution with constant mean curvature in Lorentz-Minkowski space, Tohoku Math. J., 36 (1984), 427-437.

[2] T. Ikawa, On curves and submanifolds in an indefinite-Riemannian manifold, Tsukuba J. Math., 9 (1985), 353-371.

[3] T. Kubota, Ueber die Bewegung einer Kurve, die auf einer anderen Kurve rollt, Science Rep. of the Tôhoku Imperial Univ., 9 (1920), 251-274.

[4] B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, 1983.

Nihon University, Dep. of Math., School of Medicine,

Itabashi, Tokyo, Japan, 173

email:tikawa@med.nihon-u.ac.jp