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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
Weconsider 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 Classi¯cations: 53A35, 53B30
Key words: base curve, curvature, Euler - Savary’s formula, rolling curve, roulette.
1 Introduction
2
On the Euclidean plane E , we consider two curves c and c . Let P be a point
B R
relative to c . When c rolles without splitting along c , the locus of the point P
R R B
makes a curve, say c . On this set of curves, c , c c are called the base curve,
L B R L
rolling curve and roulette, respectively. For example, if cB is a straight line, cR is a
quadratic curve and P is a focus of c , then c is the Delaunay curve that are used
R L
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 ¯nd 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 con-
stant mean curvature [1]. The purpose of this paper is to give answers to the above-
mentioned 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. Sec-
tion 4 is devoted to give the Euler - Savary’s formula on the Minkowski plane. In the
¯nal section, we determine the third curve from other two.
∗
Balkan Journal of Geometry and Its Applications, Vol.8, No.2, 2003, pp. 31-36.
c
°Balkan Society of Geometers, Geometry Balkan Press 2003.
32 T. Ikawa
2 Preliminaries
2 2
Let L be the Minkowski plane with metric g = (+,−). A vector X of L 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 6= 0. 2 2
Acurve c is a smooth mapping c : I → L from an open interval I into L . 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 ¯eld dc = µdx, dy¶ =: X is called the tangent
dt dt dt
vector ¯eld of the curve c(t). If the tangent vector ¯eld 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 ¯eld X is non-null, we can have the arc length parameter s and have the Frenet
formula
(2.1) dX =kY, dY =kX,
ds ds
where k is the curvature of c(s) (cf. [2]). The vector ¯eld Y is called the normal vector
¯eld 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 dφ = k.
ds
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
(3.1) c =c(s)+u (s)X +u (s)Y,
A 1 2
then cA(s) generally makes a curve. This curve is called the associated curve of c(s).
Remark that {u (s),u (s)} is a relative coordinate of c (s) with respect to
1 2 A
{c(s),X,Y}.
If we put
dc δu δu
A = 1X+ 2Y,
ds ds ds
then, since
dc dc du dX du dY µ du ¶ µ du ¶
A = + 1X+u + 2Y+u = 1+ 1+ku X+ ku + 2Y ,
ds ds ds 1 ds ds 2 ds ds 2 1 ds
by virtue of (2.1), we have
δu1 = du1 +ku +1,
ds ds 2
(3.2)
δu2 = du2 +ku .
ds ds 1
Let s be the arc length parameter of c . Then, from
A A
Euler - Savary’s Formula on Minkowski Geometry 33
dc dc ds
A = A A =v X+v Y,
ds ds ds 1 2
A
du du
v := 1 +ku +1, v := 2 +ku ,
1 ds 2 2 ds 1
the Frenet frame {Z,W} of cA has following equations;
dZ =k W,
ds A
(3.3) A
dW =k Z,
ds A
A
where kA is the curvature of cA.
Let θ (resp. ω) be the slope angle of c (resp. cA). Then
dω dω ds µ dφ¶p 1
(3.4) k = = = k+ ,
A ds ds ds ds 2 2
A A A |v −v |
1 2
where φ = ω −θ.
If cA is space-like, then we can put
v
coshφ = p 1 ,
v2 −v2
1 2
v
sinhφ = p 2 .
v2 −v2
1 2
Since à !
dφ = d cosh−1 p v1 ,
ds ds v2 −v2
1 2
(3.4) reduces to µ ¶
v1v′ −v′v2 1
k = k+ 2 1 p ,
A v2 −v2 v2 −v2
1 2 1 2
where dash represents the derivative with respect to s.
If c is time-like, since sinhφ = p v1 , we have
A v2 −v2
2 1
µ v′v −v v′ ¶ 1
k = k+ 1 2 1 2 p ,
A v2 −v2 v2 −v2
2 1 2 1
4 Euler - Savary’s formula
In this section, we consider the roulette and give the Euler - Savary’s formula.
Let cB (resp. cR ) be the base (resp. rolling) curve and kB (resp. kR) the curvature
of c (resp. c ). Let P be a point relative to c . By c , we denote the roulette of the
B R R L
locus of P.
Wecanconsider that c is an associated curve of c , then the relative coordinate
L B
{x,y} of c with respect to c satis¯es
L B
34 T. Ikawa
δx = dx +k y+1,
ds ds B
(4.1) B B
δy = dy +k x,
ds ds B
B B
by virtue of (3.2).
Since c rolles without splitting along c , at each point of contact, we can consider
R B
{x,y} is a relative coordinate of cL with respect to cR for a suitable parameter sR.
In this case, the associated curve is reduced to a point P. Hence it follows that
δx = dx +k y+1=0,
ds ds R
(4.2) R R
δx = dx +k y=0.
ds ds R
R R
Substituting these equations into (4.1), we have
(4.3) δx =(k −k )y, δy =(k −k )x,
ds B R ds B R
B B
so
(4.4) δx = x.
δy y
Proposition 4.1 Let cR rolles without splitting along cB from the starting time
t = 0. Then at each time t = t of this motion, the normal at the point c (t ) passes
0 L 0
through the point of contact cB(t0) = cR(t0).
Suppose that cL is spacelike. Then, from (4.3),
µ ¶ µ ¶
δx 2 δy 2
2 2 2
(4.5) 0 < − =(k −k ) (y −x ).
ds ds B R
B B
Hence we can put
x=sinhφ, y = coshφ.
Di®erentiating these equations, we have
dx = dr sinhφ+rcoshφ dφ =−1−k rcoshφ,
ds ds ds R
R R R
dy = dr coshφ+rsinhφ dφ =−k rsinhφ,
ds ds ds R
R R R
by virtue of (4.2). From these equations, it follows that
r dφ =−coshφ−k r.
ds r
R
Therefore, substituting this equation into (3.4), we have
rk =±1− coshφ .
L r|k −k |
B R
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