ON THE VOLUME OF THE TRAJECTORY SURFACES UNDER THE HOMOTHETIC
MOTIONS
M. DÜLDÜL and N. KURUOĞLU
Abstract. The volumes of the surfaces of 3-dimensional Euclidean Space which are traced by a fixed point as a
trajectory surface during 3-parametric motions are given by H. R. Müller [3], [4], [5] and W. Blaschke [1].
In this paper, the volumes of the trajectory surfaces of fixed points under3-parametric homothetic motions are
computed. Also, using a certain pseudo-Euclidean metric we generalized the well-known classical Holditch Theorem, [2],
to the trajectory surfaces.
1.
Introduction
Let R and R0 be moving and fixed spaces and {O; e 1 , e 2 , e 3 } and {O0 ; e 1 0 , e 2 0 , e 3 0 } be their orthonormal coordinate
u = u (t1 , t2 , t3 ), then a 3-parameter motion B3 of R with respect
systems, respectively. If e j = e j (t1 , t2 , t3 ) andu
−−0→
0
to R is defined, where u = O O and t1 , t2 , t3 are the real parameters. For the rotation part of B3 , we have the
anti-symmetric system of differentiation equations (Ableitungsgleichungen)
deei = e k ωj − e j ωk ,
i, j, k = 1, 2, 3 (cyclic)
with the conditions of integration (Integrierbarkeitsbedingungen)
dωi = ωj ∧ ωk ,
Received April 05, 2006.
2000 Mathematics Subject Classification. Primary 53A17.
Key words and phrases. homothetic motion, Holditch theorem, volume of trajectory surface.
The author thanks TÜBİTAK-BAYG for their financial supports during his doctorate studies.
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where “d” is the exterior derivative and “ ∧ ” is the wedge product of the differential forms. For the translation
part of B3
−−→
dO0 O = σ = σ1e 1 + σ2e 2 + σ3e 3 ,
where the conditions of integration are
dσi = σj ∧ ωk − σk ∧ ωj .
During B3 , ωi and σi are the linear differential forms with respect to t1 , t2 , t3 . We assume that ωi , i = 1, 2, 3 are
linear independent, i.e., ω1 ∧ ω2 ∧ ω3 6= 0.
2.
The volume of the trajectory surface
under the homothetic motions
I.
Now, let us consider the 3-parametric homothetic motion of the fixed point X = (xi ) with respect to arbitrary
moving Euclidean space. We may write
x0 = u + hx
x,
where x and x 0 are the position vectors of the point X with respect to the moving and fixed coordinate systems,
respectively, and h = h(t1 , t2 , t3 ) is the homothetic scale of the motion. Then, we get
x0 = σ + x dh + hx
x × ω,
dx
P
where ω = ωie i is P
the rotation vector and “ × ” denotes the vector product.
x0 = τie i , we get
If we write dx
(1)
τi = σi + xi dh + h(xj ωk − xk ωj ).
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The volume element of the trajectory surface of X is
dJX = τ1 ∧ τ2 ∧ τ3 .
(2)
Thus, the integration of the volume
element over the region G of the parameter space yields the volume of the
R
trajectory surface, i.e., JX = G dJX . Let Γ be the closed and orientated edge surface of G.
If we replace (1) in (2), for the volume of the trajectory surface of X we get
! 3
!
3
3
3
X
X
X
X
X
2
2
(3)
JX = JO +
Ãi xi +
Aij xi xj +
Bi xi +
Ci xi ,
xi
i=1
i6=j
i=1
i=1
i=1
where
Z
h2 σi ∧ ωj ∧ ωk + hdh ∧ σj ∧ ωj + hdh ∧ σk ∧ ωk
Ãi =
G
(4)
1
=
2
Z
Aij =
Z
h2 σ j ∧ ω j + h2 σ k ∧ ω k ,
Γ
hdh ∧ ωi ∧ σj + hdh ∧ ωj ∧ σi + h2 σj ∧ ωj ∧ ωk + h2 σi ∧ ωk ∧ ωi
G
Z
1
=
h2 ω i ∧ σ j + h2 ω j ∧ σ i ,
2
Z Γ
Z
Bi =
(hσi ∧ σk ∧ ωk + dh ∧ σj ∧ σk + hσi ∧ σj ∧ ωj ) =
hσj ∧ σk ,
Γ
ZG
Z
1
Ci =
h2 dh ∧ ωj ∧ ωk =
h3 ω j ∧ ω k
3
G
Γ
and JO =
R
G
σ1 ∧ σ2 ∧ σ3 is the volume of the trajectory surface of the origin point O.
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Let us suppose that σi ∧ ωi , i = 1, 2, 3, have the same sign when integrated over any consistently orientated
simplex from Γ. Then, using the mean-value theorem for double integrals, we obtain
Z
Z
(5)
h2 σi ∧ ωi = h2 (ui , vi ) σi ∧ ωi ,
i = 1, 2, 3,
Γ
Γ
where ui and vi are the parameters. If we assume that
h2 (u1 , v1 ) = h2 (u2 , v2 ) = h2 (u3 , v3 ),
then using (4) and (5) we can find the parameters u0 and v0 such that
JX = JO + h2 (u0 , v0 )
(6)
3
X
i=1
Ai x2i +
X
Aij xi xj +
3
X
Bi xi
i=1
i6=j
+
3
X
!
x2i
i=1
3
X
!
Ci xi
,
i=1
where
Z
1
Ai =
(σj ∧ ωj + σk ∧ ωk ) .
2 Γ
Now, let us consider the plane P : C1 x + C2 y + C3 z = 0. The volumes of the trajectory surfaces of points on P
are quadratic polynomial with respect to xi . If we choose the moving coordinate system such that the coefficients
of the mixture quadratic terms vanish, i.e. Aij = 0, then we get for a point X ∈ P
(7)
JX = JO + h2 (u0 , v0 )
3
X
i=1
Ai x2i +
3
X
Bi xi .
i=1
Hence, we may give the following theorem:
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Theorem 1. All the fixed points of P whose trajectory surfaces have equal volume during the homothetic
motion lie on the same quadric.
II.
Let X and Y be two fixed points on P and Z be another point on the line segment XY , that is,
zi = λxi + µyi ,
λ + µ = 1.
Using (7), we get
JZ = λ2 JX + 2λµJXY + µ2 JY ,
(8)
where
JXY = JY X = JO + h2 (u0 , v0 )
3
X
3
Ai xi yi +
i=1
1X
Bi (xi + yi )
2 i=1
is called the mixture trajectory surface volume. It is clearly seen that JXX = JX . Since
(9)
JX − 2JXY + JY = h2 (u0 , v0 )
3
X
Ai (xi − yi )2 ,
i=1
we can rewrite (8) as follows:
(10)
JZ = λJX + µJY − h2 (u0 , v0 )λµ
3
X
Ai (xi − yi )2 .
i=1
We will define the distance D(X, Y ) between the points X, Y of P by
(11)
D2 (X, Y ) = ε
3
X
Ai (xi − yi )2 ,
ε = ±1,
[4].
i=1
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By the orientation of the line XY we will distinguish D(X, Y ) = −D(Y, X). Therefore, from (10) we have
JZ = λJX + µJY − εh2 (u0 , v0 )λµD2 (X, Y ).
(12)
Since X, Y and Z are collinear, we may write
D(X, Z) + D(Z, Y ) = D(X, Y ).
Thus, if we denote
λ=
D(Z, Y )
,
D(X, Y )
µ=
D(X, Z)
,
D(X, Y )
from (12) we get
(13)
JZ =
1
[D(Z, Y )JX + D(X, Z)JY ]
D(X, Y )
− εh2 (u0 , v0 )D(X, Z)D(Z, Y ).
Now, we consider that the points X and Y trace the same trajectory surface. In this case, we get JX = JY .
Then, from (13) we obtain
(14)
JX − JZ = εh2 (u0 , v0 )D(X, Z)D(Z, Y )
which is the generalization of Holditch’s result, [2], for trajectory surfaces during the homothetic motions. (14)
is also equivalent to the result given by [6]. We may give the following theorem:
Theorem 2. Let XY be a line segment with the constant length on P and the endpoints of this line segment
have the same trajectory surface. Then, the point Z on this line segment traces another trajectory surface. The
volume between these trajectory surfaces depends on the distances (in the sense of (11)) of Z from the endpoints
and the homothetic scale h.
Special case: In the case of h ≡ 1, we have the result given by H. R. Müller, [3].
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III.
III
Let X1 = (xi ), X2 = (yi ) and X3 = (zi ), i=1,2,3 be noncollinear points on P and Q = (qi ) be another point
Let we
X1 =
(xi ),write
X2 = (yi ) and X3 = (zi ), i=1,2,3 be noncollinear points on P and Q = (qi ) be
on P (Fig. 1). Then,
may
another point on P (Fig.1). Then, we may write
qi = λ1 xi + λ2 yi + λ3 zi ,
λ1 + λ2 + λ3 = 1.
qi = λ1 xi + λ2 yi + λ3 zi ,
Xu1
X2
λ1 + λ2 + λ3 = 1.
c
AA c
A c
c
A
c
Q3 a
A
u
c uQ2
a
aa
A Q
!c
a
A u!!
c
aa
!
c
!! Aaaa
c
!
A
a
!
aa c
A
!!
aac
!
!!
A
ac
ac
cuX
Au
u!
a
!
3
Q1
Figure
1.
Fig.1
If obtain
we use (7), we obtain
If we use (7), we
2
2
2 JX +
2 λ JX 2+ 2λ1 λ2 JX X + 2λ1 λ3 JX X + 2λ2 λ3 JX X .
JQ = λ21 JJXQ1=+λλ
2X1 +
2 λ2 JX3
3
1 2
1 3
2 3
2 + λ3 JX3 + 2λ1 λ2 JX1 X2 + 2λ1 λ3 JX1 X3 + 2λ2 λ3 JX2 X3 .
1J
After eliminating the mixture trajectory surface volumes by using (9), we get
After eliminating the mixture trajectory surface volumes by using (9), we get
(15)
JQ =λ1 JX1 + λ2 JX2 + λ3 JX3 − h2 (u0 , v0 )
JQ = λ1 JX1 + λ2 JX2 + λ3 JX3 −
n
o
2
2
2
2
v0 ) 1 ,εX
(Xλ11, λ
X32D
) +2 (X
ε131λ,1 λ
(Xε1 ,23
Xλ3 )2 λ
+3εD
D, 2X
(X
· ε12 λ−h
D02, (X
X3 D
. 3 ) . (15)
12 λ
2 Dε13
23 λ(X
2 λ32
2, X
1 λ2(u
2 1)λ+
3) +
3)
On the other hand,Onif the
we other
consider
point
Q1 the
= (a
write
i ), we
hand,the
if we
consider
point
Q1 may
= (ai ),
we may write
ai = µ1 yi + µ2 zi ,
qi = µ3 xi + µ4 ai ,
µ1 + µ2 = µ3 + µ4 = 1.
ai = µ1 yi + µ2 zi , qi = µ3 xi + µ4 ai , µ1 + µ2 = µ3 + µ4 = 1.
Thus, we have λ1 = µ3 , λ2 = µ1 µ4 , λ3 = µ2 µ4 i.e.
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Thus, we have λ1 = µ3 , λ2 = µ1 µ4 , λ3 = µ2 µ4 i.e.
λ1 =
D(Q, Q1 )
,
D(X1 , Q1 )
λ2 =
D(X1 , Q)D(Q1 , X3 )
,
D(X1 , Q1 )D(X2 , X3 )
λ3 =
D(X1 , Q)D(X2 , Q1 )
.
D(X1 , Q1 )D(X2 , X3 )
Similarly, considering the points Q2 and Q3 , respectively, we find
D(Q, Qi )
D(Xj , Q)D(Xk , Qj )
=
D(Xi , Qi )
D(Xj , Qj )D(Xk , Xi )
D(Xk , Q)D(Qk , Xj )
=
,
i, j, k = 1, 2, 3 (cyclic).
D(Xk , Qk )D(Xi , Xj )
λi =
Then, from (15) the generalization of (12) is found as
2
X
X D(Q, Qi )
D(Xk , Q)
2
JX − h (u0 , v0 )
εij
D(Qk , Xj )D(Xi , Qk ).
JQ =
D(Xi , Qi ) i
D(Xk , Qk )
If X1 , X2 , X3 trace the same trajectory surface, then the difference between the volumes is
2
X
D(Xk , Q)
JX1 − JQ = h2 (u0 , v0 )
εij
D(Qk , Xj )D(Xi , Qk ).
D(Xk , Qk )
Then, we can give the following theorem:
Theorem 3. Let us consider a triangle on the plane P . If the vertices of this triangle trace the same trajectory
surface, then a different point on P traces another surface. The volume between these trajectory surfaces depends
on the distances (in the sense of (11)) of the moving triangle and the homothetic scale h.
Acknowledgment. The authors are very grateful to the referee for the helpful comments and valuable
suggestions.
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1.
2.
3.
4.
5.
6.
Blaschke W., Über Integrale in der Kinematik, Arch. Math. 1 (1948), 18–22.
Holditch H., Geometrical Theorem, Q. J. Pure Appl. Math. 2 (1858), 38.
Müller H. R., Räumliche Gegenstücke zum Satz von Holditch, Abh. Braunschw. Wiss. Ges. 30 (1979), 54–61.
, Über den Rauminhalt kinematisch erzeugter, geschlossener Flächen, Arch. Math. 38 (1982), 43–49.
, Ein Holditch Satz für Flächenstücke im R3 , Abh. Braunschw. Wiss. Ges. 39 (1987), 37–42.
Tutar A. and Kuruoğlu, N., The Steiner formula and the Holditch theorem for the homothetic motions on the planar kinematics,
Mech. Mach. Theory 34 (1999), 1–6.
M. Düldül, Sinop University, Science and Arts Faculty, Department of Mathematics, 57000, Sinop, Turkey,
e-mail: mduldul@omu.edu.tr
N. Kuruoğlu, Bahçeşehir University, Science and Arts Faculty, Department of Mathematics and Computer Sciences, Bahçeşehir 34538,
Istanbul, Turkey,
e-mail: kuruoglu@bahcesehir.edu.tr
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