241
Acta Math. Univ. Comenianae
Vol. LXXVI, 2(2007), pp. 241–246
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 under
3-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 systems, respectively. If e j = e j (t1 , t2 , t3 ) and
u = u (t1 , t2 , t3 ), then a 3-parameter motion B3 of R with respect to R0 is defined,
−−→
where u = O0 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 ,
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.
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.
242
M. DÜLDÜL and N. KURUOĞLU
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
x,
x 0 = u + hx
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
τi = σi + xi dh + h(xj ωk − xk ωj ).
(1)
The volume element of the trajectory surface of X is
dJX = τ1 ∧ τ2 ∧ τ3 .
(2)
Thus, the integration of the volume element over the regionR G of the parameter
space yields the volume of the 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
Ãi x2i +
Aij xi xj +
Bi xi +
x2i
Ci xi ,
(3) JX = JO +
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 + h 2 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 Γ
G
R
and JO = G σ1 ∧ σ2 ∧ σ3 is the volume of the trajectory surface of the origin
point O.
243
ON THE VOLUME OF THE TRAJECTORY
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
2
2
(5)
h σi ∧ ωi = h (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 )
3
X
Ai x2i +
i=1
(6)
X
Aij xi xj +
3
X
Bi xi
i=1
i6=j
+
3
X
!
x2i
i=1
3
X
!
Ci xi
,
i=1
where
Ai =
1
2
Z
(σj ∧ ωj + σk ∧ ωk ) .
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
JX = JO + h2 (u0 , v0 )
(7)
3
X
Ai x2i +
i=1
3
X
Bi xi .
i=1
Hence, we may give the following theorem:
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
(8)
JZ = λ2 JX + 2λµJXY + µ2 JY ,
where
JXY = JY X = JO + h2 (u0 , v0 )
3
X
i=1
3
Ai xi yi +
1X
Bi (xi + yi )
2 i=1
244
M. DÜLDÜL and N. KURUOĞLU
is called the mixture trajectory surface volume. It is clearly seen that JXX = JX .
Since
3
X
(9)
JX − 2JXY + JY = h2 (u0 , v0 )
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
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].
245
ON THE VOLUME OF THE TRAJECTORY
III.
Let X1 = (xi ), X2 = (yi ) and X3 = (zIII
i ), i=1,2,3 be noncollinear points on P
andLetQX=1 =
(qi(x
) ibe
another
point
on
P
(Fig.
1).beThen,
we may
write
), X2 = (yi ) and X3 = (zi ), i=1,2,3
noncollinear
points
on P and Q = (qi ) be
another point on qP (Fig.1).
Then, we may write
i = 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
aaa
A Q
c
aa A !!! c
u
!
a
c
A
a
!
a
!
c
A aa
!!
c
a
!
A
aa c
!
!
aac
!!
A
ac
cuX
Au
u!
a
!
3
Q1
Figure
1.
Fig.1
If
wewe
obtain
If we
weuse
use(7),
(7),
obtain
22
JX2λ
+2λ
2λ11 λ33JJXX1 X1 X
2λ2 λ32JλX32 X
JQ = λ21 JJXQ1=+λλ
J1X+2 λ+22 JλX232 J+Xλ3 23+
λ21JλX
J3X. 2 X3 .
2 J1X
3 +12λ
3 +
1J
X12X2+
2X
3 + 2λ
After
eliminating
the mixture
mixturetrajectory
trajectory
surface
volumes
by(9),
using
(9), we get
After
eliminating the
surface
volumes
by using
we get
JQ =λ
λ3 JX − h2 (u0 , v0 )
X2λ +
JQ1 JX
=1 +
λ1 JλX21J+
2 JX2 + λ33JX3 −
n
o
(15)
−h2 (u02, v0 ) ε12 λ1 λ2 D2 (X1 , X2 ) +2 ε13 λ1 λ3 D2 (X1 , X3 ) + ε232λ2 λ3 D2 (X2 ,X3 ) . (15)
· ε12 λ1 λ2 D (X1 , X2 ) + ε13 λ1 λ3 D (X1 , X3 ) + ε23 λ2 λ3 D (X2 , X3 ) .
otherhand,
hand, if
thethe
point
Q1 =Q(a
), we
OnOn
thetheother
if we
weconsider
consider
point
(aimay
), wewrite
may write
1 i=
ai = µ1 yi +
zi1,yi + µ2qzii ,=qµ
µ2µ3=+µµ34 +
µ4 = 1.
ai µ
=2µ
xi µ
+4µa4ia,i , µ1 µ
+1µ+
= 1.
3 xµi 3+
i =
2 =
Thus,
= µµ33, ,λλ
2 = µ1 µ4 , λ3 = µ2 µ4 i.e.
Thus,wewehave
have λλ11 =
2 = µ1 µ4 , λ3 = µ2 µ4 i.e.
D(Q, Q1 )
D(X1 , Q)D(Q1 , X3 )
D(X1 , Q)D(X2 , Q1 )
D(X1 , Q)D(Q1 , X3 ), λ3 = D(X1 , Q)D(X2 , Q1 )
D(Q,
λ1 =
, Qλ12) =
.
, λD(X
λ11 ,=Q1 )
2 = 1 , Q1 )D(X2 , X3 ) , λ3 = D(X1 , Q1 )D(X2 ,.X3 )
D(X
D(X1 , Q1 )
D(X1 , Q1 )D(X2 , X3 )
D(X1 , Q1 )D(X2 , X3 )
Similarly, considering the points Q2 and Q3 , respectively, we find
Similarly, considering the points Q2 and Q3 , respectively, we find
D(Q, Qi )
D(Xj , Q)D(Xk , Qj )
λD(Q,
=
i = Qi )
D(X
D(Xk , Q)D(Qk , Xj )
j , Q)D(X
k , Qj )
D(X
,
Q
)
D(X
i
i
j , Qj )D(X
λi =
=
= k , Xi )
, i, j, k = 1, 2, 3(cyclic).
D(Xi , Qi )
D(Xj , Qj )D(Xk , Xi )
D(Xk , Qk )D(Xi , Xj )
D(Xk , Q)D(Qk , Xj )
,
i, j, k = 1, 2, 3 (cyclic).
=
Then, from (15)
thek ,generalization
D(X
Qk )D(Xi , Xofj )(12) is found as
!2
Then, from (15)
generalization
of (12)
Xthe
X is found
D(Q,
Qi )
D(Xk ,as
Q)
JQ =
JXi − h2 (u0 , v0 ) εij
k , Xj )D(Xi , Qk ).
D(Q
2
D(Xk ,,QQ)
X D(Q, QD(X
X
i , Qi )
k)
D(X
i)
k
2
JQ =
JX − h (u0 , v0 )
εij
D(Qk , Xj )D(Xi , Qk ).
D(Xi , Qi ) i
D(Xk , Qk )
5
If X1 , X2 , X3 trace the same trajectory surface, then the difference between the
volumes is
2
X
D(Xk , Q)
D(Qk , Xj )D(Xi , Qk ).
JX1 − JQ = h2 (u0 , v0 )
εij
D(Xk , Qk )
246
M. DÜLDÜL and N. KURUOĞLU
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.
References
1. Blaschke W., Über Integrale in der Kinematik, Arch. Math. 1 (1948), 18–22.
2. Holditch H., Geometrical Theorem, Q. J. Pure Appl. Math. 2 (1858), 38.
3. Müller H. R., Räumliche Gegenstücke zum Satz von Holditch, Abh. Braunschw. Wiss. Ges.
30 (1979), 54–61.
4.
, Über den Rauminhalt kinematisch erzeugter, geschlossener Flächen, Arch. Math. 38
(1982), 43–49.
5.
, Ein Holditch Satz für Flächenstücke im R3 , Abh. Braunschw. Wiss. Ges. 39 (1987),
37–42.
6. 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