On The Volume Of The Trajectory Surface Under The Galilean Motions In The Galilean Space

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1 Applied Mathematics E-Notes, 17(2017), c ISSN Available free at mirror sites of amen/ On The Volume Of The Trajectory Surface Under The alilean Motions In The alilean Space Mücahit Akbıyık, Salim Yüce Received 15 Feb 2017 Abstract In this work, the volumes of the trajectory surfaces which are traced by fixed points during 3-parameter alilean space motions are studied. Also, the wellknown classical Holditch theorem [3] is generalized for the volumes of the trajectory surfaces in the alilean space. 1 Introduction In 1958, H. Holditch, [3], came up with the following outstanding classical theorem: If the endpoints A and B of a fixed line segment AB with length a + b are rotated once along an oval k in the Euclidean plane E 2, then a given fixed point X (AX = a, XB = b) of AB describes a closed not necessarily convex curve k X. The area F of the Holditch- Ring bounded by the curves k and k X is F = πab. Later, this theorem was studied by different methods [1,2,6 10] and in different spaces [13 15]. One of the generalizations of this theorem is on the volumes of the surfaces of 3-dimensional Euclidean space which are traced by fixed points during 3- parameter motions are given by H. R. Müller [6 8] and W. Blashke [1]. In this paper, the volumes of the trajectory surfaces of fixed points under 3- parameter alilean space motions are calculated. Also, by the help of a special distance that we have defined, we generalize the well-known classical Holditch theorem for the volumes of the trajectory surfaces of fixed points under 3-parameter alilean space motions. 2 Preliminaries alilean geometry 3 can be described as the study of properties of 3-dimensional space with coordinates that are invariant under alilean transformations x = x + a, y = (v cos α) x + (cos ϕ) y + (sin ϕ) z + b, z = (v sin α) x + ( sin ϕ) y + (cos ϕ) z + d. Mathematics Subject Classifications: 53A17, 53A35. Department of Mathematics, Yildiz Technical University, Esenler, Istanbul 34220, Turkey Department of Mathematics, Yildiz Technical University, Esenler, Istanbul 34220, Turkey 297

2 298 On the Volume of the Trajectory Surface in 3 The alilean transformations consist of translation, rotation and shear motions, and are described by I. M. Yaglom in [11]. In the literature, the basic information about alilean eometry is firstly given by I. M. Yaglom. Then, the differential geometry of curves and surfaces in the alilean space 3 is worked in detail by O. Röshcel in [9]. Also, the quadrics in the alilean space are examined by Kamenarović in [4]. Now, let s give some basic information about the alilean space 3. Let a = (x, y, z) and b = (x 1, y 1, z 1 ) be two vectors in the alilean space. The scalar product of a and b is defined by < a, b > = xx 1. The vectors in the alilean space are divided into two classes as non-isotropic vectors and isotropic vectors which are of the form a = (x, y, z), x 0 and p = (0, y, z), respectively. Moreover, the special scalar product of isotropic vectors p = (0, y, z) and q = (0, y 1, z 1 ) is defined by < p, q > δ = yy 1 + zz 1. If a = (x, y, z) and b = (x 1, y 1, z 1 ) are vectors in alilean space, the vector product of a and b is defined as the following in [5]: 0 e 2 e 3 a b = x y z x 1 y 1 z 1. Let g 1 be a nonisotropic vector, g 2 and g 3 be isotropic vectors in the alilean space. If the vectors g 1, g 2, and g 3 satisfy that < g 1, g 1 > =< g 2, g 2 > δ =< g 3, g 3 > δ = 1 and < g 1, g 2 > =< g 1, g 3 > =< g 2, g 3 > δ = 0, then the vector system {g 1, g 2, g 3 } is called an orthonormal frame of alilean space. More information about the alilean geometry can be found in [9, 11]. 3 One Parameter alilean Space Motion Let R and R be two 3 dimensional alilean spaces. Let {O; g 1, g 2, g 3 } and {O ; g 1, g 2, g 3} are orthonormal frames of spaces R and R, respectively. Assume that the frame {O; g 1, g 2, g 3 } moves with respect to frame {O ; g 1, g 2, g 3}. Then, it is accepted that the space R moves according to the space R. The spaces R and R are called moving space and fixed space, respectively. Moreover, the frames {O; g 1, g 2, g 3 } and {O ; g 1, g 2, g 3} are called moving frame and fixed frame, respectively. This motion is called one parameter alilean space motion and is denoted by B = R/R. Here, the spaces R and R are orientated in the same direction. During the motion B = R/R, it is clear that x = u + x, where x and x correspond to the position vectors of any point X R according to the rectangular coordinate systems of R, R, respectively, and u = OO =u 1 g 1 + u 2 g 2 + u 3 g 3.

3 M. Akbiyik and S. Yüce 299 Here, the vector u is called translation vector. In the motion B = R/R, the vectors x, x and u are continuously differentiable functions of a real parameter t. If the frames {g 1, g 2, g 3 } and {g 1, g 2, g 3} are written as = g 1 g 2 g 3 in the matrix form, respectively, then and = g 1 g 2 g 3 = A and = A 1 (1) are hold. Here, A is an invertible matrix. In this case, by differentiating both sides of the equation (1) and considering that is fixed frame, we get d = Ω, (2) where Ω = daa 1. If we calculate Ω from above equation (2) and by the necessary operations with the basis vectors g i, 1 i 3, we get 0 ω 3 ω 2 Ω = 0 0 ω 1, (3) 0 ω 1 0 where ω i, 1 i 3 are the linear differential forms with respect to t, that is, ω i = f i (t) dt. The vector ω =ω 1 g 1 + ω 2 g 2 + ω 3 g 3 which is defined by nonzero components of Ω is called the instantaneous Pfaffi an vector of the motion B = R/R. Especially, if ω 1 = 0, then, the motion of B consists of only translation and shear motions. The motion doesn t contain the rotation. We will, therefore, accept ω 1 0 in this work. If it is used the equality (3) given for Ω, we get dg 1 = ω 3 g 2 + ω 2 g 3, dg 2 = ω 1 g 3, dg 3 = ω 1 g 2. (4) Moreover, if we calculate the exterior derivation of equation (4), by considering that basis vectors g i, i = 1, 2, 3, are linearly independent, we obtain dω 1 = 0, dω 2 = ω 3 ω 1, dω 3 = ω 2 ω 1, where " is the wedge product of the differential forms. Hence, the conditions of integration for components of the pfaffi an vector of the motion R/R are found as dω 1 = 0, dω 2 = ω 3 ω 1, dω 3 = ω 2 ω 1. (5) On the other hand, by differentiating of translation vector O O = u = u 1 g 1 u 2 g 2 u 3 g 3

4 300 On the Volume of the Trajectory Surface in 3 and by the aid of using the equation (4), we have σ = du =σ 1 g 1 + σ 2 g 2 + σ 3 g 3, where and The equations σ = du σ 1 = du 1, σ 2 = du 2 +u 1 ω 3 u 3 ω 1, σ 3 = du 3 u 1 ω 2 + u 2 ω 1. dg 1 = ω 2 g 3 ω 3 g 2, dg 2 = ω 1 g 3, dg 3 = ω 1 g 2, σ = du =σ 1 g 1 + σ 2 g 2 + σ 3 g 3, are called derivative equations of motion R/R. Furthermore, (6) 0=dσ = dσ 1 g 1 + (dσ 2 σ 1 ω 3 + σ 3 ω 1 ) g 2 + (dσ 3 + σ 1 ω 2 σ 2 ω 1 ) g 3 and because of the fact that {g 1, g 2, g 3 } are linearly independent, we find dσ 1 = 0, dσ 2 = σ 1 ω 3 σ 3 ω 1, dσ 3 = σ 1 ω 2 + σ 2 ω 1. (7) So, the conditions of integration obtained for the translation vector of the motion R/R are equations (7). Now, let s examine the velocity vectors of the point X under the motion R/R. Let X be any point in R. So, we can write x = x i g i with respect to the moving frame {O; g 1, g 2, g 3 }. Since we have x = u + x for position vector x of the point X with respect to fixed frame {O ; g 1, differential of position vector x can be expressed as g 2, g 3}, the dx = du+dx. By (6), it is calculated as dx = σ 1 g 1 + (σ 2 x 1 ω 3 + x 3 ω 1 ) g 2 + (σ 3 + x 1 ω 2 x 2 ω 1 ) g 3 + dx 1 g 1 + dx 2 g 2 + dx 3 g 3. During the motion R/R, the velocity vector of any point X R with respect to fixed space R and moving space R is called absolute velocity X A and relative velocity X R, respectively. X A and X R are expressed by X A = dx

5 M. Akbiyik and S. Yüce 301 and X R = dx 1 g 1 + dx 2 g 2 + dx 3 g 3. The difference between the absolute and the relative velocities is called sliding velocity X F of the point X and it is stated as X F = σ 1 g 1 + (σ 2 x 1 ω 3 + x 3 ω 1 ) g 2 + (σ 3 + x 1 ω 2 x 2 ω 1 ) g 3. In this way, the following theorem can be given: THEOREM 1. Let X be a point in R and X A, X F, and X R be the absolute, the sliding and the relative velocity of the point X under the motion R/R, respectively. Then, the relation between the velocities is given by X A = X F + X R. If any point X in R is fixed, then the relative velocity X R = 0 and X A = X F under the motion R/R. Furthermore, by considering derivative equations (6), the following equation holds: or dx = ( x 1 ω 3 + x 3 ω 1 ) g 2 + (x 1 ω 2 x 2 w 1 ) g 3 dx = x ω for any fixed point X in R. So, for any fixed point X during the motion R/R, one can state dx = σ + x ω. Also, one can rewrite where dx = τ 1 g 1 + τ 2 g 2 + τ 3 g 3, τ 1 = σ 1, τ 2 = σ 2 + x 3 ω 1 ω 3 x 1, τ 3 = σ 3 + x 1 ω 2 ω 1 x 2. (8) 4 The Volume of the Trajectory Surface in 3 I. Until now, we have considered that the translation vector u and basis vectors g i for 1 i 3 of the motion B are functions of a real parameter t. From now on, we assume that the translation vector u and basis vectors g i for 1 i 3 of the motion B are functions of real parameters t 1, t 2 and t 3. And this motion is called 3-parameter alilean space motion and we shall denote this 3-parameter motion by B 3. During the motion B 3, ω i and σ i are the linear differential forms with respect to t 1, t 2 and t 3. So, the equations (5), (7) and (8) are not changed for the motion B 3.

6 302 On the Volume of the Trajectory Surface in 3 Under the motion B 3, any fixed point X in the moving space R determines a volumetric trajectory surface in R. The volume element of the trajectory surface of X under the motion B 3, is defined by dj X = τ 1 τ 2 τ 3. (9) Thus, the integration of the volume element over a region determined by the fixed point X of the parameter space during the motion B 3 yields the volume of the trajectory surface, i.e., J X = dj X. By putting equation (8) into (9) and after making necessary arrangement, the volume of the trajectory surface of fixed point X in R during the motion B 3 is calculated as J X = J O + ax b i x i x 1 + c i x i, (10) where J O = σ 1 σ 2 σ 3, a = σ 1 ω 2 ω 3, b 2 = (σ 1 ω 1 ω 3 ), b 3 = (σ 1 ω 1 ω 2 ), c 1 = σ 1 σ 2 ω 2 σ 1 ω 3 σ 3, c 2 = (σ 1 σ 2 ω 1 ), c 3 = σ 1 ω 1 σ 3. J O is the volume of trajectory surface of origin point O. So, the volume J X of trajectory surface is a quadratic polynomial of x i. THEOREM 2. All fixed points in R whose trajectory surfaces have equal volume under the motion B 3 lie on the same quadric. II. Let X = (x i ) and Y = (y i ) be two fixed points in R and Z = (z i ) be another point on the line segment XY, that is, z i = λx i + µy i, λ + µ = 1 in barycentric coordinates. Then, the volume of the region in R determined by the fixed point Z under the motion B 3, by using equation (10), is obtained as where J XY = J O + ax 1 y J Z = λ 2 J X + 2λµJ XY + µ 2 J Y, b i (x 1 y i + y 1 x i ) c i (x i + y i ). (11) Also, J XY is called the mixture trajectory surface volume. It is clearly seen that J XX = J X and J XY = J Y X. Since J X 2J XY + J Y = a (x 1 y 1 ) 2 + b i (x 1 y 1 ) (x i y i ), (12)

7 M. Akbiyik and S. Yüce 303 we ( can restate the volume trajectory surface ) of the fixed point Z as J Z = λj X + µj Y λµ a (x 1 y 1 ) b i (x 1 y 1 ) (x i y i ). So, we may give following theorem: THEOREM 3. Let X and Y be two different fixed points in R, and Z be another point on the segment XY. During the motion B 3, the relation between the volumes of the trajectory surfaces of fixed points X, Y and Z is as follows: ( ) J Z = λj X + µj Y λµ a (x 1 y 1 ) 2 + b i (x 1 y 1 ) (x i y i ). (13) We will define the distance D (X, Y ) between the fixed points X, Y R, by D 2 (X, Y ) = ε ( a (x 1 y 1 ) 2 + ) b i (x 1 y 1 ) (x i y i ), ε = ±1. (14) Therefore, by the help of the distance (14), the equation (13) can be restated as follows: J Z = λj X + µj Y ελµd 2 (X, Y ). (15) Since X, Y and Z are collinear, we can write D (X, Z) + D (Z, Y ) = D (X, Y ). Hence, if we represent λ = D(Z,Y ) D(X,Y ), µ = D(X,Z) D(X,Y ), where D (X, Y ) 0, i.e., x 1 y 1, then from equation (15), we have J Z = 1 D (X, Y ) [D (Z, Y ) J X + D (X, Z) J Y ] εd (Z, Y ) D (X, Z). (16) Now, we consider that the fixed points X and Y trace the same trajectory surface. In this case, we get J X = J Y. Then, from the above equation (16), we get So, we may give the following theorem: J X J Z = εd (Z, Y ) D (X, Z). THEOREM 4. Let X = (x 1, x 2, x 3 ) and Y = (y 1, y 2, y 3 ), where x 1 y 1, be two different fixed points in R and Z be another point on the segment XY. Let the fixed points X and Y trace the same trajectory surface and the fixed point Z trace the different trajectory surface during the motion B 3. Then, the difference between the volumes of these two trajectory surfaces depends on the distances of Z from the endpoints which are defined in (14) with respect to the motion B 3. In case of x 1 = y 1, then the equation (13) is arranged as J Z = λj X + µj Y. If the fixed points X and Y trace the same trajectory surface during the motion B 3, we obtain J X J Z = 0.

8 304 On the Volume of the Trajectory Surface in 3 COROLLARY. Let X = (x 1, x 2, x 3 ) and Y = (y 1, y 2, y 3 ), where x 1 = y 1 be two different fixed points in R and Z be another point on the segment XY. If the fixed points X and Y trace the same trajectory surface, the fixed point Z trace the trajectory surface with same volume during the motion B 3. III. THEOREM 5. Let us consider a triangle in R whose vertices are points X 1 = (x 1, x 2, x 3 ), X 2 = (y 1, y 2, y 3 ) and X 3 = (z 1, z 2, z 3 ), where x 1 y 1, x 1 z 1 and y 1 z 1. If the vertices of this triangle trace the same trajectory surface in R, then a different point Q on the plane which is determined by X 1, X 2 and X 3 traces another surface. The difference between the volumes of these two trajectory surfaces depends on the distances D (X k, Q), D (X k, Q k ), D (Q k, X j ) and D (X i, Q k ) which are measured with respect to the motion B 3. Here, the point Q i is a intersection point of the segments X i Q and X j X k, i, j, k = 1, 2, 3; 2, 3, 1; 3, 1, 2. PROOF. Let X 1 = (x i ), X 2 = (y i ) and X 3 = (z i ), x 1 y 1, x 1 z 1 and y 1 z 1 be three non- collinear fixed points in R, and Q = (q i ) be another fixed point on the plane which is determined by X 1 = (x i ), X 2 = (y i ) and X 3 = (z i ). Then, for the point Q, we can write q i = λ 1 x 1 + λ 2 y i + λ 3 z i, λ 1 + λ 2 + λ 3 = 1. (17) Under the motion B 3, by the help of equations (10), (11) and (17), the volume of the region determined by the point Q can be calculated as J Q = λ 2 1J X1 + λ 2 2J X2 + λ 2 3J X3 + 2λ 1 λ 2 J X1X 2 + 2λ 1 λ 3 J X1X 3 + 2λ 2 λ 3 J X2X 3. Also, by considering the equations (12) and (14), we get J Q = λ 1 J X1 + λ 2 J X2 + λ 3 J X3 { ε 12 λ 1 λ 2 D 2 (X 1, X 2 ) + ε 13 λ 1 λ 3 D 2 (X 1, X 3 ) + ε 23 λ 2 λ 3 D 2 (X 2, X 3 ) }. Let the points Q 1 = (q 1i ) are intersection point of the segments X 1 Q and X 2 X 3. Then, we can write q 1i = ξ 1 y i + ξ 2 z i, q i = ξ 3 x i + ξ 4 a i, where ξ 1 + ξ 2 = ξ 3 + ξ 4 = 1. Hence, we have λ 1 = ξ 3, λ 2 = ξ 1 ξ 4, λ 2 = ξ 2 ξ 4 i.e., λ 1 = D (Q, Q 1) D (X 1, Q 1 ), λ 2 = D (Q 1, X 3 ) D (X 1, Q) D (X 2, X 3 ) D (X 1, Q 1 ), λ 3 = D (X 2, Q 1 ) D (X 1, Q) D (X 2, X 3 ) D (X 1, Q 1 ). Similarly, for Q 2 and Q 3, we calculate λ i = D (Q, Q i) D (X i, Q i ) = D (X j, Q) D (X k, Q j ) D (X j, Q j ) D (X k, X i ) = D (X k, Q) D (Q k, X j ) D (X k, Q k ) D (X i, X j ) for i, j, k = 1, 2, 3; 2, 3, 1; 3, 1, 2. So, the volume of the trajectory surface determined by the point Q can be rearranged as J Q = D (Q, Q i ) D (X, Q i ) J X i ( ) 2 D (Xk, Q) ε ij D (Q k, X j ) D (X i, Q k ). (18) D (X k, Q k )

9 M. Akbiyik and S. Yüce 305 During the motion B 3, since the points X 1, X 2 and X 3 trace the same trajectory surface in R, we can write J X1 = J X2 = J X3. So, from the equation λ 1 + λ 2 + λ 3 = 1 and equation (18), we can get J Xi J Q = ( ) 2 D (Xk, Q) ε ij D (Q k, X j ) D (X i, Q k ), D (X k, Q k ) for i, j, k (cyclic). Finally, the difference between the volumes J X1 and J Q only depends on distances on triangle X 1 X 2 X 3 defined in (14) according to the motion B 3. References [1] W. Blaschke, Über Integrale in der Kinematik. Arch. Math., 1(1948) [2] M. Düldül and N. Kuruoğlu, On the volume of the trajectory surfaces under the homothetic motions, Acta Math. Univ. Comenian., 76(2007), [3] H. Holditch, eometrical Theorem, Q. J. Pure Appl. Math., 2(1858), 38. [4] I. Kamenarović, Quadrics the alilean Space 3, Rad Hrvatske Akad. Znan. Umjet. No. 470 (1995), [5] Z. Milin-Sipus, Ruled Weingarten surfaces in alilean space, Period. Math. Hungar., 56(2008), [6] H. R. Müller, Raumliche egenstücke zum Satz von Holditch, Abh. Braunschw. Wiss. es., 30(1979), [7] H. R. Müller, Über den Räuminhalt kinematisch erzeugter, geschlossener Flächen, Arch. Math. 38(1982), [8] H. R. Müller, Ein Holditch Satz für Flä chenstücke im R 3, Abh. Braunschw. Wiss. es, 39(1987), [9] O. Röschel, Die eometrie des alileischen Raumes, Berichte [Reports], 256. Forschungszentrum raz, Mathematisch-Statistische Sektion, raz, [10] A. Tutar and N. Kuruoğlu, The Steiner formula and the Holditch theorem for the homothetic motions on the planar kinematics, Mech. Mach. Theory, 34(1999), 1 6. [11] I. M. Yaglom, A simple non-euclidean eometry and its Physical Basis, New York, USA: Springer-Verlag, [12] H. Yıldırım, S. Yüce and N. Kuruoğlu, Holditch theorem for the closed space curves in Lorentzian 3-space, Acta Math. Sci. Ser. B Engl. Ed., 31(2011),

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