/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 7, No. 3, 2014, 335-342 ISSN 1307-5543 – www.ejpam.com One-Parameter Planar Motions in Affine Cayley-Klein Planes N. (Bayrak) Gürses∗, S. Yüce Department of Mathematics, Faculty of Arts And Sciences, Yıldız Technical University, Istanbul, Turkey Abstract. In 1956, W. Blaschke and H.R. Müller introduced the one-parameter planar motions and obtained the relation between absolute, relative, sliding velocities and accelerations in the Euclidean plane E2 [3]. A. A. Ergin [4] considering the Lorentzian plane L2, instead of the Euclidean plane E2, introduced the one-parameter planar motions in the Lorentzian plane L2 and also gave the relations between the velocities and accelerations in 1991. In addition to this, in 2013, M. Akar and S. Yüce [1] introduced the one-parameter motions in the Galilean plane G2 and gave same concepts stated above. In this paper, we will introduce one parameter planar motions in affine Cayley-Klein (CK) planes Pε and we will discuss the relations between absolute, relative, sliding velocities and accelerations. 2010 Mathematics Subject Classifications: 53A17, 53A35, 53A40. Key Words and Phrases: Cayley-Klein planes, one-parameter planar motion, kinematics 1. Introduction The geometrical systems have a significant role in plane geometries. Cayley-Klein (CK) geometries, first introduced by Klein in 1871 and Cayley, are number of geometries including Euclidean, Galilean, Minkowskian and Bolyai-Lobachevsikan [8, 9]. Following Cayley and Klein, Yaglom distinguished these geometries with choosing one of three ways of measuring length (parabolic, elliptic, or hyperbolic) between two points on a line and one of the three ways of measuring angles between two lines (parabolic, elliptic, or hyperbolic) [14]. This gives nine ways of measuring lengths and angles and thus the nine plane geometries listed in Table 1. A great deal of studies are conducted in CK-planes [5–7, 10–13]. There is a known (but not well-known) relationship between the plane geometries which have parabolic measure of distance: Euclidean, Galilean and Minkowskian (Lorentz) geometries. They are called affine CK-plane geometries [14]. ∗Corresponding author. Email addresses: nbayrak@yildiz.edu.tr (N. Gürses ), sayuce@yildiz.edu.tr (S. Yüce) http://www.ejpam.com 335 c© 2014 EJPAM All rights reserved. N. Gürses , S. Yüce / Eur. J. Pure Appl. Math, 7 (2014), 335-342 336 Table 1: Nine CK-geometries in the Plane Measure of length between two points Elliptic Parabolic Hyperbolic Elliptic Elliptic Euclidean Hyperbolic Geometry Geometry Geometry Measure of angles Parabolic co-Euclidean Galilean co-Minkowskian between two lines (Euclidean) Geometry Geometry Geometry (Anti-Newton Hooke) (Newton-Hooke) Hyperbolic co-Hyperbolic Minkowskian doubly-Hyperbolic Geometry Geometry Geometry (Anti-De-Sitter) (De-Sitter) In kinematics, the one-parameter planar motions introduced by W. Blaschke and H.R. Müller and the relation between absolute, relative and sliding velocities (accelerations) are examined on the Euclidean plane E2 [3]. Then, the one-parameter planar motions on the Lorentzian (Minkowskian) plane L2 were given by [4]. In addition to this, same concept are investigated on the Galilean plane G2 by [1] and [2]. In this paper, we will introduce and focus on one parameter planar motions in affine CK- planes with generalizing the notations introduced by above scientists. Also, we will discuss the relations between absolute, relative and sliding velocities (accelerations). 2. Basic Notations of Affine CK-Planes In this section, we will investigate the basic notations of affine CK-planes [6, 14]. These planes are denoted by Pε. Let us consider R2 with the bilinear form 〈x,y〉ε = x1 y1 + εx2 y2 where ε may be 1,0 or −1 and x = (x1, x2), y = (y1, y2). The matrix of this bilinear form is given as below: B = � 1 0 0 ε � . For all x and y in Pε we can write 〈x,y〉 = xT By. For ε = 1 we have Euclidean plane E2, for ε= 0 we have Galilean plane G2 and for ε= −1 we have Lorentzian plane L2. If 〈x,y〉ε = 0, then the vectors x and y in Pε are orthogonal. Self-orthogonal vectors are called isotropic. N. Gürses , S. Yüce / Eur. J. Pure Appl. Math, 7 (2014), 335-342 337 The norm of the vector x= (x1, x2) in Pε is defined by ‖x‖ε = Ç � �〈x,x〉ε � �= Ç � �x2 1 + εx2 2 � �. The vector system � c1 = (1,0) ,c2 = (0,1) is orthonormal basis for Pε. The distance be- tween two points A= � x1, x2 � and B = � y1, y2 � is defined by ‖AB‖= Ç � �〈AB,AB〉ε � �= dAB = s � � � � y1 − x1 �2 + ε � y2 − x2 �2 � � �. For ε= 1 only the zero vector, for ε= 0 zero vectors and vertical vectors are isotropic and for ε= −1 zero vectors and vectors parallel to (±1,1) are isotropic [6]. A circle is the locus of points equidistant from a given fixed point, the center of the circle. The unit circle in Pε is the set of points with ‖P‖ = 1, for all P ∈ Pε. The equation of the unit circle in Pε is x2 + εy2 = ±1. They are shown in the Figure 1. Figure 1: Unit Circles in Pε The linear transformation J : Pε→ Pε with matrix, also denoted by J and given as below: J = � 0 −ε 1 0 � . This linear transformation converts any vector x to an orthogonal vector Jx. If x is a nonisotropic and y is orthogonal to x, then it is written such that y = kJx for some real number k [6]. It is not difficult to verify directly from the definition of the matrix exponential as eJϕ = ∑∞ n=0 (Jϕ)n n! that eJϕ = cos εϕ + J sin εϕ = � cos εϕ −ε sin εϕ sin εϕ cosεϕ � N. Gürses , S. Yüce / Eur. J. Pure Appl. Math, 7 (2014), 335-342 338 where cos εϕ = ∞ ∑ n=0 (−εn)ϕ2n (2n)! sin εϕ = ∞ ∑ n=0 (−εn)ϕ2n+1 (2n+1)! . For ε = 1 these are usual cosine and sine functions, for ε = −1 they are hyperbolic cosine and sine functions, and for ε= 0 they are just cos 0ϕ = 1 and cos 0ϕ = ϕ for all ϕ. In all cases, we obtain cos2 εϕ + ε sin2 εϕ = 1 and ∂ϕ cos εϕ = −ε sin εϕ,∂ϕ sin εϕ = cos εϕ. By equating corresponding entries of the matrix equation eJ(ϕ+θ ) = eJϕeJθ , we can find the sum formulae [14] as follows: cos ε(ϕ + θ ) = cos εϕ cos εθ − ε sin εϕ sin εθ sin ε(ϕ + θ ) = sin εϕ cos εθ + cos εϕ sin εθ . 3. One-Parameter Planar Motions in Affine CK-Planes 3.1. Derivative Formulae, Velocities and Pole Point Notation In this section, we stated that the one-parameter planar motions in affine CK-planes is an extension of the one-parameter planar motions in the Euclidean plane, Lorentzian plane, and Galilean plane, respectively given [3], [4] and [1]. We will define the one-parameter planar motions in affine CK-planes and we will obtain the relations between velocities and accelerations of a point under these motions. Let Pε and P′ε be moving and fixed affine CK-planes and {O;c1,c2} and {O′;c′1,c′2} be their orthonormal coordinate systems, respectively. Let us take the vector OO′ = u= u1c1 + u2c2 for u1,u2 ∈ R. (1) Let us define a transformation as given below: x′ = x− u, (2) where x, x′ are coordinate vectors with respect to the moving and fixed rectangular coordinate system of a point X = (x1, x2) ∈ Pε, respectively. By the equation (2), one-parameter planar motions in affine CK-planes are defined. These motions denoted by Pε/P ′ ε. Moreover, ϕ, the angle between the vectors c1 and c′1, is the rotation angle of the motions Pε/P ′ ε and x, x′, u are continuously differentiable functions of a time parameter t ∈ I ⊂ R. For t = 0, the coordinate systems are coincident. By taking ϕ = ϕ(t), we can write � c1 = cos εϕc′1 + sin εϕc′2 c2 = −ε sin εϕc′1 + cos εϕc′2 (3) N. Gürses , S. Yüce / Eur. J. Pure Appl. Math, 7 (2014), 335-342 339 We assume that ϕ̇ (t) = dϕ d t 6= 0, and ϕ̇ (t) is called the angular velocity of the motions Pε/P ′ ε. By differentiating the equations (1) and (3) with respect to t, the derivative formulae of the motions Pε/P ′ ε are obtained as follows:    ċ1 = ϕ̇c2 ċ2 = −εϕ̇c1 u̇ = (u̇1 − εϕ̇u2)c1 + (u̇2 + ϕ̇u1)c2. (4) By using these derivative formulae, we will define velocities of a point X = (x1, x2) ∈ Pε. The velocity of the point X with respect to Pε is called the relative velocity denoted by Vr and it is defined by dx d t = ẋ: Vr = ẋ1c1 + ẋ2c2. (5) Besides, the absolute velocity of the X with respect to Pε is obtained by differentiating the equation (2) with respect to t and using derivative formulae. It is denoted by Va and obtained as follows: Va = dx′ d t = {−u̇1 + εϕ̇(u2 − x2)}c1 + {u̇2 + ϕ̇(−u1 + x1)}c2 +Vr . (6) By using equation (6), we get the sliding velocity vector as below: V f = {−u̇1 + εϕ̇(u2 − x2)}c1 + {u̇2 + ϕ̇(−u1 + x1)}c2. (7) From equations (5), (6), and (7), the following theorem can be given. Theorem 1. Let X be a moving point on the plane Pε and Vr ,Va and V f be the relative, absolute and sliding velocities of X under the one-parameter planar motions Pε/P ′ ε, respectively. Then, the relation between the velocities are given as below: Va = V f +Vr . Proof. The proof is obvious from the calculations of velocities given in the equations (5), (6), and (7). Now, we will investigate the points that does not move during the motions Pε/P ′ ε and the sliding velocity vector V f is equal to zero for every t ∈ [t0, t1]. These points are called the pole points or the instantaneous rotation pole centers. If we use the equation (8) for a pole point P = (p1, p2) ∈ Pε of the motions Pε/P ′ ε, we have � −u̇2 + ϕ̇(−u1 + x1) = 0 −u̇1 + εϕ̇(u2 − x2) = 0 (8) N. Gürses , S. Yüce / Eur. J. Pure Appl. Math, 7 (2014), 335-342 340 So, we obtain the pole point from the solution of the system (8) as follows: ( p1(t) = x1(t) = u1(t) + u̇2(t) ϕ̇(t) εp2(t) = εx2(t) = εu2(t)− u̇1(t) ϕ̇(t) (9) Therefore, the point P is instant in the plane Pε. Let us rearrange the sliding velocity vector (7) by using the equation (9): V f = {−ε(x2 − p2)c1 + (x1 − p1)c2}ϕ̇. (10) With reference the above equation, we can give the following corollaries: Corollary 1. During the one-parameter planar motions Pε/P ′ ε in affine CK-planes, the pole ray PX and the sliding velocity V f are perpendicular vectors in the sense of affine CK-geometry, i.e., 〈PX,V f 〉ε = 0. Then, the focus of the point X of the motions Pε/P ′ ε is an orbit that its normal pass through the rotation pole P. Corollary 2. Under the motions Pε/P ′ ε, the affine CK-norm of the sliding velocity V f is written below: V f ε = ‖PX‖ε � �ϕ̇ � � . 3.2. Accelerations and Acceleration Pole Point Notation In this section, we will define relative, absolute, sliding and Coriolis acceleration vectors denoted by br,ba,bf and bc, respectively, during the one-parameter planar motions Pε/P ′ ε in affine CK-planes. Let X be a moving point in Pε. By differentiating the relative velocity vector according to t, we obtain the relative acceleration br as below: br = V̇r = ẍ= ẍ1c1 + ẍ2c2. (11) The acceleration of the point X with respect to P′ε is known as the absolute acceleration and it is defined by ba = dVa d t = V̇a. If we differentiate the equation (6) with respect to t and use the equations (4), we obtain the absolute acceleration as below: ba =ε � ϕ̇ ṗ2 − (ϕ̇) 2(x1 − p1)− ϕ̈(x2 − p2) c1 + � −ϕ̇ ṗ1 − ε(ϕ̇) 2(x2 − p2) + ϕ̈(x1 − p1) c2 + ẍ1c1 + ẍ2c2 + 2ϕ̇(−ε ẋ2c1 + ẋ1c2). (12) In the equation (12), the expression b f = ε � ϕ̇ ṗ2 − (ϕ̇) 2(x1 − p1)− ϕ̈(x2 − p2) c1 + � −ϕ̇ ṗ1 − ε(ϕ̇) 2(x2 − p2) + ϕ̈(x1 − p1) c2 (13) is called the sliding acceleration and bc = 2ϕ̇(−ε ẋ2c1 + ẋ1c2). (14) is called the Coriolis acceleration of the one-parameter planar motion Pε/P ′ ε. Consequently, we can give the following theorem and corollary with using the equations (11), (12), (13), and (14). REFERENCES 341 Theorem 2. Let X be a moving point on the plane Pε and br , ba, b f and bc be the relative, absolute, sliding and Coriolis accelerations of X , respectively. Then, the relation between the accelerations under the one-parameter planar motions Pε/P ′ ε are given as below: ba = b f + bc + br . Corollary 3. During the motions Pε/P ′ ε, the Coriolis acceleration vector bc and the relative veloc- ity vector Vr are perpendicular to each other in the sense of affine CK-geometry,i.e. 〈Vr ,bc〉ε = 0. During the one-parameter planar motions Pε/P ′ ε, the acceleration pole is characterized by b f= 0. Then, if we take the acceleration pole point Q = (q1,q2) ∈ Pε of the motions Pε/P ′ ε, we get the following equation system: ¨ ε � ϕ̇ �2 � x1 − p1 � + εϕ̈ � x2 − p2 � = εϕ̇ ṗ2 ϕ̈ � x1 − p1 � − ε � ϕ̇ �2 � x2 − p2 � = ṗ1ϕ̇ (15) If ε � ϕ̇ �4 + � ϕ̈ �2 6= 0, we obtain the pole point from the above system as follows:            � ε2 � ϕ̇(t) �4 + ε � ϕ̈(t) �2 � q1(t) = � ε2 � ϕ̇(t) �4 + ε � ϕ̈(t) �2 � p1(t) +ϕ̇(t) � εϕ̈(t)ṗ1(t) + ε 2 � ϕ̇(t) �2 ṗ2(t) � � ε2 � ϕ̇(t) �4 + ε � ϕ̈(t) �2 � q2(t) = � ε2 � ϕ̇(t) �4 + ε � ϕ̈(t) �2 � p2(t) −εϕ̇(t) �� ϕ̇(t) �2 ṗ1(t)− ϕ̈(t)ṗ2(t) � So that the point Q is instant in the plane Pε. ACKNOWLEDGEMENTS The authors thank the readers of European Journal of Pure and Applied Mathematics, for making our journal successful. References [1] M. Akar, S. Yüce, and N. Kuruoğlu. One-Parameter Planar Motion in the Galilean Plane. International Electronic Journal of Geometry (IEJG), 6(1): 79–88, 2013. [2] M. Akbıyık. 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