ON THE CHINESE CHECKERS SPHERICAL INVERSIONS IN THREE DIMENSIONAL CHINESE CHECKERS SPACE

In this paper, we study an inversion with respect to a Chinese checkers sphere in the three dimensional Chinese Checkers space, and prove several properties of this inversion. We also study cross ratio, harmonic conjugates and the inverse images of lines, planes and Chinese Checkers spheres in three dimensional Chinese Checkers space.


Introduction
In the game of Chinese checkers, checkers are allowed to move in the vertical (north and south), horizontal (east and west), and diagonal (northeast, northwest, southeast and southwest) directions. In [7], Krause asked how to develop a distance function that measures and made a suggestion for the idea of Chinese Checkers geometry and in G. Chen, introduced it by de…ning the metric in the coordinate plane [22]. The inversion was introduced by Perga and then studied and applied by Steiner about 1820s [2]. During the following decades, many physicists and mathematicians independently rediscovered inversions, proving the properties that were most useful for their particular applications (for some references see [1], [4], [10]). Many kinds of generalizations of inversion transform have been presented in literature. The inversions with respect to the central conics in real Euclidean plane was introduced in [3]. Then the inversions with respect to ellipse was studied detailed in [13]. In three-dimensional space a generalization of the spherical inversion is given in [16]. Also, the inversions with respect to the taxicab distance, alpha-distance [17], [21], [19] or in general a p-distance [20]. The circle inversion have been generalized in three-dimensional space by using a sphere as the circle of inversion [16]. ON THE CHINESE CHECKERS SPHERICAL INVERSIONS   1499 H. Minkowski was one of the developers in "non-Euclidean"geometry and found taxicab geometry [9]. The taxicab geometry has been studied and improved by some mathematicians (for some references see [5], [6], [8], [14], [18]).
An inversion in a sphere is a transformation of the space that ‡ips the sphere inside-out. That is, points outside the sphere get mapped to points inside the sphere, and points inside the sphere get mapped outside the sphere. In the present article, we de…ne a notion of inversion valid in three dimensional Chinese Checkers space. In particular, we de…ne an inversion with respect to a Chinese Checkers sphere and prove several properties of this new transformation. Also we introduce inverse points, cross ratio, harmonic conjugates and the inverse images of lines, planes and Chinese Checkers spheres in three dimensional Chinese Checkers space.

Chinese Checkers Spherical Inversions
In this section, we introduce the inversion in a Chinese Checkers sphere. The Chinese Checkers space R 3 C is almost the same as the Euclidean space R 3 . The points and lines are the same, and the angles are measured the same way, but the distance function is di¤erent. In R 3 the Chinese Checkers metric is de…ned using the distance function where d L (A; B) = max fjx 1 x 2 j ; jy 1 y 2 j ; jz 1 z 2 jg and d S (A; B) = min fjx 1 x 2 j + jy 1 y 2 j ; jx 1 x 2 j + jz 1 z 2 j ; jy 1 y 2 j + jz 1 z 2 jg where A = (x 1 ; y 1 ; z 1 ); B = (x 2 ; y 2 ; z 2 ) in R 3 . The unit ball in R 3 C is the set of points (x; y; z) in space which satisfy the equation max fjxj ; jyj ; jzjg + p 2 1 min fjxj + jyj ; jxj + jzj ; jyj + jzjg = 1 We can de…ne the notion of inversion in R 3 C as an analogue of inversion in R 3 : De…nition 1. Let S be a Chinese Checkers sphere centered at a point O with radius r in R 3 C : The inversion in the Chinese Checkers sphere S or the Chinese Checkers spherical inversion respect to S is the function such that The point P 0 is said to be the Chinese Checkers spherical inverse of P in S , S is called the sphere of inversion and O is called the center of inversion.
The Chinese Checkers spherical inversions with respect to the sphere, like re- ‡ections, are involutions. The …xed points of I (O;r) are the points on the Chinese Checkers sphere S centered at O with radius r.
Some basic properties about spherical inversion are given in the following items. Note that it is possible to extend every property of the Chinese Checkers circle inversion to Chinese Checkers spherical inversion.
Theorem 2. Let S be an Chinese Checkers sphere with the center O in the Chinese Checkers spherical inversion I (O;r) . If the point P is in the exterior of S then the point P 0 , the inverse of P , is interior to S, and conversely.
Proof. Let the point P be in the exterior of S, then d C (O; P ) > r.
. Suppose that P = (x; y; z) and P 0 = (x 0 ; y 0 ; z 0 ) are inverse points with respect to the Chinese Checkers spherical inversion I (O;r) . Since the points O; P and P 0 are collinear and the rays ! OP and ! OP 0 are same direction, Replacing the value of k in (x 0 ; y 0 ; z 0 ) = (kx; ky; kz), the equations of x 0 ; y 0 and z 0 are obtained.
Proof. Since the translation preserve distances in the Chinese Checkers space [15], [12] by translating in R 3 C (0; 0; 0) to (a; b; c) one can easily get the value of x 0 ; y 0 and z 0 .
Theorem 5. Let P; Q and O be any three collinear di¤ erent points in R 3 C . If the Chinese Checkers spherical inversion I (O;r) transform P and Q into P 0 and Q 0 respectively, then Proof. Assume …rst that O; P; Q are collinear. From the De…nition 1 we conclude Since the ratios of the Euclidean and Chinese Checkers distances along a line are same, When O; P; Q are not collinear, the theorem is not valid in Chinese Checkers space, generally R 3 C . For example, for O = (0; 0; 0); P = (1; 0; 0); Q = (2; 2; 2) and r = 2 p 2, the inversion I (O;r) transform P and Q into P 0 = (8; 0; 0) and Theorem 6. Let P; Q and O be any three non-collinear di¤ erent points in R 3 C and I (O;r) be the inversion such that transform P and Q into P 0 and Q 0 respectively. If the direction of P Q line is any element of D sets i = 1; 2; 3 such that D 1 = fu i j i 2 1; 2; :::; 18g for u 1 = (1; 0; 0); u 2 = ( 1; 0; 0); u 3 = (0; 1; 0); u 4 = (0; 1; 0); u 5 = (0; 0; 1); Proof. Since translations preserve the Chinese Checkers distances, it is enough to consider the center O of the inversion sphere as the origin. Let P , Q 2 D 1 in R 3 C . If P = (p; 0; 0) and Q = (0; q; 0), the images of P and Q respect to I (O;r) are P 0 = ( r 2 p ; 0; 0) and Q 0 = (0; r 2 q ; 0): It follows that and then on D 1 If P , Q 2 D 2 in R 3 C . then the images of P = and Q 0 = r 2 q(2 p 2 1) ; and then on D 2 : then the images of P = p; p p 2 + 1 ; 0 and Q = q; q p 2 + 1 ; 0 respect to I (O;r) are P 0 = r 2 8p ;

Cross Ratio and Harmonic Conjugates
The Chinese Checkers directed distance from the point A to the point B along a line l in R 2 C is denoted by d C [AB]. If the ray with initial point A containing B has the positive direction of orientation, d C [AB] = d C (A; B) and if the ray has the opposite direction, d C [AB] = d C (A; B) [11]. The Chinese Checkers cross ratio is preserved by the inversion in the Chinese Checkers circle as in the taxicab plane in [17]. Now, we show the properties related to the Chinese Checkers cross ratio and harmonic conjugates in R 3 C . De…nition 7. Let A; B; C and D be four distinct points on an oriented line in R 3 C . We de…ne the their Chinese Checkers cross ratio (AB; CD) C in R 3 C by It is known that the cross ratio is positive if both C and D are between A and B or if neither C nor D is between A and B, whereas the cross ratio is negative if the pairs fA; Bg and fC; Dg separate each other. Also, the cross ratio is an invariant under inversion in a sphere whose center is not any of the four points A; B; C and D in the taxicab plane; [11]. Similarly, this property is valid in Chinese Checkers sphere. Hence it is su¢ ces to show that j(A 0 B 0 ; C 0 D 0 ) C j = j(AB; CD) C j. This follows Theorem 2 Proof. Suppose that P and P 0 are Chinese Checkers harmonic conjugates with respect to A and B in R 3 C . Then Since Conversely, if P and P 0 are Chinese Checkers inverse points with respect to the Chinese Checkers spherical inversion I (O;r) , the proof is similar.

Chinese Checkers Spherical Inversions of Lines, Planes and Chinese Checkers Spheres
It is well known that inversions with respect to circle transform lines and circles into lines and/ or circles in Euclidean plane and Hyperbolic plane.
The following features are well known for inversion in Euclidean plane: i) Lines passing through the inversion center are invariant. ii) Lines that do not pass through the center of inversion transform circles passing through the center of inversion.
iii) Circles passing through the center of inversion transform lines does not pass through the center of the inversion. iv) Circles not passing through the center of inversion transform circles does not pass through the center of the inversion. v) Circles with center of inversion transform circles with center of inversion.
In this section, we study the Chinese Checkers spherical inversion of lines, planes and Chinese Checkers spheres. The Chinese Checkers spherical inversion I (O;r) maps the lines, planes passing through O onto themselves.
The Chinese Checkers spherical inversion I (O;r) maps Chinese Checkers spheres with centered O onto Chinese Checkers spheres. But the Chinese Checkers spherical inversion of a sphere not passing through the centre of inversion is another Chinese Checkers sphere that does not contain the centre of inversion.