FUNDAMENTAL THEOREMS FOR THE HYPERBOLIC GEODESIC TRIANGLES

In this work, we state and prove the sine, cosine I, cosine II, sine-cosine and cotangent rules for spherical triangles on the hyperbolic unit sphere 2 0 H in the Lorentzian space 3 1 R . KeywordsLorentzian Space, Geodesic Triangles, Sine-Cosine Rules.


INTRODUCTION
In plane Lorentzian geometry it is studied points, timelike, spacelike, and lightlike lines, triangles, etc [5].On the hyperbolic sphere, there are points, but there are no straight lines, at least not in the usual sense.However the straight timelike lines in the Lorentzian plane are characterized by the fact that they are the shortest paths between points.The curves on the hyperbolic sphere with the same property are the great hyperbolic circles.Thus it is natural to use the great hyperbolic circles as (geodesic) replacements for timelike lines.
The formulas for the sine, cosine-I, cosine-II, sine-cosine and cotangent rules for Euclidean sphere 2 S are given in [2,6].In this study, we obtain the sine, cosine-I, cosine-II, sine-cosine and cotangent rules for spherical triangles on the hyperbolic unit sphere 2 0 H .

BASIC CONCEPTS
In this section, we give a brief summary of the theory of Lorentzian concepts.Let 3 L be vector space 3 R provide with Lorentzian inner product > <, given by A vector ) , , ( There are two components of the hyperbolic unit sphere 2 0 H .The components of 2 0 H through (0,0,1) and (0,0,-1) are called the future pointing hyperbolic unit sphere and the past pointing hyperbolic unit sphere and denoted by be two vectors in 3 L , then the Lorentzian cross product of a and b is given by ) , , ( ) , , ( ) , , (

FUNDAMENTAL THEOREMS FOR HYPERBOLIC GEODESIC TRIANGLES
In this section we prove the sine, cosine-I, cosine-II, sine-cosine, cotangent rules for hyperbolic geodesic triangles.
Fundamental relations of hyperbolic spherical trigonometry can be given on a trihedron.With the aid of this trihedron, both angles and sides of the spherical triangle can be represented as the spacelike angles between the hyperbolic angles corresponding to the sides of hyperbolic geodesic triangles.The radius of the sphere is not important while getting the fundamental relations related with the hyperbolic spherical triangles.That is, these relations are independent from the radius.Therefore, we consider the unit sphere in our work.
Similarly, if we take the spacelike planes passing through the point B and perpendicular to the lines OC and OA, respectively, then we get From the equations ( 4) and ( 5), we obtain The common ratio m is called modulo of the hyperbolic spherical triangle.This value changes for different hyperbolic spherical triangles.Proof: For brevity, let X, Y and Z be a cosh , b cosh and c cosh , respectively.Then the cosine rule I yields On the other hand, since 1 sin cos We note that D is positive and symmetric in X, Y and Z. Then we obtain , If we write the formulas (20)-( 24) in the right side of the formula (19), then the equality is satisfied: .
By the same way, we can give the similar formulas for b cosh and a cosh as follows: .We note that, by changing the elements of the triangle in cyclical order, we get ■ Formulas in (34) and (35) are known as hyperbolic cotangent rules.In each of these formulas, there are four elements of spherical triangle.Furthermore, these four elements are not by chance, they follow each other in order.This property allows us to write the formulas in (34) and (35) in general.That is, by starting any sides of the hyperbolic spherical triangle, these four elements, which followed by each other, can be numbered in any direction.Therefore, hyperbolic cotangent rule can be generalized as follows: IV  This formula is equal to the last formula of (34).In a similar way, starting with the same side, but anticlockwise direction gives a new formula.For example, as in figure 2 .This formula is equal to the second formula of (35).Since the triangle has three sides, and the numeration can be made in two different directions for each side, then the six formulas of the hyperbolic cotangent rule are generalized with the formula (36).

Lemma 2 . 4 (
The Hyperbolic Sine-Cosine Rule) Let ABC be a spherical triangle on the hyperbolic unit sphere 2 0 H . Then the hyperbolic sine-cosine rule is given by A It follows from the right triangle OFP that c OP PF sinh = .(29)If we replace in the equation (29) OP by its value (10), we obtain c On the other hand, from the right triangles CNQ and OCQ in Figure2.1, we get B the corresponding values of PF, NQ and PG into (28), we deduce c way, we can find two more hyperbolic sine-cosine formulas as follows: hyperbolic cosine rule I, we can deduce different formulas of hyperbolic sine-cosine rule.From (18) it follows that We note that the hyperbolic sine-cosine rule has five elements whereas the others have four elements.Lemma 2.5 (The Hyperbolic Cotangent Rule).Let ABC be a spherical triangle on the hyperbolic unit sphere 2 0 H . Then the hyperbolic cotangent rule is given by B that, if the elements of the triangle are changed by cyclically, we get B For example, in Figure2.4,starting with the side a , the elements of hyperbolic spherical triangle are numbered in clockwise direction: the numeration by the letters corresponding to the elements of the hyperbolic triangle, we get A

Figure 2
Figure 2.4: .4, starting with the side a and replacing the angles A, B and C by the numbers I, II and III gives the hyperbolic cotangent rule as follows: