Some Extrinsic Area Formulas Of The Triangle In Spherical Geometry And Hyperbolic Geometry

In the study of the geometry, people often focus on studying the intrinsic geometric property which remain unchanged even the model for studying the geometry changes, in other words,the proof of those proprerties must have nothing to do with the model which,generally, is achieved by means of axiomatization step by step. When the model is used in the study, for example, some problems in Euclidean geometry can be solved with the help of Descartes coordinate system and some new extrinsic formulas can be obtained by using algebraic operation. Similar situations may occur in spherical geometry and hyperbolic geometry.The main work of this paper is that some new extrinsic area formulas of the triangle in spherical geometry and hyperbolic geometry are obtained by using models.The main results of this paper can be divided into two parts1.In the unit sphere model of spherical geometry, we turn the problem about the area of a spherical triangle into the one of the angle between two vectors in a plane by using the comformal stereographic projection and get a extrinsic area formula of the spherical triangle(theorem3.1.1). In addition, we obtain another extrinsic area formula of the spherical triangle in virtue of the cross-ratio of the complex number after presenting the unit sphere with quaternions(theorem3.1.2), and we make this formula generalized for the general spherical quadrilateral(theorem3.1.3).2.In the hyperboloid model of hyperbolic geometry, we, similarly, turn the problem about the area of a hyperbolic triangle into the one of the angle between two vectors in a plane by using a comformal projection and get a extrinsic area formula of the hyperbolic triangle(theorem3.2.1). In addition, we obtain another extrinsic area formula of the hyperbolic triangle in virtue of the cross-ratio of the complex number after presenting the hyperboloid with quaternions(theorem3.2.2) which formula can be generalized for the general hyperbolic quadrilateral(theorem3.2.3).In the upper half-plane model of hyperbolic geometry, we, with a more straight use of the cross-ratio, get a different extrinsic area formula of the hyperbolic triangle(theorem3.2.4),and the formula can be be generalized for the general hyperbolic quadrilateral in the upper half-plane model successfully(theorem3.2.5).