We obtain a new version of Schlafli differential formula based on edge lengths for the volume of a tetrahedron in hyperbolic and spherical 3-spaces, by using the edge matrix of a hyperbolic(or spherical) tetrahedron and its submatrix.
We introduce special Smarandache curves based on Sabban frame onS12and we investigate geodesic curvatures of Smarandache curves on de Sitter and hyperbolic spaces. The existence of duality between Smarandache curves on de Sitter space and Smarandache curves on hyperbolic space is shown. Furthermore, we give examples of our main results.
Orthogonal projection along a geodesic to the chosenk-plane is introduced using edge and Gram matrix of ann-simplex in hyperbolic or sphericaln-space. The distance from a point tok-plane is obtained by the orthogonal projection. It is also given the perpendicular foot from a point tok-plane of hyperbolic and sphericaln-space.
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