Frenet Formulas and Geodesics in Sol Geometry
نویسندگان
چکیده
In this paper we deal with one of the homogeneous 3geometries, the Sol geometry. The Frenet frame and the curvature and torsion of a curve has been determined, moreover, we have computed the parametric form of geodesics, their curvatures and torsions in Theorem 4.1.
منابع مشابه
The equiform differential geometry of curves in the pseudo - Galilean space ∗
In this paper the equiform differential geometry of curves in the pseudo-Galilean space G3 is introduced. Basic invariants and a moving trihedron are described. Frenet formulas are derived and the fundamental theorem of curves in equiform geometry of G3 is proved. The curves of constant curvatures are described.
متن کاملGeodesics on an ellipsoid in Minkowski space
We describe the geometry of geodesics on a Lorentz ellipsoid: give explicit formulas for the first integrals (pseudo-confocal coordinates), curvature, geodesically equivalent Riemannian metric, the invariant area-forms on the timeand space-like geodesics and invariant 1-form on the space of null geodesics. We prove a Poncelet-type theorem for null geodesics on the ellipsoid: if such a geodesic ...
متن کاملCrofton Formulas in Projective Finsler Spaces
We extend the classical Crofton formulas in Euclidean integral geometry to Finsler metrics on Rn whose geodesics are straight lines.
متن کاملNumerical Treatment of Geodesic Differential Equations on Two Dimensional Surfaces
This paper presents a brief instructions to nd geodesics equa-tions on two dimensional surfaces in R3. The resulting geodesic equations are solved numerically using Computer Program Matlab, the geodesics are dis-played through Figures.
متن کاملUsing Geometric Algebra for Visualizing Integral Curves
The Differential Geometry of curves is described by means of the Frenet-Serret formulas, which cast first, second and third order derivatives into curvature and torsion. While in usual vector calculus these quantities are usually considered to be scalar values, formulating the Frenet-Serret equations in the framework of Geometric Algebra exhibits that they are best described by a bivector for t...
متن کامل