7/7/2023 0 Comments Differential geometry![]() It is found that the total, or Gaussian, curvature of a surface is a bending invariant, i.e., an intrinsic property of the surface itself, independent of the space in which the surface may be considered. The results of the theory of surfaces are expressed most easily in the notation of tensors. ![]() Tangent and normal vectors are also defined for a surface, but the relationships between them are more complex than for a space curve (e.g., a surface has a whole circle of unit vectors tangent to it at a given point). The curves on the surface that locally represent the shortest distances between points on the surface are called geodesics geodesics on a plane are straight lines. In the analysis of surfaces, points on a surface may be described not only with respect to the three-dimensional coordinates of the space in which the surface is considered but also with respect to an intrinsic coordinate system defined in terms of a system of curves on the surface itself. Of special interest are the curves called evolutes and involutes the evolute of a curve is another curve whose tangents are the normals to the original curve, and an involute of a curve is a curve whose evolute is the given curve. These three vectors are related by the three formulas of the French mathematician Jean Frédéric Frenet, which are fundamental to the study of space curves: dt/ ds = κn dn/ ds = -κt + τb db/ ds = -τn, where the constants κ and τ are the curvature and the torsion of the curve, respectively. The normal vector n is perpendicular to the curve at the point and indicates the direction of the rate of change of t, i.e., the tendency of r to bend in the plane containing both r and t, and the binormal vector b is perpendicular to both t and n and indicates the tendency of the curve to twist out of the plane of t and n. If a point r moves along a curve at arc length s from some fixed point, then t = dr/ ds is a unit tangent vector to the curve at r. the methods of differential geometry have been applied in other areas of geometry, e.g., in projective geometry. The approach in classical differential geometry involves the use of coordinate geometry (see analytic geometry Cartesian coordinates), although in the 20th cent. Differential geometry, branch of geometry in which the concepts of the calculus are applied to curves, surfaces, and other geometric entities. ![]()
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