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In classical differential geometry, Clairaut's relation, named after Alexis Claude de Clairaut, is a formula that characterizes the great circle paths on the unit sphere. The formula states that if γ is a parametrization of a great circle then. where ρ ( P) is the distance from a point P on the great circle to the z -axis, and ψ ( P) is the ...
Torsion of a curve. In the differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting out of the osculating plane. Taken together, the curvature and the torsion of a space curve are analogous to the curvature of a plane curve. For example, they are coefficients in the system of differential ...
In differential geometry, an equiareal map, sometimes called an authalic map, is a smooth map from one surface to another that preserves the areas of figures. Properties [ edit ] If M and N are two Riemannian (or pseudo-Riemannian ) surfaces, then an equiareal map f from M to N can be characterized by any of the following equivalent conditions:
e. Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity.
Tangent developable. In the mathematical study of the differential geometry of surfaces, a tangent developable is a particular kind of developable surface obtained from a curve in Euclidean space as the surface swept out by the tangent lines to the curve. Such a surface is also the envelope of the tangent planes to the curve.
Carl Friedrich Gauss in 1828. In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric . Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and ...
In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( / ˈriːtʃi / REE-chee, Italian: [ˈrittʃi] ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation ...
Spherical image. In differential geometry, the spherical image of a unit-speed curve is given by taking the curve's tangent vectors as points, all of which must lie on the unit sphere. The movement of the spherical image describes the changes in the original curve's direction [1] If is a unit-speed curve, that is , and is the unit tangent ...
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