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The definition of the tangent cone can be extended to abstract algebraic varieties, and even to general Noetherian schemes. Let X be an algebraic variety, x a point of X, and (O X,x, m) be the local ring of X at x. Then the tangent cone to X at x is the spectrum of the associated graded ring of O X,x with respect to the m-adic filtration:
Tangent developable of a curve with zero torsion. The tangent developable is a developable surface; that is, it is a surface with zero Gaussian curvature.It is one of three fundamental types of developable surface; the other two are the generalized cones (the surface traced out by a one-dimensional family of lines through a fixed point), and the cylinders (surfaces traced out by a one ...
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R n. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in terms of ...
In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of as the intersection of two " infinitesimally adjacent" curves, meaning the limit of intersections of ...
Every continuously differentiable curve in M can be lifted to a curve in F in such a way that the tangent vector field of the lifted curve is the lift of the tangent vector field of the original curve. This statement means that any frame on a curve can be parallelly transported along the curve. This is precisely the idea of "moving frames".
Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, [4] and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
In classical differential geometry, development is the rolling one smooth surface over another in Euclidean space. For example, the tangent plane to a surface (such as the sphere or the cylinder ) at a point can be rolled around the surface to obtain the tangent plane at other points.
That is, since is parallelizable, the pullback of its tangent bundle to M is trivial; since this pullback is the direct sum of the (intrinsically defined) tangent bundle on M, TM, which has dimension m, and of the normal bundle ν of the immersion i, which has dimension n − m, for there to be a codimension k immersion of M, there must ...