Tangent - Surfaces and Higher-dimensional Manifolds

Surfaces and Higher-dimensional Manifolds

The tangent plane to a surface at a given point p is defined in an analogous way to the tangent line in the case of curves. It is the best approximation of the surface by a plane at p, and can be obtained as the limiting position of the planes passing through 3 distinct points on the surface close to p as these points converge to p. More generally, there is a k-dimensional tangent space at each point of a k-dimensional manifold in the n-dimensional Euclidean space.