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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 .
The tangent plane at a regular point is the affine plane in R 3 spanned by these vectors and passing through the point r(u, v) on the surface determined by the parameters. Any tangent vector can be uniquely decomposed into a linear combination of r u {\displaystyle \mathbf {r} _{u}} and r v . {\displaystyle \mathbf {r} _{v}.}
In mathematics, the tangent space of a manifold is a generalization of tangent lines to curves in two-dimensional space and tangent planes to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on ...
It is an irregular point that remains irregular, whichever parametrization is chosen (otherwise, there would exist a unique tangent plane). Such an irregular point, where the tangent plane is undefined, is said singular. There is another kind of singular points. There are the self-crossing points, that is the points where the surface crosses ...
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.
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 this situation, although the planes being rolled over the surface are tangent planes in a naive sense, the notion of a tangent space is really an infinitesimal notion, [e] whereas the planes, as affine subspaces of R 3, are infinite in extent. However these affine planes all have a marked point, the point of contact with the surface, and ...
At each point p of a differentiable surface in 3-dimensional Euclidean space one may choose a unit normal vector.A normal plane at p is one that contains the normal vector, and will therefore also contain a unique direction tangent to the surface and cut the surface in a plane curve, called normal section.