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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}.}
The second fundamental form of a parametric surface S in R 3 was introduced and studied by Gauss.First suppose that the surface is the graph of a twice continuously differentiable function, z = f(x,y), and that the plane z = 0 is tangent to the surface at the origin.
In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R 3. It permits the calculation of curvature and metric properties of a surface such as length and area in a manner consistent with the ambient space.
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 ...
An encapsulation of surface curvature can be found in the shape operator, S, which is a self-adjoint linear operator from the tangent plane to itself (specifically, the differential of the Gauss map). For a surface with tangent vectors X and normal N, the shape operator can be expressed compactly in index summation notation as
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 ...
The tangent plane is an affine concept, because its definition is independent of the choice of a metric. In other words, any affine transformation maps the tangent plane to the surface at a point to the tangent plane to the image of the surface at the image of the point.