# tangent space

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## tangent space

[′tan·jənt ‚spās]
(mathematics)
The vector space of all tangent vectors at a given point of a differentiable manifold.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
References in periodicals archive ?
In order to correlate the tangent space with the Grassmann manifold, we review a theorem about geodesics and horizontal lifts .
for any v [member of] TxS, and hence grad(f) = d[f.sup.#] is the projection of grad(F) = d[F.sup.#] on the tangent space [T.sub.x]S.
q [member of] G is an element of the Lie group, and its tangent space is [T.sub.q]G.
For the manifold [S.sub.n], the horizontal tangent space at x [member of] [S.sub.n] is [H.sub.x][S.sub.n] = [span.sub.R]{Re Z(x), Im Z(x)}.
where X*, Y* [member of] [chi](M*) form an orthogonal basis {X*([alpha]*(t*)), Y *([alpha]*(t*)) of a tangent space at each point [alpha]*(t*) of M*.
Let {[e.sub.i]}, i = 1, 2, ..., n be an orthonormal basis of the tangent space at any point.
Another important concept is the tangent space to a manifold in a certain point, which is basically a first-order (vector space) approximation of the manifold at this point.
Let [M.sup.n] be a complete hypersurface and {[e.sub.1],...,[e.sub.n]} an orthonormal basis of the tangent space [T.sub.x](M) at a point x [member of] M such that A[e.sub.i] = [[lambda].sub.i][e.sub.i], 1 [??] i [??] n.
For any vector field X of [??], we define a mapping J from the tangent space of [??] onto itself is given by JX = [J.sub.m]PX + [J.sub.n]QX, then we see that
The components of the metric tensor g ([x.sub.N]) = [[eta].sub.ik]d[X.sup.i] [cross product] d[X.sup.k] describing the locally flat tangent space [T.sub.x](M) of rigid frames at a point [x.sub.N] = [x.sub.N] ([x.sup.a]) are given by
The normal map image is usually blue because the normals are captured in tangent space. Later on, using a DOT3 texture operation we are able to render the low resolution model and make it look like the high quality one.
By finite type m we mean that the tangent space of M at the origin is spanned by commutators of length m of sections of [T.sup.1,0]M [symmetry] [T.sup.0,1]M and it is not spanned by commutators of length at most m - 1.

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