Tangent Plane


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tangent plane

[′tan·jənt ′plān]
(mathematics)
The tangent plane to a surface at a point is the plane having every line in it tangent to some curve on the surface at that point.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Tangent Plane

 

The tangent plane to a surface S at a point M is the plane that passes through the point M and that is characterized by the property that the distance from this plane to the variable point M’ on the surface S is infinitesimal in comparison with the distance MM’ as M ’ approaches M. If a surface S has the equation z = f(x, y), then the equation of the tangent plane at the point (x0, y0, z0), where z0 = f(x0, y0), has the form

z − z0 = A(x - x0) + B(y - y0)

if and only if the function f(x, y) has a total differential at thepoint (x0, y0). In this case, A and B are the values of the partialderivatives ∂f/∂x and ∂f/∂x at the point (x0y0) (seeDIFFER-ENTIAL CALCULUS).

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
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The tangent plane to w at the point [X.sub.0] = (x([u.sub.0], [v.sub.0]), y([u.sub.0], [v.sub.0]), z([u.sub.0], [v.sub.0]) on w is the plane that contains the tangent vectors [w.sub.u], [w.sub.v] and the point [X.sub.0], and therefore [w.sub.u] x [w.sub.v] is a normal vector of the tangent plane.
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In particular, this construction guarantees tangent plane continuity, the strong convex hull property, locality and affine invariance.
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