Tangent Plane

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tangent plane

[′tan·jənt ′plān]
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 It might be outdated or ideologically biased.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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where [U.sup.[??]] is the projection of U on the tangent plane of M.
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.
Because 3D facial animation is composed of various different expression models, the tangent plane and normal vector of the same vertex point also vary with changes in expressions, as shown in Figure 6.