# 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 It might be outdated or ideologically biased.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).

References in periodicals archive ?
When a finger is close to a CG object, this system calculates a tangent plane slope angles on the surface which the finger is close to.
Similarly the tangent plane equation of the pipe at the point Q is expressed as
The tangent plane at any point in the equatorial plane is parallel to the x-axis so the d[p.sub.ik] at these points lie in the tangent plane.
Caption: Figure 5: The vectors [r.sub.[epsilon]] and [r.sub.[epsilon]] in the local tangent plane to the surface.
The properties of the function f show that this surface does not remain, round the point a, on the same side of the tangent plane, though each parametrized curve [alpha] [member of] [[GAMMA].sup.m.sub.F] (a) stays, around the point a, on the same side of the tangent plane.
constitute the contravariant basis at the point [theta](y), where [[delta].sup.i.sub.j] is the Kronecker symbol (note that [mathematical expression not reproducible] and the vector [[??].sup.[alpha]](y) is also in the tangent plane to S at [??](y)) (cf.
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.
a) Sliding type, when the displace is produced in the tangent plane at interface,
The reflection model known as the ray method, is based on the assumption that incident light is specularly reflected by the local tangent plane on the surface.
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.

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