A tangent plane is a plane that passes through a given point and has the same slope as the surface at that point. The tangent plane can be thought of as the generalization of tangent lines to surfaces.

Formula

Explicit Formula

Given a surface defined explicitly as we can define the tangent plane at the point as follows:

Implicit Formula

Given a surface defined implicitly as , because the function is defined as a level set, meaning the value of never changes as we "walk" along it's surface. The tangent plane must be plane must therefore be the direction of zero change, using this fact and the definition of the directional derivative we can determine that the gradient of the function is perpendicular to the tangent plane. Hence the tangent plane can be easily defined as follows: