Consider the surface, S={x in R^n: h(x)=0}, where h is in C^1. A differentiable curve passing thru x* in S is {x(t): x(0)=x* and h(x(t))=0 for all t in (-e,e)}, for which the derivative, x'(t), exists, where e 0. The tangent plane at x* is the set of all initial derivatives: {x'(0)}. (This is a misnomer, except in the special case of one function and two variables at a non-stationary point.) An important fact that underlies the classical Lagrange multiplier theorem when the rank of grad_h(x*) is full row (x* is then called a regular point): the tangent plane is {d: grad_h(x*)d = 0}. Extending this to allow inequalities, the equivalent of the tangent plane for a regular point (x*) is the set of directions that satisfy first-order conditions to be feasible: {d: grad_h(x*)d=0 and grad_g_i(x*)d = 0 for all i: g_i(x*)=0}.

the plane that contains all the lines tangent to a specific point on a surface

a plane such that there is in the plane section a double point in addition to the nodes or cusps at the intersections with the singular lines onthe surface