arbitrary point of in place of the origin. The tangent cone serves as the extension of the notion of the tangent space to X at a regular point, where X most closely resembles a differentiable manifold, to all of X. (The tangent cone at a point of that is not contained in X is empty.) For example, the nodal curve is singular at the origin, because both partial derivatives of f(x, y) = y2 − x3 − x2 vanish at (0, 0). Thus the Zariski tangent space to C at the origin is the whole plane, and has higher dimension than the curve itself (two versus one). On the other hand, the tangent cone is the union of the tangent lines to the two branches of C at the origin, Its defining ideal is the principal ideal of k[x] generated by the initial term of f, namely y2 − x2 = 0. The definition of the tangent cone can be extended to abstract algebraic varieties, and even to general Noetherian schemes. Let X be an algebraic variety, x a point of X, and (OX,x, m) be the local ring of X at x. Then the tangent cone to X at x is the spectrum of the associated graded ring of OX,x with respect to the m-adic filtration: If we look at our previous example, then we can see that graded pieces contain the same information. So let then if we expand out the associated graded ring we can see that the polynomial defining our variety in See also Cone Monge cone Normal cone References Convex geometry Algebraic geometry Variational analysis