classification of conics/quadrics

A conic in the plane (ellipse, parabola, hyperbola) and a quadric in space (ellipsoid, paraboloid, hyperboloid, cone, cylinder) are the zero sets of a degree-2 equation. The quadratic part is a quadratic form x^T B x; the principal axis theorem rotates away its cross terms, and the resulting signs of the squared coefficients — the signature — tell you which named shape you have.

In two variables, after diagonalizing the pure-quadratic part you read the signs of the two coefficients. Both the same sign gives an ellipse (signature (2,0) or (0,2)); opposite signs give a hyperbola (signature (1,1)); a missing square (a zero coefficient, so a degenerate quadratic part) paired with a surviving linear term gives a parabola. Signs of the eigenvalues, not their magnitudes, decide the type.

In three variables the same idea expands the menu: signature (3,0) is an ellipsoid, (2,1) a hyperboloid (of one or two sheets depending on the constant), a zero eigenvalue with the right linear term gives a paraboloid, and so on. The leftover linear and constant terms only translate and shift the surface; the quadratic form's signature fixes the fundamental shape.

Honesty about the full procedure: classifying a general conic or quadric requires handling the linear and constant terms too, which is cleanest with the (n+1)-by-(n+1) augmented matrix and its rank, distinguishing genuine curves from degenerate cases (a point, a pair of lines, the empty set). But the headline is that signature of the quadratic part is the primary invariant that names the geometry.