Quadric surface

Quadrics are the three-dimensional analogue of the conic sections in the plane. After a rotation and a translation every quadric reduces to a standard form, and the non-degenerate ones are six: ellipsoid \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\), hyperboloid of one sheet, hyperboloid of two sheets, elliptic cone, elliptic paraboloid and hyperbolic paraboloid. If \(a=b=c\) the ellipsoid is a sphere.

The tool for recognizing them is traces, the curves obtained by cutting with planes \(x=k\), \(y=k\) or \(z=k\). On the paraboloid \(z=x^2+y^2\) the horizontal traces are circles and the vertical ones are parabolas; on the cone \(z^2=x^2+y^2\) the horizontal traces are circles and, at \(z=0\), degenerate to a point. The signs of the quadratic terms decide the shape: three positive give an ellipsoid, one negative gives hyperboloids, and when a variable appears only to the first power the result is a paraboloid.

Take care with missing variables: in \(\mathbb R^3\), the equation \(x^2+y^2=4\) does not mention \(z\), so it describes a cylinder, not a circle. If the center is not at the origin, complete the square, as in \(x^2-2x+y^2=3\), which is equivalent to \((x-1)^2+y^2=4\). In multivariable calculus they will appear as graphs of functions of two variables and as level surfaces of functions of three.