A conic section is what you get when a second-degree equation in x and y is drawn in the plane: a circle, an ellipse, a parabola, a hyperbola. Step up one dimension — allow a z as well, and let every term be at most second degree — and the graph is no longer a curve but a surface floating in space. These surfaces are the quadric surfaces, and they are the exact 3-D analogues of the conics.

The general quadric is the most lavish second-degree equation three variables can carry,

but after a rotation and a shift of the axes (the same tidying-up that turns a messy conic into x^2/a^2 + y^2/b^2 = 1) every quadric lands in one of a short list of standard forms. Learn the handful below and you can name almost any surface at a glance. The magic word for reading them is traces.

The standard family

Here is the whole cast, each centred at the origin. The positive constants a, b, c are the scales along the three axes; the interesting thing to watch is how many terms are negative.

• Ellipsoid — \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1. A squashed ball; three positive terms. (All equal → a sphere.)
• Elliptic paraboloid — z = \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}. A bowl — a satellite dish. One variable is linear, the other two squared and added.
• Hyperbolic paraboloid — z = \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2}. A saddle — a Pringle. Same as the bowl but with the two squared terms subtracted.
• Hyperboloid of one sheet — \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} - \dfrac{z^2}{c^2} = 1. A single connected surface pinched at the waist — a cooling tower. Exactly one negative term.
• Hyperboloid of two sheets — -\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1. Two separate bowls facing away from each other — two negative terms.
• Elliptic cone — \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = \dfrac{z^2}{c^2}. The double cone whose slices are the conic sections; the borderline case between the two hyperboloids.
• Cylinder — one variable is missing, e.g. x^2 + y^2 = 1. The 2-D conic is simply extruded straight along the absent axis (here an infinite circular pipe along z).

Notice the pattern in the hyperboloids and the cone: as the number of minus signs climbs from zero (ellipsoid) to one (one sheet) to two (two sheets), the surface tears itself apart — first pinching, then splitting. The signs are the whole story, which is why the next thing we do is read a surface through its traces rather than trying to picture it whole.

Worked example 1 — read the bowl z = x^2 + y^2

We can't draw in three dimensions on this flat page — but we don't need to. We take the surface apart into three traces, one for each coordinate plane, and each is an ordinary 2-D conic you already know.

Horizontal traces, z = k. Substituting a constant height gives

For k > 0 this is a circle of radius \sqrt{k}; at k = 0 it shrinks to the single point at the origin (the bottom of the bowl); for k < 0 it is empty — the surface never dips below z = 0. So the horizontal slices are circles that grow as we climb.

Vertical traces. Set y = 0 (the x–z plane) and you get z = x^2, an upward parabola. Set x = 0 and you get z = y^2, the same parabola in the other vertical plane.

Assemble the verdict. Growing circles going up, parabolas going across: that is a bowl opening upward — an elliptic paraboloid (a circular one, since the two scales match). Step through its three traces below.

Worked example 2 — the cooling tower x^2 + y^2 - z^2 = 1

Now a surface with a minus sign. Same method — fix one variable at a time.

Horizontal traces, z = k. Move the z term across:

The right-hand side is always positive, so every horizontal slice is a real circle — of radius \sqrt{1 + k^2}. It is smallest (radius 1) at the waist z = 0 and grows without bound as you move up or down. There is no gap: the surface is one connected piece.

Vertical trace, y = 0.

a hyperbola opening left and right. Its two branches are the flaring walls of the tower; the closest approach of the branches is the waist. Circles that pinch to a minimum and then flare, cut by hyperbolas — that is the hyperboloid of one sheet, the shape of a power-station cooling tower and of many a wastepaper basket. Compare its traces with the bowl's below.

How the signs sort the family

For the equations of the form \pm x^2 \pm y^2 \pm z^2 = 1, you can read off the surface just by counting minus signs:

And for the two paraboloids (a lone linear z facing two squared terms), the sign between the squares is the whole difference: z = x^2 + y^2 adds and gives the bowl; z = x^2 - y^2 subtracts and gives the saddle. The saddle is worth savouring: its horizontal traces x^2 - y^2 = k are hyperbolas (opening one way for k > 0, the other for k < 0), while both vertical traces are parabolas — one opening up (z = x^2) and one opening down (z = -y^2). Up one way, down the other: that is exactly the mountain-pass feel of a Pringle.