Math Core

Planes are the surfaces given by first-degree equations. The next step up is second-degree equations in $x$ , $y$ and $z$ , which produce the quadric surfaces: ellipsoids, paraboloids, cones and hyperboloids. These are the three-dimensional cousins of the conic sections, and they will be your standard examples for graphs of functions of two variables, for tangent planes, and as boundaries of solids in triple integrals. The key skill is to picture a surface from its equation, and the tool for that is slicing.

Traces

A trace (or cross-section) of a surface is its intersection with a plane. The most useful slices are the planes parallel to the coordinate planes: $x = k$ , $y = k$ and $z = k$ . Setting one variable equal to a constant turns the equation into a curve in two variables, usually a conic you already know how to draw. Stacking several traces reveals the shape of the whole surface.

For example, slice $z = x^2 + y^2$ with horizontal planes $z = k$ . For $k > 0$ the trace is the circle $x^2 + y^2 = k$ of radius $\sqrt{k}$ ; for $k = 0$ it is a single point; for $k < 0$ it is empty. Vertical slices $x = 0$ and $y = 0$ give the parabolas $z = y^2$ and $z = x^2$ . Together these describe a bowl opening upward, a circular paraboloid.

The paraboloid z = x² + y². Horizontal traces are circles; the traces in the planes x = 0 and y = 0 are parabolas.

Cylinders

If one variable is missing from an equation, that variable is free, and the surface is made of lines parallel to that variable's axis.

Definition

Cylinder

A cylinder is a surface consisting of all lines (called rulings) that are parallel to a given line and pass through a given plane curve. An equation in only two of the three variables is a cylinder whose rulings are parallel to the axis of the missing variable.

So $x^2 + y^2 = 4$ is a circular cylinder of radius 2 around the $z$ -axis, $z = y^2$ is a parabolic cylinder (a trough) whose rulings run parallel to the $x$ -axis, and $y = \sin x$ is a corrugated sheet parallel to the $z$ -axis. "Cylinder" here doesn't require circles: any curve extended straight along the missing axis counts.

The six quadric surfaces

A quadric surface is the graph of a second-degree equation in $x$ , $y$ , $z$ . By translating and rotating axes, every nondegenerate quadric that isn't a cylinder can be put into one of the standard forms below (with $a, b, c > 0$ ). Each is listed with the axis playing the special role being the $z$ -axis; swapping variables moves that axis.

surface	standard equation	horizontal traces ( $z = k$ )	vertical traces
Ellipsoid	$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1$	ellipses	ellipses
Elliptic paraboloid	$\dfrac{z}{c} = \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}$	ellipses	parabolas
Hyperbolic paraboloid	$\dfrac{z}{c} = \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2}$	hyperbolas	parabolas
Cone	$\dfrac{z^2}{c^2} = \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}$	ellipses	hyperbolas (lines if through the axis)
Hyperboloid of one sheet	$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} - \dfrac{z^2}{c^2} = 1$	ellipses	hyperbolas
Hyperboloid of two sheets	$-\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1$	ellipses for $\lvert k \rvert > c$ , nothing for $\lvert k \rvert < c$	hyperbolas

Some quick descriptions to attach to each picture:

The ellipsoid is a stretched sphere with intercepts $\pm a$ , $\pm b$ , $\pm c$ on the three axes.
The elliptic paraboloid is a bowl; the variable appearing to the first power is its axis.
The hyperbolic paraboloid is a saddle (a Pringle): it curves up along one direction and down along the perpendicular one.
The cone is two nappes meeting at the origin.
The hyperboloid of one sheet is a connected "cooling tower." Count minus signs when the equation equals $+1$ : one minus sign means one sheet.
The hyperboloid of two sheets has two minus signs and splits into two separate bowls facing away from each other; the variable with the plus sign is its axis.

How to identify a quadric

Move everything to one side and complete the square in any variable that appears both squared and to the first power.
If some variable appears only to the first power, the surface is a paraboloid (elliptic if the squared terms have the same sign, hyperbolic if opposite).
Otherwise, rewrite so the constant is $1$ (or $0$ ) and count signs: all positive is an ellipsoid; one negative is a hyperboloid of one sheet; two negatives is a hyperboloid of two sheets; constant $0$ is a cone.
Confirm with a trace or two.

Worked example: Traces of an ellipsoid

Describe the surface $\dfrac{x^2}{4} + \dfrac{y^2}{9} + z^2 = 1$ and its trace in the plane $z = \dfrac12$ .

Solution. All three squared terms are positive and the constant is $1$ , so this is an ellipsoid with intercepts $\pm 2$ on the $x$ -axis, $\pm 3$ on the $y$ -axis, and $\pm 1$ on the $z$ -axis. Setting $z = \dfrac12$ gives $\dfrac{x^2}{4} + \dfrac{y^2}{9} = \dfrac34$ , or $\dfrac{x^2}{3} + \dfrac{y^2}{27/4} = 1$ : an ellipse with semi-axes $\sqrt3$ and $\dfrac{3\sqrt3}{2}$ . Horizontal traces shrink as $|z|$ grows and disappear for $|z| > 1$ .

Worked example: Counting signs

Identify the surface $4x^2 - y^2 + 2z^2 + 4 = 0$ .

Solution. Move the constant and divide by $-4$ :

-x^2 + \frac{y^2}{4} - \frac{z^2}{2} = 1.

There are two negative terms, so this is a hyperboloid of two sheets. The positive term is $y^2$ , so its axis is the $y$ -axis. Check with a trace: setting $y = k$ gives $x^2 + \dfrac{z^2}{2} = \dfrac{k^2}{4} - 1$ , which is empty when $|k| < 2$ . The two sheets start at $(0, \pm 2, 0)$ .

Worked example: Completing the square

Identify $x^2 + 2z^2 - 6x - y + 10 = 0$ and find its vertex.

Solution. The variable $y$ appears only to the first power, so this is a paraboloid. Complete the square in $x$ :

(x^2 - 6x + 9) + 2z^2 - y + 10 - 9 = 0 \quad\Longrightarrow\quad y - 1 = (x - 3)^2 + 2z^2.

The squared terms have the same sign, so this is an elliptic paraboloid with vertex $(3, 1, 0)$ , opening in the positive $y$ -direction along the line $x = 3$ , $z = 0$ .

Common mistake

Don't decide "one sheet or two" before the constant is positive. The equation $x^2 + y^2 - z^2 = -1$ has one minus sign as written, but multiplying by $-1$ gives $-x^2 - y^2 + z^2 = 1$ , which has two: it is a hyperboloid of two sheets. Always normalize to $= 1$ first.

Practice

Practice 1

What surface does $x^2 + z^2 = 4$ describe in $\mathbb{R}^3$ ?

Practice 2

Identify the surface $z^2 = x^2 + \dfrac{y^2}{4}$ .

Practice 3

Identify the surface $x^2 - y^2 + z^2 = 1$ .

Practice 4

Identify the surface $y = z^2 - x^2$ .

Practice 5

The trace of the ellipsoid $\dfrac{x^2}{16} + \dfrac{y^2}{9} + \dfrac{z^2}{4} = 1$ in the plane $z = 1$ is an ellipse. Find the length of its major axis.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

The surface $y^2 + z^2 - 4y + 2z - x + 3 = 0$ is a circular paraboloid. Find its vertex.

Enter a point like (2, -3)

Practice 7

The trace of the hyperboloid $-x^2 - y^2 + z^2 = 1$ in the plane $z = 3$ is a circle. Find its radius.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

Identify the surface $x^2 - 4y^2 - z^2 - 4 = 0$ .