Quadric Surfaces

Main idea

Second-degree equations in $x$ , $y$ and $z$ only make a handful of shapes. To tell which one you have, slice it with flat planes and look at the curves you get: ellipses, parabolas and hyperbolas you already know.

\text{Ellipsoid: } \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \qquad \text{Cone: } \frac{z^2}{c^2} = \frac{x^2}{a^2} + \frac{y^2}{b^2}

\text{Elliptic paraboloid: } \frac zc = \frac{x^2}{a^2} + \frac{y^2}{b^2} \qquad \text{Hyperbolic paraboloid: } \frac zc = \frac{x^2}{a^2} - \frac{y^2}{b^2}

\text{Hyperboloid of one sheet: } \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \qquad \text{two sheets: } -\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1

Explanation

A quadric surface is simply a curved surface whose equation has squares in it but nothing higher: bowls, saddles, eggs, cones and cooling towers. The name comes from the Latin quadratus, "squared," the same root as "quadratic." They show up in satellite dishes, cooling towers and lenses. The formulas above look like a wall of fractions, but each is just $x^2$ , $y^2$ and $z^2$ (or one of them left unsquared) with some stretching numbers $a$ , $b$ , $c$ underneath (how wide the shape is in each direction). What tells them apart is mostly the plus and minus signs. Once you see that, the list gets short.

Planes came from first-degree equations. The next step up is second degree: equations with $x^2$ , $y^2$ , $z^2$ in them, like the sphere and cylinder from earlier in this unit. These are the quadric surfaces, and there are only a few kinds, which is good news.

Traces. A 3D surface is hard to picture from its equation, but a 2D curve isn't. So slice the surface with planes $z = k$ (and $x = k$ , $y = k$ ) and look at the curves you get. Setting one variable to a constant leaves an equation in two variables, which is a conic you met back in algebra. Stack the slices up in your head and the surface appears:

Surface	Horizontal slices	Vertical slices
Ellipsoid	ellipses	ellipses
Elliptic paraboloid	ellipses	parabolas
Cone	ellipses	lines through the vertex (at $k = 0$ ) or hyperbolas
Hyperbolic paraboloid (saddle)	hyperbolas	parabolas, opening up one way and down the other
Hyperboloid of one sheet	ellipses	hyperbolas

Quick identification. Once you've seen why the slices look the way they do, you can shortcut it by counting the squared terms and their signs. One unsquared variable means a paraboloid: it grows like a parabola in that direction. All three squared with one minus sign and $= 1$ is a one-sheet hyperboloid; with two minus signs, two sheets. And just like $x^2 + y^2 = 4$ earlier, if a variable is missing entirely, the surface is a cylinder along that axis.

See it

The surface z = x² + ky² as k slides from −1 to 1. At k = 1 it's a round bowl (an elliptic paraboloid). At k = 0 the y direction goes flat and you get a trough (a parabolic cylinder). Once k goes negative, the y direction curves down instead of up, and the bowl becomes a saddle. One sign flip changes the whole shape.

The saddle z = x² − y², a hyperbolic paraboloid. Walk along the x direction and you go uphill; walk along y and you go downhill. The origin is the low point of one parabola and the high point of the other.

Worked examples

Example 1

Identify $4x^2 + y^2 + 9z^2 = 36$ .

1
All three variables are squared with plus signs, so it's heading toward an ellipsoid. Divide by 36 to get the standard form: $\frac{x^2}{9} + \frac{y^2}{36} + \frac{z^2}{4} = 1$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Identify $z = x^2 + y^2$ .

Problem 2+5 XP

Identify $z^2 = x^2 + y^2$ .

Problem 3+5 XP

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

Problem 4+5 XP

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

Problem 5+5 XP

Identify $y^2 + z^2 = 4$ .

Problem 6+5 XP

A satellite dish is a paraboloid $z = \frac{x^2 + y^2}{4f}$ , which focuses incoming signals at the point $(0, 0, f)$ . A dish $2$ m across is $0.25$ m deep at its rim. How high above the center should the receiver sit?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Identify the surface $x^2 - 4y^2 + z^2 - 2x + 8y = 4$ .

Where it's used

Satellite dishes are paraboloids, cooling towers are one-sheet hyperboloids (strong and cheap to build from straight beams), and Pringles chips are saddles.

Satellite Dishes and SaddlesCalc III · Electrical Engineering

Equations of Planes Functions of Several Variables