Satellite Dishes and Saddles

A satellite dish is a paraboloid: every slice through its center is a parabola, so incoming signals bounce to a single focus. Change one sign in its equation and the dish turns into a saddle. Both are quadric surfaces.

Drag on the surface to move the point, or click a red marker to jump to it.

How it works

Quadric surfaces

In one sentence

The graph of $z = (x^2 + ey^2)/(4F)$ is a quadric surface: a circular paraboloid when $e = 1$ , an elliptic paraboloid when $e > 0$ , and a hyperbolic paraboloid when $e < 0$ .

General formula

{\htmlData{sym=2}{z}} = \frac{x^2 + {\htmlData{sym=1}{e}}\,y^2}{4{\htmlData{sym=0}{F}}}

Each squared term bends the surface along one axis. Both positive: a bowl. Opposite signs: up one way and down the other, a saddle.

What the symbols mean

$F$
Focal length
The height of the focus above the dish's center, where reflected signals meet.
$e$
Shape number
1 for a round dish, other positive values for an elliptic dish, negative for a saddle.
$z$
Depth
How high the surface rises at the point (x, y).

With $e = 1$ the surface is a circular paraboloid: horizontal slices are circles and vertical slices are parabolas with focus height $F$ .
With $0 < e \ne 1$ the slices become ellipses: an elliptic paraboloid, like a dish that focuses more in one direction.
With $e < 0$ the surface curves up along $x$ and down along $y$ : a hyperbolic paraboloid, the saddle shape.
Smaller $F$ makes a deeper, more sharply curved dish with its focus closer to the surface.

Learn the math

Quadric SurfacesCalc III · Lesson 8, Three-Dimensional Space