The Mathematics of a Satellite Dish

Jay Kim / Independent Math / 20 May 1997
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It is obvious that the geometric shape of a satellite dish is too complicated to define its dimensions only on a simple two-dimensional coordinate plane. Though not very complicated at all, it is required to look into the matter using the three-dimensional coordinate system.

This system is constructed by passing a z-axis perpendicular to both the x- and y-axes at the origin. The three axes determine three coordinate planes called the xy-plane, the xz-plane, and the yz-plane, which in turn separate three-space into eight octants. In this system, a point P in space is determined by an ordered triple (x, y, z), where:

x = directed distance from yz-plane to P

y = directed distance from xz-plane to P

z = directed distance from xy-plane to P

Now the shape of a satellite dish could be called an elliptic paraboloid. By observing the shape through the perspective of each coordinate plane, one can see the two-dimensional curves that the elliptic paraboloid consists of. Parallel to the xy-plane is and ellipse, parallel to the xz-plane is a parabola, and parallel to the yz-plane is another parabola. The standard variable equations to these functions are:

Parabola: y = ax*2 + bx +c

Ellipse: x *2+ y *2 = 1

a *2 b*2

One might assume that the standard variable equation of an elliptic paraboloid is the integration of both the equations stated above. In fact…it is very similar to both equations. Yet, since a third axis is present, the variable z must be added. Therefore, the equation of an elliptic paraboloid is:

z = x*2 + y *2

a*2 b*2

However, the structure of a satellite dish does not consist of an ellipse but a circle. In the equation of an ellipse, the square roots of a and b constitute the length of the lines parallel to their corresponding x- and y-axes, starting from its center. In order to form a circular paraboloid, a and b would have to be equal.

Reflection of Light from Satellite Dishes

The parabolic shape of a satellite dish serves a very special purpose. Very carefully shaped parabolic mirrors will reflect light rays to focus at a single point. The curved mirror is concave, and the reflection surface of such a mirror could be compared to a small portion of a hollow sphere. The center of this sphere is called the center of curvature and a line joining the center of curvature to the vertex of the mirror is called the principle axis, as in the z-axis of the three-dimensional system. The focus of the mirror is where the rays of light converge and is located on the principal axis, midway between the vertex and the center of curvature. So rays of light that hit the mirror will reflect and hit the focus of the mirror.