Notes on Laying Out a Parabola

On occasion I've wondered how to lay out a parabola. I'm usually thinking about something solar, but radio energy and sound can also collected or projected with a parabola. And of course, there are other uses of parabolas.

Mathematical Method

x y
0 0.0000
3 0.1875
6 0.7500
9 1.6875
12 3.0000
15 4.6875
18 6.7500
21 9.1875
24 12.0000
27 15.1875
30 18.7500
33 22.6875
36 27.0000
39 31.6875
42 36.7500
45 42.1875
48 48.0000
The formula for a parabola is:

y = x² ÷ (4 × p)

where p is the distance from the bottom of the parabola to the focal point, and x and y are cartesian coordinates of points along the parabola.

For a parabola that is to be 96 inches across (that is, -48 inches to +48 inches relative to the focal point) and 48 inches deep, with a focal point 12 inches above the bottom of the parabola, the formula generates the numbers shown in the table on the right.

Sometimes it is useful to be able to locate the focal point after the fact. Rearranging the above formula

p = x² ÷ (4 × y)

where x is the width (from the focal point) of the parabola, y is the depth of the parabola, and f is the distance ahead of the bottom of the parabola of the focal point. For our above 96 inch wide and 48 inch deep parabola, f solves to 12 inches.

Geometric Method

There is a much easier method. You'll need a T-square and a piece of string.
The focal point is drawn on the plywood. In this case, the parabola will be 96 inches wide and 48 inches deep, with the focal (marked with a circle) point 12" in front of the bottom.
A T-square is plunked on the plywood aligned with the focal point. The string is run from the focal point down to the bottom of the parabola and up to the end of the T-square. A pencil is stuck in the bend of the string and the plywood is marked.
The T-square is slowly pulled away from the focal point, and the string draws the pencil up from the bottom of the plywood. The pencil MUST be kept tight against the T-square.
Keep going...
Almost there...
Ta-da! A parabola!

References

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© 2003 W. E. Johns