STRICTLY PARABOLIC

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If you'll study Fig. 9, you'll further note that the squares of each vertical line (both evenly numbered and fractional) form a very neat sequence, in which each square advances by exactly 4. (That Is: The square of vertical line 2 is 4, the square of vertical line 2.82842 is 8, the square of vertical line 3.4641 is 12, etc.)

Furthermore, if you're familiar with right triangles, you've probably already noticed that the hypotenuses beyond f also form a series, each one greater than the last by 1. (That Is: The hypotenuse-shown as a dotted line-of the triangle running from f along OX to vertical line 2.82842, up that vertical line and back down to f . . . is 3. The hypotenuse-again shown as a dotted line-of the triangle running from 1 along OX to vertical line 3.4641, up that vertical line and back down to f . . . Is 4. And so on.)

By computing and plotting out the squares for the vertical lines involved, then, you will have the precise distances from the base line to the curvature for a parabolic curve extending as many units out as you care to take it.

It's also very easy to double check any of these computations, since every right triangle within the curve will have a hypotenuse two units greater than its base. (That Is: Point A in Fig. 9 is a total of 7 units from f and is located at the base of a vertical line measuring 5.65685 units. Seven times seven equals 49 and 5.65685 squared is 32. Forty-nine plus 32 equals 81 and 9-which is two units greater than 7-is the root of 81.)

This order of numbers is a constant factor to any parabola. Or, to put it another way, should you want a parabola of ANY measurement, you have only to multiply or divide by the necessary number to obtain the new dimensions. (Example: I want a parabola with a focus of seven feet and I know that every number above was calculated with the focal length Of given as one (1) unit. Therefore, all I have to do to figure those same numbers for a focal length of seven feet is multiply by seven. And if I don't want feet, I can just as easily convert those numbers to meters, miles, or anything else I do want. The ratio will always be the same.)

OK. You can relax. The hard part is over. Now that you know HOW to calculate a parabolic curve of any size and/or width, I'm going to save you the trouble. The chart accompanying this article contains a list of measurements based on 10 verticals between each unit (this is, in short, a list of coordinates for perfectionists who want to lay out an absolutely accurate curve).

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