Pilot's Rule of 60

Date: 06/07/2002 at 17:31:53
From: Charlie
Subject: trig/piloting: the rule of 60

There is a famous piloting rule, called the "rule of 60," that 
applies when you know your position and you have an obstacle ahead. 
Point your craft at the obstacle and think this, "60 times the 
desired distance off over the distance ahead equals my course change 
in degrees if I want to avoid the obstacle."

Is there a formula describing "the opposite" of this process? You 
know your position, distance and degrees away from the obstacle, what 
will be the "closest point of approach"?  Algebraic reconfiguring of 
the "rule of 60" results in wild innacuracies and I'm just not good 
enough to figure it out.

Charlie

Date: 06/07/2002 at 23:29:50
From: Doctor Peterson
Subject: Re: trig/piloting: the rule of 60

Hi, Charlie.

I think the "rule of 60" is an estimate of the following problem:

      B
      +
      |  \ d
      |     \
      |        + C
      |       /
      |      /
     r|     /
      |    /
      |   /
      |A /
      | /
      |/
      +
      A

Here you are at A, r units away from an obstacle at B, and you want 
to turn A degrees to the side so you will be d units from B at 
closest approach, which is when C is a right angle.

The exact solution is trigonometric:

    sin(A) = d/r

so you want the angle

    A = sin^-1(d/r)

using the inverse sine.

Since, for SMALL ANGLES, the sine of an angle is approximately equal 
to (but slightly smaller than) the angle expressed in radians, we can 
say

    d/r =~ A * pi/180

and, solving for A,

    A =~ 180/pi d/r

Since 180/pi is about 57 (the number of degrees in one radian), this 
gives essentially your formula,

    A =~ 60 d/r

Using 60 makes A a slight overestimate, compensating for the fact 
that our sine approximation is a slight underestimate. In fact, the 
angle will reach exactly 60 d/r when d/r is exactly 1/2, and the 
angle is 30 degrees. Up to that angle, this is a pretty good 
approximation, never off by much more than 0.5 degree, and is off in 
the direction you want to be off for safety. Past 30 degrees, 
however, it starts to get pretty bad!

Now, you'd like to reverse this and find d given r and A. Again, the 
exact solution is

    d = r sin(A)

If we instead solve our approximation above for d, we get

    d =~ rA/60

which, as before, is an appropriate estimate, underestimating your 
closest approach (by no more than 5%) for angles under 30 degrees.

So the rule is: multiply your distance from the obstacle by your 
course (relative to the obstacle) in degrees, and divide by 60 to get 
your distance at closest approach. As long as the angle is less than 
60 degrees, this will work well.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 06/08/2002 at 14:17:24
From: Charlie
Subject: Thank you (trig/piloting: the rule of 60)

Thanks ever so! I am a 36 year-old dyslexic, always 
terrible at math and strong on concept. You explained it 
perfectly! I will memorize it as a word problem and test it 
on a flight simulator. Then I'll take it sailing! My best 
to you who output such good into the world!