Average of angles (headings)

[yee388]

I have data sent to me describing angles as integers from -180 to 180. I need to average these angles together. For example, the average of 10 and -2 should be 4. Great. However, the average of 175 and -175 should be 180.

Has anyone ever worked with this type of issue?

(I've thought about converting -175 to 185, but then I have trouble with 5 and -5, which becomes 355).

Thoughts?

EDIT: I should probably note that I have a list of angles, not just 2. The analysis I need to do is 1) average of all angles and 2) difference from the average for each angle.

 

[GlennUK]

Explain the logic of this:

the average of 175 and -175 should be 180

 

[yee388]

Imagine 175 degrees being almost directly behind you, just a little off to the right.

-175 degress would be also almost directly behind you, just a little off to the left.

Therefore, the average of the two angles would be, directly behind you, 180 degrees.

 

[GlennUK]

That's the average of 2 angles, where you are asking for the angle half-way between the 2 angles in the sector that is between the other 2 angles.

So the average of 91 and -91 would be 180, and the average of 89 and -89 would be 0.

How can this logic ever be applied to a list of numbers?

 

[mortgageman]

Imagine 175 degrees being almost directly behind you, just a little off to the right.

-175 degress would be also almost directly behind you, just a little off to the left.

Therefore, the average of the two angles would be, directly behind you, 180 degrees.

I really don't agree. The refrence angle for -175 is 5. The average of 175 and 5 is 90.

 

[tusharm]

The problem working with angles is that the values -- and the result of applying functions like average (or sum or whatever) -- "circle around." There's ambiguity about how to interpret any angle as you've already pointed out. -175 is also 185 just as -5 is also 355.

One way to resolve your problem is to realize that you always have two possible answers, one 180 degrees away from the other. So, the average of 175 and -175 is 0 or 180. Similarly, the average of 5 and -5 is also 0 or 180. Which answer you pick will depend on which is closer to (one of) the original value. Since ABS(180-175) < ABS(0-175), pick 180. Similarly, since ABS(0-5) < ABS(180-5), you pick 0.

This still leaves unanswered which answer one picks when the original angles are precisely 180 degrees apart. What's the average of 0 and 180? Is it 90 or -90 degrees? Similarly, what's the average of 45 and 225? Is it 135? Or -45?

I don't know if the above helps you figure out how to work with multiple (i.e., >2) angles but I'll leave that to you.

I have data sent to me describing angles as integers from -180 to 180. I need to average these angles together. For example, the average of 10 and -2 should be 4. Great. However, the average of 175 and -175 should be 180.

Has anyone ever worked with this type of issue?

(I've thought about converting -175 to 185, but then I have trouble with 5 and -5, which becomes 355).

Thoughts?

EDIT: I should probably note that I have a list of angles, not just 2. The analysis I need to do is 1) average of all angles and 2) difference from the average for each angle.

 

[tusharm]

Imagine 175 degrees being almost directly behind you, just a little off to the right.

-175 degress would be also almost directly behind you, just a little off to the left.

Therefore, the average of the two angles would be, directly behind you, 180 degrees.

I really don't agree. The refrence angle for -175 is 5. The average of 175 and 5 is 90.

Not quite sure what you mean by "reference angle." One guide to measuring +ve and -ve angles is http://www4.nau.edu/ifwfd/ts_lessons/angle/angle_upload/angle/A_posneg.htm

 

[mortgageman]

Imagine 175 degrees being almost directly behind you, just a little off to the right.

-175 degress would be also almost directly behind you, just a little off to the left.

Therefore, the average of the two angles would be, directly behind you, 180 degrees.

I really don't agree. The refrence angle for -175 is 5. The average of 175 and 5 is 90.

Not quite sure what you mean by "reference angle." One guide to measuring +ve and -ve angles is http://www4.nau.edu/ifwfd/ts_lessons/angle/angle_upload/angle/A_posneg.htm

It's standard stuff from trig 101:

http://home.xnet.com/~fidler/triton/math/review/mat114/refang/refang1.htm

 

[mortgageman]

Imagine 175 degrees being almost directly behind you, just a little off to the right.

-175 degress would be also almost directly behind you, just a little off to the left.

Therefore, the average of the two angles would be, directly behind you, 180 degrees.

I really don't agree. The refrence angle for -175 is 5. The average of 175 and 5 is 90.

Not quite sure what you mean by "reference angle." One guide to measuring +ve and -ve angles is http://www4.nau.edu/ifwfd/ts_lessons/angle/angle_upload/angle/A_posneg.htm

It's standard stuff from trig 101:

http://home.xnet.com/~fidler/triton/math/review/mat114/refang/refang1.htm

 

[mikerickson]

May I suggest a terminology shift. It sounds as if the OP is looking for the "bisecting angle between two angles" rather than the "average of two angles."

Given two angles A and B, if sin(A-B)>0 then (A+B)/2 will be the bisecting angle, otherwise (A+B+360)/2.