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.

 

[mortgageman]

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.

I know what the bisector of an angle is. I don't have any clear idea what the "bisecting angle between two angles" is. Can you define it? (Without just restating that formula above)

 

[tusharm]

This tangential discussion is going further off target than might be appropriate. So, I will not post any more on the subject of "reference angle" after this post. Others can have the final say.

- - - - - -

No, you are extrapolating "standard stuff" that works because of the rules governing certain trig functions to outside of those trig functions. Unfortunately, it doesn't work that way.

-175 degrees does not have an *universal* "reference angle" of 5 degrees. It has a "reference angle" *only* in the context of certain trig functions. That does not mean one can substitute 5 for -175 when applying other mathematical operators / functions.

As far as "reference angle" being "standard stuff" goes, it's far from universal. I use trig -- far more advanced stuff than computing COS or SIN -- when appropriate and this is the first I can remember encountering the phrase. And, personal knowledge and limitations aside, *neither* wikipedia nor mathworld have any material on the subject. I suspect it's left over from the days of printed tables and slide rules when it was important to map any angle into an acute angle, a process that made further computations practical.

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]

You said:

-175 degrees does not have an *universal* "reference angle" of 5 degrees. It has a "reference angle" *only* in the context of certain trig functions. That does not mean one can substitute 5 for -175 when applying other mathematical operators / functions.

Me:
I agree. I was only trying to guess at the OP's meaning/logic.



You said:
As far as "reference angle" being "standard stuff" goes, it's far from universal. I use trig -- far more advanced stuff than computing COS or SIN -- when appropriate and this is the first I can remember encountering the phrase. And, personal knowledge and limitations aside, *neither* wikipedia nor mathworld have any material on the subject. I suspect it's left over from the days of printed tables and slide rules when it was important to map any angle into an acute angle, a process that made further computations practical.

Me:
I'm sure you do advanced trig stuff - but I can tell you that in any high school trig class - today, not years ago - , the first thing that is taught is reference angles.

 

[yee388]

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.

This ambiguity is the problem I'm facing, I just wasn't as eloquent. I will take all of this feedback back to the guy I'm helping and we'll think through the significance of such an 'average', even assuming it's possible.

Thanks for all the responses, for my part, consider the matter closed. If we reach another level of clarity on what we're trying to achieve, I'll be back with a new post.

 

[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.

I know what the bisector of an angle is. I don't have any clear idea what the "bisecting angle between two angles" is. Can you define it? (Without just restating that formula above)

An angle is two rays originating from the same point. If a third ray from that point,interior to the angle, forms congruent angles with each of two original rays, then it is said to bisect that angle.

 

[mortgageman]

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.

I know what the bisector of an angle is. I don't have any clear idea what the "bisecting angle between two angles" is. Can you define it? (Without just restating that formula above)

An angle is two rays originating from the same point. If a third ray from that point,interior to the angle, forms congruent angles with each of two original rays, then it is said to bisect that angle.

Thanks Mike - Now can you tell me what is the "bisecting angle between two angles"

 

[pgc01]

Hi guys

I was just going away and happened to browse this thread and I see lots of people having great fun with this angle stuff.

So I just have time to give a small contribution (and then run away).

Bisecting angle. Instead of trigonometry let's try pastry.

You have a nice circular pumpkin pie. The 2 angles define a slice of the pie.

You want to share the slice with me and, being nice guys, divide the slice in 2 equal parts.

To do that you place the point of the knife on the center of the pie and the edge of the knife on the slice, at such an angle that the slice will be cut into 2 exactly equal slices.

That angle is the bisecting angle.

So, if the slice was limited by the angles 80 and 120 degrees, the slice would be cut at 100 degrees.

Or, with the OP's example, if the slice was limited by the angles -175 and 175, the slice would be cut at 180 degrees.

I'm sure this highly scientific explanation cleared all the doubts !

 

[mortgageman]

Hi guys

I was just going away and happened to browse this thread and I see lots of people having great fun with this angle stuff.

So I just have time to give a small contribution (and then run away).

Bisecting angle. Instead of trigonometry let's try pastry.

You have a nice circular pumpkin pie. The 2 angles define a slice of the pie.

You want to share the slice with me and, being nice guys, divide the slice in 2 equal parts.

To do that you place the point of the knife on the center of the pie and the edge of the knife on the slice, at such an angle that the slice will be cut into 2 exactly equal slices.

That angle is the bisecting angle.

So, if the slice was limited by the angles 80 and 120 degrees, the slice would be cut at 100 degrees.

Or, with the OP's example, if the slice was limited by the angles -175 and 175, the slice would be cut at 180 degrees.

I'm sure this highly scientific explanation cleared all the doubts !

This sounds like semantics. What you are calling the "bisecting angle" I would call the angel bisector. I sorta like mine better because the angle bisector is not an angle - it is a line (okay okay - a ray) that creates two angles. But anyway - problem solved (ours not the OP's)

 

[pgc01]

Hi Gene

This sounds like semantics. What you are calling the "bisecting angle" I would call the angle bisector.

Well, that's something I wouldn't dream discussing with you.
You are a native English speaker, and I'm not.

So, angle bisector is good for me.

Now, as you said, the OP's problem is not yet solved. I can't try it now, but I'm sure that, among us, we'll figure out a nice solution. If no one else has the time I'll try to do it on Sunday.

Cheers.

 

[mortgageman]

Hi Gene

This sounds like semantics. What you are calling the "bisecting angle" I would call the angle bisector.

Well, that's something I wouldn't dream discussing with you.
You are a native English speaker, and I'm not.

Cheers.

Didn't realize that you were not a native English speaker. I wish I could speak any 2nd language as well as you seem to handle English. And BTW - in my case certaintly - native does not imply quality!