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In Hahn's Classical Complex Analysis text, he states a conformal mapping satisfies the condition

(b) the angle between two arcs at $z_0$ is also preserved in sense as well as in size

  • pg. 85

He later states about reflections on the Riemann Sphere

These two reflections certainly preserve the size but reverse the sense of the angle between great circles that pass through the origin and the north pole. However, reversing the sense twice amounts to preserving the sense.

  • pg. 93

It seems to me that there is some definition of "sense" that he is using, but I can't seem to glean it from his writing. I didn't see a formal definition thus far, and I've never heard the word before. Even a cursory Google search for "sense of an angle" turns up nothing particularly useful.

My best guess for what he means is "direction" or "sign", but it's honestly just a shot in the dark at this point. Does anyone know what he means?

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Alex Jones
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2 Answers2

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It's simply the orientation (clockwise vs counterclockwise) of the angle. For example, the function $z\mapsto \overline{z}$ does NOT preserve the sense of angles, but it does preserve their size.

MPW
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  • Do you have any source for this, or know of any other sources that use this term? I've read quite a bit of mathematical literature and never seen or heard of this term, so I'm curious about it. – Alex Jones Jul 26 '17 at 17:06
  • It's very commonly used. I don't have a source, though--I doubt there is one. You didn't have any trouble with the word "size", did you? For that matter, why do you accept "orientation" but not "sense"? – MPW Jul 26 '17 at 17:09
  • I accept it, and as I stated in the question, it was my guess (though I used the words "sign" and "direction" as they are the common vernacular I hear/use). I don't have any problem with the word "sense", only that I've never seen it used before in any other literature.

    Moreover, the word "orientation" is commonly used in mathematics and is usually defined. There is even a Wikipedia page about it that shows up when I Google "orientation of an angle". This is not the case for "sense".

     – Alex Jones Jul 26 '17 at 17:12
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    @AlexanderJ93: Perhaps it would help to Google for clockwise counterclockwise sense? For me that produces copious real-world evidence that the word "sense" is indeed used with that meaning. – hmakholm left over Monica Jul 26 '17 at 17:27
  • Thanks, this is helpful. It's possible that it's use is more common in some countries than others. – Alex Jones Jul 26 '17 at 17:32
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It's indeed a very rare expression (also abroad: "sens d'un angle" in french; "senso di un angolo" in italian) meaning the sense in which the angle is measured, as you correctly guessed.

I just found these sources, hoping they're enough for you:

trying
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