Averages/Mean angle

When calculating the average or mean of an angle one has to take into account how angles wrap around so that any angle in degrees plus any integer multiple of 360 degrees is a measure of the same angle.

If one wanted an average direction of the wind over two readings where the first reading was of 350 degrees and the second was of 10 degrees then the average of the numbers is 180 degrees, whereas if you can note that 350 degrees is equivalent to -10 degrees and so you have two readings at 10 degrees either side of zero degrees leading to a more fitting mean angle of zero degrees.

To calculate the mean angle of several angles:

Assume all angles are on the unit circle and convert them to complex numbers expressed in real and imaginary form.
Compute the mean of the complex numbers.
Convert the complex mean to polar coordinates whereupon the phase of the complex mean is the required angular mean.

(Note that, since the mean is the sum divided by the number of numbers, and division by a positive real number does not affect the angle, you can also simply compute the sum for step 2.)

You can alternatively use this formula:

Given the angles

\alpha _{1},\dots ,\alpha _{n}

the mean is computed by

{\bar {\alpha }}=\operatorname {atan2} \left({\frac {1}{n}}\cdot \sum _{j=1}^{n}\sin \alpha _{j},{\frac {1}{n}}\cdot \sum _{j=1}^{n}\cos \alpha _{j}\right)

Task

write a function/method/subroutine/... that given a list of angles in degrees returns their mean angle.
(You should use a built-in function if you have one that does this for degrees or radians).
Use the function to compute the means of these lists of angles (in degrees):
- [350, 10]
- [90, 180, 270, 360]
- [10, 20, 30]
Show your output here.

11l

Translation of: C#

<lang 11l>F mean_angle(angles)

  V x = sum(angles.map(a -> cos(radians(a)))) / angles.len
  V y = sum(angles.map(a -> sin(radians(a)))) / angles.len
  R degrees(atan2(y, x))

print(mean_angle([350, 10])) print(mean_angle([90, 180, 270, 360])) print(mean_angle([10, 20, 30]))</lang>

Output:

-1.61481e-15
-90
20

Ada

An implementation based on the formula using the "Arctan" (atan2) function, thus avoiding complex numbers: <lang Ada>with Ada.Text_IO, Ada.Numerics.Generic_Elementary_Functions;

procedure Mean_Angles is

  type X_Real is digits 4;      -- or more digits for improved precision
  subtype Real is X_Real range 0.0 .. 360.0; -- the range of interest
  type Angles is array(Positive range <>) of Real;

  procedure Put(R: Real) is
     package IO is new Ada.Text_IO.Float_IO(Real);
  begin
     IO.Put(R, Fore => 3, Aft => 2, Exp => 0);
  end Put;

  function Mean_Angle(A: Angles) return Real is
     Sin_Sum, Cos_Sum: X_Real := 0.0; -- X_Real since sums might exceed 360.0
     package Math is new Ada.Numerics.Generic_Elementary_Functions(Real);
     use Math;
  begin
     for I in A'Range loop
       Sin_Sum := Sin_Sum + Sin(A(I), Cycle => 360.0);
       Cos_Sum := Cos_Sum + Cos(A(I), Cycle => 360.0);
     end loop;
     return Arctan(Sin_Sum / X_Real(A'Length), Cos_Sum / X_Real(A'Length),
                   Cycle => 360.0);
       -- may raise Ada.Numerics.Argument_Error if inputs are
       -- numerically instable, e.g., when Cos_Sum is 0.0
  end Mean_Angle;

begin

  Put(Mean_Angle((10.0, 20.0, 30.0)));     Ada.Text_IO.New_Line;    -- 20.00
  Put(Mean_Angle((10.0, 350.0)));          Ada.Text_IO.New_Line;    --  0.00
  Put(Mean_Angle((90.0, 180.0, 270.0, 360.0))); -- Ada.Numerics.Argument_Error!

end Mean_Angles;</lang>

Output:

 20.00
  0.00

raised ADA.NUMERICS.ARGUMENT_ERROR : a-ngelfu.adb:427 instantiated at mean_angles.adb:17

Aime

<lang aime>real mean(list l) {

   integer i;
   real x, y;

   x = y = 0;

   i = 0;
   while (i < l_length(l)) {
       x += Gcos(l[i]);
       y += Gsin(l[i]);
       i += 1;
   }

   return Gatan2(y / l_length(l), x / l_length(l));

}

integer main(void) {

   o_form("mean of 1st set: /d6/\n", mean(l_effect(350, 10)));
   o_form("mean of 2nd set: /d6/\n", mean(l_effect(90, 180, 270, 360)));
   o_form("mean of 3rd set: /d6/\n", mean(l_effect(10, 20, 30)));

   return 0;

}</lang>

Output:

mean of 1st set: -.000000
mean of 2nd set: -90
mean of 3rd set: 19.999999

ALGOL 68

Works with: ALGOL 68 version Revision 1

Works with: ALGOL 68G version Any - tested with release algol68g-2.8.3.

Translation of: C – Note: This specimen retains the original C coding style

File: Averages_Mean_angle.a68<lang algol68>#!/usr/bin/a68g --script #

-*- coding: utf-8 -*- #

PROC mean angle = ([]#LONG# REAL angles)#LONG# REAL: (

 INT size = UPB angles - LWB angles + 1;
 #LONG# REAL y part := 0, x part := 0;
 FOR i FROM LWB angles TO UPB angles DO
     x part +:= #long# cos (angles[i] * #long# pi / 180);
     y part +:= #long# sin (angles[i] * #long# pi / 180)
 OD;

 #long# arc tan2 (y part / size, x part / size) * 180 / #long# pi

);

main: (

 []#LONG# REAL angle set 1 = ( 350, 10 );
 []#LONG# REAL angle set 2 = ( 90, 180, 270, 360);
 []#LONG# REAL angle set 3 = ( 10, 20, 30);

 FORMAT summary fmt=$"Mean angle for "g" set :"-zd.ddddd" degrees"l$;
 printf ((summary fmt,"1st", mean angle (angle set 1)));
 printf ((summary fmt,"2nd", mean angle (angle set 2)));
 printf ((summary fmt,"3rd", mean angle (angle set 3)))

)</lang>

Output:

Mean angle for 1st set : -0.00000 degrees
Mean angle for 2nd set :-90.00000 degrees
Mean angle for 3rd set : 20.00000 degrees

AutoHotkey

Works with: AutoHotkey 1.1

<lang AutoHotkey>Angles := [[350, 10], [90, 180, 270, 360], [10, 20, 30]] MsgBox, % MeanAngle(Angles[1]) "`n" . MeanAngle(Angles[2]) "`n" . MeanAngle(Angles[3])

MeanAngle(a, x=0, y=0) { static c := ATan(1) / 45 for k, v in a { x += Cos(v * c) / a.MaxIndex() y += Sin(v * c) / a.MaxIndex() } return atan2(x, y) / c }

atan2(x, y) {

  return dllcall("msvcrt\atan2", "Double",y, "Double",x, "CDECL Double")

}</lang> Output:

-0.000000
-90.000000
20.000000

AWK

<lang AWK>#!/usr/bin/awk -f {

   PI = atan2(0,-1);
   x=0.0; y=0.0;
   for (i=1; i<=NF; i++) {

p = $i*PI/180.0; x += sin(p); y += cos(p);

   }
   p = atan2(x,y)*180.0/PI;	
   if (p<0) p += 360;
   print p;

}</lang>

 echo 350 10 | ./mean_angle.awk 
360
 echo 10 20 30 | ./mean_angle.awk 
20
 echo 90 180 270 360  | ./mean_angle.awk 
270

BBC BASIC

Works with: BBC BASIC for Windows

<lang bbcbasic> *FLOAT 64

     DIM angles(3)
     angles() = 350,10
     PRINT FNmeanangle(angles(), 2)
     angles() = 90,180,270,360
     PRINT FNmeanangle(angles(), 4)
     angles() = 10,20,30
     PRINT FNmeanangle(angles(), 3)
     END
     
     DEF FNmeanangle(angles(), N%)
     LOCAL I%, sumsin, sumcos
     FOR I% = 0 TO N%-1
       sumsin += SINRADangles(I%)
       sumcos += COSRADangles(I%)
     NEXT
     = DEGFNatan2(sumsin, sumcos)
     
     DEF FNatan2(y,x) : ON ERROR LOCAL = SGN(y)*PI/2
     IF x>0 THEN = ATN(y/x) ELSE IF y>0 THEN = ATN(y/x)+PI ELSE = ATN(y/x)-PI</lang>

Output:

-1.61480993E-15
       -90
        20

C

<lang c>#include<math.h>

include<stdio.h>

double meanAngle (double *angles, int size) {

 double y_part = 0, x_part = 0;
 int i;

 for (i = 0; i < size; i++)
   {
     x_part += cos (angles[i] * M_PI / 180);
     y_part += sin (angles[i] * M_PI / 180);
   }

 return atan2 (y_part / size, x_part / size) * 180 / M_PI;

}

int main () {

 double angleSet1[] = { 350, 10 };
 double angleSet2[] = { 90, 180, 270, 360};
 double angleSet3[] = { 10, 20, 30};

 printf ("\nMean Angle for 1st set : %lf degrees", meanAngle (angleSet1, 2));
 printf ("\nMean Angle for 2nd set : %lf degrees", meanAngle (angleSet2, 4));
 printf ("\nMean Angle for 3rd set : %lf degrees\n", meanAngle (angleSet3, 3));
 return 0;

}</lang>

Output:

Mean Angle for 1st set : -0.000000 degrees
Mean Angle for 2nd set : -90.000000 degrees
Mean Angle for 3rd set : 20.000000 degrees

C#

<lang csharp>using System; using System.Linq; using static System.Math; class Program {

   static double MeanAngle(double[] angles)
   {
       var x = angles.Sum(a => Cos(a * PI / 180)) / angles.Length;
       var y = angles.Sum(a => Sin(a * PI / 180)) / angles.Length;
       return Atan2(y, x) * 180 / PI;
   }
   static void Main()
   {
       Action<double[]> printMean = x => Console.WriteLine("{0:0.###}", MeanAngle(x));
       printMean(new double[] { 350, 10 });
       printMean(new double[] { 90, 180, 270, 360 });
       printMean(new double[] { 10, 20, 30 });
   }

}</lang>

Output:

0
-90
20

C++

Translation of: C#

<lang cpp>#include <iomanip>

include <iostream>
include <vector>

define _USE_MATH_DEFINES
include <math.h>

template<typename C> double meanAngle(const C& c) {

   auto it = std::cbegin(c);
   auto end = std::cend(c);

   double x = 0.0;
   double y = 0.0;
   double len = 0.0;
   while (it != end) {
       x += cos(*it * M_PI / 180);
       y += sin(*it * M_PI / 180);
       len++;

       it = std::next(it);
   }

   return atan2(y, x) * 180 / M_PI;

}

void printMean(std::initializer_list<double> init) {

   std::cout << std::fixed << std::setprecision(3) << meanAngle(init) << '\n';

}

int main() {

   printMean({ 350, 10 });
   printMean({ 90, 180, 270, 360 });
   printMean({ 10, 20, 30 });

   return 0;

}</lang>

Output:

-0.000
-90.000
20.000

Clojure

<lang clojure>(defn mean-fn

 [k coll]
 (let [n (count coll)
       trig (get {:sin #(Math/sin %) :cos #(Math/cos %)} k)]
   (* (/ 1 n) (reduce + (map trig coll)))))

(defn mean-angle

 [degrees]
 (let [radians (map #(Math/toRadians %) degrees)
       a (mean-fn :sin radians)
       b (mean-fn :cos radians)]
   (Math/toDegrees (Math/atan2 a b))))</lang>

Example: <lang clojure>(mean-angle [350 10])

=> -1.614809932057922E-15

(mean-angle [90 180 270 360])

=> -90.0

(mean-angle [10 20 30])

=> 19.999999999999996</lang>

Common Lisp

<lang lisp>(defun average (list)

 (/ (reduce #'+ list) (length list)))

(defun radians (angle)

 (* pi 1/180 angle))

(defun degrees (angle)

 (* 180 (/ 1 pi) angle))

(defun mean-angle (angles)

 (let* ((angles (map 'list #'radians angles))

(cosines (map 'list #'cos angles)) (sines (map 'list #'sin angles)))

   (degrees (atan (average sines) (average cosines)))))

(loop for angles in '((350 10) (90 180 270 360) (10 20 30))

  do (format t "~&The mean angle of ~a is ~$°." angles (mean-angle angles)))</lang>

Output:

The mean angle of (350 10) is -0.00°.
The mean angle of (90 180 270 360) is -90.00°.
The mean angle of (10 20 30) is 20.00°.

D

<lang d>import std.stdio, std.algorithm, std.complex; import std.math: PI;

auto radians(T)(in T d) pure nothrow @nogc { return d * PI / 180; } auto degrees(T)(in T r) pure nothrow @nogc { return r * 180 / PI; }

real meanAngle(T)(in T[] D) pure nothrow @nogc {

   immutable t = reduce!((a, d) => a + d.radians.expi)(0.complex, D);
   return (t / D.length).arg.degrees;

}

void main() {

   foreach (angles; [[350, 10], [90, 180, 270, 360], [10, 20, 30]])
       writefln("The mean angle of %s is: %.2f degrees",
                angles, angles.meanAngle);

}</lang>

Output:

The mean angle of [350, 10] is: -0.00 degrees
The mean angle of [90, 180, 270, 360] is: 90.00 degrees
The mean angle of [10, 20, 30] is: 20.00 degrees

EchoLisp

<lang scheme> (define-syntax-rule (deg->radian deg) (* deg 1/180 PI)) (define-syntax-rule (radian->deg rad) (* 180 (/ PI) rad))

(define (mean-angles angles) (radian->deg

         (angle

(for/sum ((a angles)) (make-polar 1 (deg->radian a))))))

(mean-angles '( 350 10))

   → -0

(mean-angles '[90 180 270 360])

   → -90

(mean-angles '[10 20 30])

   → 20

</lang>

Elixir

<lang Elixir> defmodule MeanAngle do

 def mean_angle(angles) do
   rad_angles = Enum.map(angles, &deg_to_rad/1)
   sines = rad_angles |> Enum.map(&:math.sin/1) |> Enum.sum
   cosines = rad_angles |> Enum.map(&:math.cos/1) |> Enum.sum

   rad_to_deg(:math.atan2(sines, cosines))
 end

 defp deg_to_rad(a) do
   (:math.pi/180) * a
 end

 defp rad_to_deg(a) do
   (180/:math.pi) * a
 end

end

IO.inspect MeanAngle.mean_angle([10, 350]) IO.inspect MeanAngle.mean_angle([90, 180, 270, 360]) IO.inspect MeanAngle.mean_angle([10, 20, 30]) </lang>

Output:

-1.614809932057922e-15
-90.0
19.999999999999996

Erlang

The function from_degrees/1 is used to solve Averages/Mean_time_of_day. Please keep backwards compatibility when editing. Or update the other module, too. <lang Erlang> -module( mean_angle ). -export( [from_degrees/1, task/0] ).

from_degrees( Angles ) -> Radians = [radians(X) || X <- Angles], Sines = [math:sin(X) || X <- Radians], Coses = [math:cos(X) || X <- Radians], degrees( math:atan2( average(Sines), average(Coses) ) ).

task() -> Angles = [[350, 10], [90, 180, 270, 360], [10, 20, 30]], [io:fwrite( "Mean angle of ~p is: ~p~n", [X, erlang:round(from_degrees(X))] ) || X <- Angles].

average( List ) -> lists:sum( List ) / erlang:length( List ).

degrees( Radians ) -> Radians * 180 / math:pi().

radians( Degrees ) -> Degrees * math:pi() / 180. </lang>

Output:

16> mean_angle:task().
Mean angle of [350,10] is: 0
Mean angle of [90,180,270,360] is: -90
Mean angle of [10,20,30] is: 20

Euler Math Toolbox

<lang EulerMathToolbox>>function meanangle (a) ... $ z=sum(exp(rad(a)*I)); $ if z~=0 then error("Not meaningful"); $ else return deg(arg(z)) $ endfunction >meanangle([350,10])

>meanangle([90,180,270,360])

Error : Not meaningful

Error generated by error() command

Error in function meanangle in line
if z~=0 then error("Not meaningful");

>meanangle([10,20,30])

20</lang>

Euphoria

Works with: OpenEuphoria

<lang Euphoria> include std/console.e include std/mathcons.e

sequence AngleList = {{350,10},{90,180,270,360},{10,20,30}}

function atan2(atom y, atom x) return 2*arctan((sqrt(power(x,2)+power(y,2)) - x)/y) end function

function MeanAngle(sequence angles) atom x = 0, y = 0 integer l = length(angles)

for i = 1 to length(angles) do x += cos(angles[i] * PI / 180) y += sin(angles[i] * PI / 180) end for

return atan2(y / l, x / l) * 180 / PI end function

for i = 1 to length(AngleList) do printf(1,"Mean Angle for set %d: %3.5f\n",{i,MeanAngle(AngleList[i])}) end for

if getc(0) then end if </lang>

Output:

Mean Angle for set 1:  0.00000
Mean Angle for set 2:  -90.00000
Mean Angle for set 3:  20.00000

F#

<lang fsharp>open System open System.Numerics

let deg2rad d = d * Math.PI / 180. let rad2deg r = r * 180. / Math.PI

[<EntryPoint>] let main argv =

   let makeComplex = fun r ->  Complex.FromPolarCoordinates(1., r)
   argv
   |> Seq.map (Double.Parse >> deg2rad >> makeComplex)
   |> Seq.fold (fun x y -> Complex.Add(x,y)) Complex.Zero
   |> fun c -> c.Phase |> rad2deg
   |> printfn "Mean angle for [%s]: %g°" (String.Join("; ",argv))
   0</lang>

Output:

>RosettaCode 350 10
Mean angle for [350; 10]: -2.74518e-14°

>RosettaCode 10 20 30
Mean angle for [10; 20; 30]: 20°

>RosettaCode 90 180 270 360
Mean angle for [90; 180; 270; 360]: -90°

Factor

<lang factor>USING: formatting kernel math math.functions math.libm math.trig sequences ;

mean-angle ( seq -- x )

   [ deg>rad ] map [ [ sin ] map-sum ] [ [ cos ] map-sum ]
   [ length ] tri recip [ * ] curry bi@ fatan2 rad>deg ;

show ( seq -- )

   dup mean-angle "The mean angle of %u is: %f°\n" printf ;

{ { 350 10 } { 90 180 270 360 } { 10 20 30 } } [ show ] each</lang>

Output:

The mean angle of { 350 10 } is: -0.000000°
The mean angle of { 90 180 270 360 } is: -90.000000°
The mean angle of { 10 20 30 } is: 20.000000°

Fortran

Please find the example output along with the build instructions in the comments at the start of the FORTRAN 2008 source. Compiler: gfortran from the GNU compiler collection. Command interpreter: bash. <lang FORTRAN> !-*- mode: compilation; default-directory: "/tmp/" -*- !Compilation started at Mon Jun 3 18:07:59 ! !a=./f && make $a && OMP_NUM_THREADS=2 $a !gfortran -std=f2008 -Wall -fopenmp -ffree-form -fall-intrinsics -fimplicit-none f.f08 -o f ! -7.80250048E-06 350 10 ! 90.0000000 90 180 270 360 ! 19.9999962 10 20 30 ! !Compilation finished at Mon Jun 3 18:07:59

program average_angles

 !real(kind=8), parameter :: TAU = 6.283185307179586232 ! http://tauday.com/
 !integer, dimension(13), parameter :: test_data = (/2,350,10, 4,90,180,270,360, 3,10,20,30, 0/)
 !integer :: i, j, n
 !complex(kind=16) :: some
 !real(kind=8) :: angle
 real, parameter :: TAU = 6.283185307179586232 ! http://tauday.com/
 integer, dimension(13), parameter :: test_data = (/2,350,10, 4,90,180,270,360, 3,10,20,30, 0/)
 integer :: i, j, n
 complex :: some
 real :: angle
 i = 1
 n = int(test_data(i))
 do while (0 .lt. n)
   some = 0
   do j = 1, n
     angle = (TAU/360)*test_data(i+j)
     some = some + cmplx(cos(angle), sin(angle))
   end do
   some = some / n
   write(6,*)(360/TAU)*atan2(aimag(some), real(some)),test_data(i+1:i+n)
   i = i + n + 1
   n = int(test_data(i))
 end do

end program average_angles </lang>

FreeBASIC

<lang freebasic> ' FB 1.05.0 Win64

Const PI As Double = 3.1415926535897932

Function MeanAngle(angles() As Double) As Double

 Dim As Integer length = Ubound(angles) - Lbound(angles) + 1
 Dim As Double sinSum = 0.0
 Dim As Double cosSum = 0.0
 For i As Integer = LBound(angles) To UBound(angles)
   sinSum += Sin(angles(i) * PI / 180.0)
   cosSum += Cos(angles(i) * PI / 180.0)    
 Next
 Return Atan2(sinSum / length, cosSum / length) * 180.0 / PI

End Function

Dim As Double angles1(1 To 2) = {350, 10} Dim As Double angles2(1 To 4) = {90, 180, 270, 360} Dim As Double angles3(1 To 3) = {10, 20, 30}

Print Using "Mean for angles 1 is : ####.## degrees"; MeanAngle(angles1()) Print Using "Mean for angles 2 is : ####.## degrees"; MeanAngle(angles2()) Print Using "Mean for angles 3 is : ####.## degrees"; MeanAngle(angles3()) Print Print "Press any key to quit the program" Sleep </lang>

Output:

Mean for angles 1 is :   -0.00 degrees
Mean for angles 2 is :  -90.00 degrees
Mean for angles 3 is :   20.00 degrees

Go

Complex

<lang go>package main

import ( "fmt" "math" "math/cmplx" )

func deg2rad(d float64) float64 { return d * math.Pi / 180 } func rad2deg(r float64) float64 { return r * 180 / math.Pi }

func mean_angle(deg []float64) float64 { sum := 0i for _, x := range deg { sum += cmplx.Rect(1, deg2rad(x)) } return rad2deg(cmplx.Phase(sum)) }

func main() { for _, angles := range [][]float64{ {350, 10}, {90, 180, 270, 360}, {10, 20, 30}, } { fmt.Printf("The mean angle of %v is: %f degrees\n", angles, mean_angle(angles)) } }</lang>

Output:

The mean angle of [350 10] is: -0.000000 degrees
The mean angle of [90 180 270 360] is: -90.000000 degrees
The mean angle of [10 20 30] is: 20.000000 degrees

Atan2

A mean_angle function that could be substituted above. Functions deg2rad and rad2deg are not used here but there is no runtime advantage either way to using them or not. Inlining should result in eqivalent code being generated. Also the Go Atan2 library function has no limits on the arguments so there is no need to divide by the number of elements.

<lang go>func mean_angle(deg []float64) float64 { var ss, sc float64 for _, x := range deg { s, c := math.Sincos(x * math.Pi / 180) ss += s sc += c } return math.Atan2(ss, sc) * 180 / math.Pi }</lang>

Groovy

<lang groovy>import static java.lang.Math.* def meanAngle = {

   atan2( it.sum { sin(it * PI / 180) } / it.size(), it.sum { cos(it * PI / 180) } / it.size()) * 180 / PI

}</lang> Test: <lang groovy>def verifyAngle = { angles ->

   def ma = meanAngle(angles)
   printf("Mean Angle for $angles: %5.2f%n", ma)
   round(ma * 100) / 100.0

} assert verifyAngle([350, 10]) == -0 assert verifyAngle([90, 180, 270, 360]) == -90 assert verifyAngle([10, 20, 30]) == 20</lang>

Output:

Mean Angle for [350, 10]: -0.00
Mean Angle for [90, 180, 270, 360]: -90.00
Mean Angle for [10, 20, 30]: 20.00

Haskell

<lang haskell>import Data.Complex (cis, phase)

meanAngle

 :: RealFloat c
 => [c] -> c

meanAngle = (/ pi) . (* 180) . phase . sum . map (cis . (/ 180) . (* pi))

main :: IO () main =

 mapM_
   (\angles ->
       putStrLn $
       "The mean angle of " ++
       show angles ++ " is: " ++ show (meanAngle angles) ++ " degrees")
   [[350, 10], [90, 180, 270, 360], [10, 20, 30]]</lang>

Output:

The mean angle of [350.0,10.0] is: -2.745176884498468e-14 degrees
The mean angle of [90.0,180.0,270.0,360.0] is: -90.0 degrees
The mean angle of [10.0,20.0,30.0] is: 19.999999999999996 degrees

Alternative Solution: This solution gives an insight about using factoring, many small functions like Forth and using function composition.

-- file: trigdeg.fs

deg2rad deg = deg*pi/180.0 rad2deg rad = rad*180.0/pi

sind = sin . deg2rad cosd = cos . deg2rad tand = tan . deg2rad atand = rad2deg . atan atan2d y x = rad2deg (atan2 y x )

avg_angle angles = atan2d y x

   where
   y = mean (map sind angles) 
   x = mean (map cosd angles)

-- End of trigdeg.fs -------- </lang>

Output:


-- GHCI shell  $ ghci

Prelude> :load trigdeg.hs 
[1 of 1] Compiling Main             ( trigdeg.hs, interpreted )
Ok, modules loaded: Main.
*Main> 
*Main> avg_angle [350.0, 10.0]
-2.745176884498468e-14
*Main> 
*Main> avg_angle [90.0, 180.0, 270.0, 360.0 ]
-90.0
*Main> mean [10.0, 20.0, 30.0]
20.0
*Main>

Icon and Unicon

<lang unicon>procedure main(A)

   write("Mean angle is ",meanAngle(A))

end

procedure meanAngle(A)

   every (sumSines := 0.0) +:= sin(dtor(!A))
   every (sumCosines := 0.0) +:= cos(dtor(!A))
   return rtod(atan(sumSines/*A,sumCosines/*A))

end</lang>

Sample runs:

->ama 10 350
Mean angle is -2.745176884498468e-14
->ama 90 180 270 360
Mean angle is -90.0
->ama 10 20 30
Mean angle is 20.0

J

<lang J>avgAngleD=: 360|(_1 { [: (**|)&.+.@(+/ % #)&.(*.inv) 1,.])&.(1r180p1&*)</lang> This verb can be represented as simpler component verbs, for example: <lang J>rfd=: 1r180p1&* NB. convert angle to radians from degrees toComplex=: *.inv NB. maps integer pairs as length, complex angle (in radians) mean=: +/ % # NB. calculate arithmetic mean roundComplex=: (* * |)&.+. NB. discard an extraneous least significant bit of precision from a complex value whose magnitude is in the vicinity of 1 avgAngleR=: _1 { [: roundComplex@mean&.toComplex 1 ,. ] NB. calculate average angle in radians avgAngleD=: 360|avgAngleR&.rfd NB. calculate average angle in degrees</lang> Example use: <lang J> avgAngleD 10 350 0

  avgAngleD 90 180 270 360 NB. result not meaningful

0

  avgAngleD 10 20 30

20

  avgAngleD 20 350

5

  avgAngleD 10 340

355</lang>

Notes:

(**|)&.+. is an expression to discard an extraneous least significant bit of precision from a complex value whose magnitude is in the vicinity of 1.

(+/ % #) finds the (Pythagorean) mean

verb&.(*.inv) maps integer pairs as length,complex angle (in radians) and uses the verb in the domain of complex numbers, and then maps the result back to length,angle.

(1,.]) prefixes every value in a list with 1 (forming a two column table)

(_1 { verb) takes the last item from the result of the verb (and note that after we average our complex values and convert them back to length/angle format, we will be working with a list of two elements: length and angle -- and we do not care about length, which will usually be less than 1).

verb&.(1r180p1&*) converts its argument from degrees to radians, uses the verb in the radian domain, then converts the result of that argument back to degrees.

360|verb ensures that the result is not negative and is also less than 360

Java

Translation of: NetRexx

Works with: Java version 7+

<lang java5>import java.util.Arrays;

public class AverageMeanAngle {

   public static void main(String[] args) {
       printAverageAngle(350.0, 10.0);
       printAverageAngle(90.0, 180.0, 270.0, 360.0);
       printAverageAngle(10.0, 20.0, 30.0);
       printAverageAngle(370.0);
       printAverageAngle(180.0);
   }

   private static void printAverageAngle(double... sample) {
       double meanAngle = getMeanAngle(sample);
       System.out.printf("The mean angle of %s is %s%n", Arrays.toString(sample), meanAngle);
   }

   public static double getMeanAngle(double... anglesDeg) {
       double x = 0.0;
       double y = 0.0;

       for (double angleD : anglesDeg) {
           double angleR = Math.toRadians(angleD);
           x += Math.cos(angleR);
           y += Math.sin(angleR);
       }
       double avgR = Math.atan2(y / anglesDeg.length, x / anglesDeg.length);
       return Math.toDegrees(avgR);
   }

}</lang>

Output:

The mean angle of [350.0, 10.0] is -1.614809932057922E-15
The mean angle of [90.0, 180.0, 270.0, 360.0] is -90.0
The mean angle of [10.0, 20.0, 30.0] is 19.999999999999996
The mean angle of [370.0] is 9.999999999999977
The mean angle of [180.0] is 180.0

JavaScript

atan2

<lang javascript>function sum(a) {

   var s = 0;
   for (var i = 0; i < a.length; i++) s += a[i];
   return s;

}

function degToRad(a) {

   return Math.PI / 180 * a;

}

function meanAngleDeg(a) {

   return 180 / Math.PI * Math.atan2(
       sum(a.map(degToRad).map(Math.sin)) / a.length,
       sum(a.map(degToRad).map(Math.cos)) / a.length
   );

}

var a = [350, 10], b = [90, 180, 270, 360], c = [10, 20, 30]; console.log(meanAngleDeg(a)); console.log(meanAngleDeg(b)); console.log(meanAngleDeg(c));</lang>

Output:

-1.614809932057922e-15
-90
19.999999999999996

jq

Works with: jq version 1.4

To avoid anomalies, the following is designed to assign the null value as the mean angle in cases such as [0, 180].

Generic Infrastructure <lang jq>def pi: 4 * (1|atan);

def deg2rad: . * pi / 180;

def rad2deg: if . == null then null else . * 180 / pi end;

Input: [x,y] (special handling of x==0)
Output: [r, theta] where theta may be null

def to_polar:

 if .[0] == 0
 then [1, if .[1] > 5e-14 then pi/2 elif .[1] < -5e-14 then -pi/2 else null end]
 else [1, ((.[1]/.[0]) | atan)]
 end;

def from_polar: .[1] | [ cos, sin];

def abs: if . < 0 then - . else . end;

def summation(f): map(f) | add;</lang>

Mean Angle <lang jq># input: degrees def mean_angle:

 def round:
   if . == null then null
   elif . < 0 then -1 * ((- .) | round) | if . == -0 then 0 else . end
   else ((. + 3e-14) | floor) as $x
   | if ($x - .) | abs < 3e-14 then $x else . end
   end;

 map( [1, deg2rad] | from_polar)
 | [ summation(.[0]), summation(.[1]) ]
 | to_polar
 | .[1]
 | rad2deg
 | round;</lang>

Examples <lang jq>([350, 10], [90, 180, 270, 360], [10, 20, 30]) | "The mean angle of \(.) is: \(mean_angle)"</lang>

Output:

<lang sh>jq -r -n -f Mean_angle.jq The mean angle of [350,10] is: 0 The mean angle of [90,180,270,360] is: null The mean angle of [10,20,30] is: 20</lang>

Julia

Julia has built-in functions sind and cosd to compute the sine and cosine of angles specified in degrees accurately (avoiding the roundoff errors incurred in conversion to radians), and a built-in function to convert radians to degrees (or vice versa). Using these: <lang julia>using Statistics meandegrees(degrees) = rad2deg(atan(mean(sind.(degrees)), mean(cosd.(degrees))))</lang> The output is: <lang julia>julia> meandegrees([350, 10]) 0.0

julia> meandegrees([90, 180, 270, 360]]) 0.0

julia> meandegrees([10, 20, 30]]) 19.999999999999996</lang> (Note that the mean of 90°, 180°, 270°, and 360° gives zero because of the lack of roundoff errors in the sind function, since the standard-library atan2(0,0) value is zero. Many of the other languages report an average of 90° or –90° in this case due to rounding errors.)

Kotlin

<lang scala>// version 1.0.5-2

fun meanAngle(angles: DoubleArray): Double {

   val sinSum = angles.sumByDouble {  Math.sin(it * Math.PI / 180.0) }
   val cosSum = angles.sumByDouble {  Math.cos(it * Math.PI / 180.0) }
   return Math.atan2(sinSum / angles.size, cosSum / angles.size) * 180.0 / Math.PI

}

fun main(args: Array<String>) {

   val angles1 = doubleArrayOf(350.0, 10.0)
   val angles2 = doubleArrayOf(90.0, 180.0, 270.0, 360.0)
   val angles3 = doubleArrayOf(10.0, 20.0, 30.0)
   val fmt  = "%.2f degrees" // format results to 2 decimal places
   println("Mean for angles 1 is ${fmt.format(meanAngle(angles1))}")
   println("Mean for angles 2 is ${fmt.format(meanAngle(angles2))}")
   println("Mean for angles 3 is ${fmt.format(meanAngle(angles3))}")

}</lang>

Output:

Mean for angles 1 is -0.00 degrees
Mean for angles 2 is -90.00 degrees
Mean for angles 3 is 20.00 degrees

Liberty BASIC

<lang lb>global Pi Pi =3.1415926535

print "Mean Angle( "; chr$( 34); "350,10"; chr$( 34); ") = "; using( "###.#", meanAngle( "350,10")); " degrees." ' 0 print "Mean Angle( "; chr$( 34); "90,180,270,360"; chr$( 34); ") = "; using( "###.#", meanAngle( "90,180,270,360")); " degrees." ' -90 print "Mean Angle( "; chr$( 34); "10,20,30"; chr$( 34); ") = "; using( "###.#", meanAngle( "10,20,30")); " degrees." ' 20

end

function meanAngle( angles$)

  term =1
  while word$( angles$, term, ",") <>""
     angle  =val( word$( angles$, term, ","))
     sumSin = sumSin +sin( degRad( angle))
     sumCos = sumCos +cos( degRad( angle))
     term =term +1
  wend
  meanAngle= radDeg( atan2( sumSin, sumCos))
  if abs( sumSin) +abs( sumCos) <0.001 then notice "Not Available." +_
    chr$( 13) +"(" +angles$ +")" +_
    chr$( 13) +"Result nearly equals zero vector." +_
    chr$( 13) +"Displaying '666'.": meanAngle =666

end function

function degRad( theta)

   degRad =theta *Pi /180

end function

function radDeg( theta)

   radDeg =theta *180 /Pi

end function

function atan2( y, x)

   if x >0           then at =atn( y /x)
   if y >=0 and x <0 then at =atn( y /x) +pi
   if y <0  and x <0 then at =atn( y /x) -pi
   if y >0  and x =0 then at =    pi /2
   if y <0  and x =0 then at = 0 -pi /2
   if y =0  and x =0 then notice "undefined": end
   atan2 =at

end function</lang>

Output:

Mean Angle( "350,10") = 0.0 degrees.
Mean Angle( "90,180,270,360") = 666.0 degrees.
Mean Angle( "10,20,30") = 20.0 degrees.

Logo

<lang logo>to mean_angle :angles

  local "avgsin
  make "avgsin quotient apply "sum map "sin :angles count :angles
  local "avgcos
  make "avgcos quotient apply "sum map "cos :angles count :angles
  output (arctan :avgcos :avgsin)

end

foreach [[350 10] [90 180 270 360] [10 20 30]] [

 print (sentence [The average of \(] ? [\) is] (mean_angle ?))

]

bye </lang>

Output:

The average of ( 350 10 ) is 0
The average of ( 90 180 270 360 ) is 0
The average of ( 10 20 30 ) is 20

Lua

Translation of: Tcl

Works with: Lua version 5.1

<lang Lua>function meanAngle (angleList)

 local sumSin, sumCos = 0, 0
 for i, angle in pairs(angleList) do
   sumSin = sumSin + math.sin(math.rad(angle))
   sumCos = sumCos + math.cos(math.rad(angle))
 end
 local result = math.deg(math.atan2(sumSin, sumCos))
 return string.format("%.2f", result)

end

print(meanAngle({350, 10})) print(meanAngle({90, 180, 270, 360})) print(meanAngle({10, 20, 30}))</lang>

Output:

-0.00
-90.00
20.00

Maple

The following procedure takes a list of numeric degrees (with attached units) such as <lang Maple>> [ 350, 10 ] *~ Unit(arcdeg);

                     [350 [arcdeg], 10 [arcdeg]]</lang>

as input. (We could use "degree" instead of "arcdeg", since "degree" is taken, by default, to mean angle measure, but it seems best to avoid the ambiguity.) <lang Maple>MeanAngle := proc( L )

       uses Units:-Standard; # for unit-awareness
       local   u;
       evalf( convert( argument( add( u, u = exp~( I *~ L ) ) ), 'units', 'radian', 'degree' ) )

end proc:</lang> Applying this to the given data sets, we obtain: <lang Maple>> MeanAngle( [ 350, 10 ] *~ Unit(arcdeg) );

0.

> MeanAngle( [ 90, 180, 270, 360 ] *~ Unit(arcdeg) );

0.

> MeanAngle( [ 10, 20, 30 ] *~ Unit(arcdeg) );

                             20.00000000</lang>

Mathematica / Wolfram Language

<lang Mathematica>meanAngle[data_List] := N@Arg[Mean[Exp[I data Degree]]]/Degree</lang>

Output:

meanAngle /@ {{350, 10}, {90, 180, 270, 360}, {10, 20, 30}}
{0., Interval[{-180., 180.}], 20.}

MATLAB / Octave

<lang MATLAB>function u = mean_angle(phi) u = angle(mean(exp(i*pi*phi/180)))*180/pi; end</lang>

 mean_angle([350, 10])
ans = -2.7452e-14
 mean_angle([90, 180, 270, 360])
ans = -90
 mean_angle([10, 20, 30])
ans =  20.000

NetRexx

Translation of: C

<lang NetRexx>/* NetRexx */ options replace format comments java crossref symbols nobinary numeric digits 80

runSample(arg) return

-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ method getMeanAngle(angles) private static binary

 x_component = double 0.0
 y_component = double 0.0
 aK = int angles.words()
 loop a_ = 1 to aK
   angle_d = double angles.word(a_)
   angle_r = double Math.toRadians(angle_d)
   x_component = x_component + Math.cos(angle_r)
   y_component = y_component + Math.sin(angle_r)
   end a_
 x_component = x_component / aK
 y_component = y_component / aK
 avg_r = Math.atan2(y_component, x_component)
 avg_d = Math.toDegrees(avg_r)
 return avg_d

-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ method runSample(arg) private static

 angleSet = [ '350 10', '90 180 270 360', '10 20 30', '370', '180' ]
 loop angles over angleSet
   say 'The mean angle of' angles.space(1, ',') 'is:'
   say '  'getMeanAngle(angles).format(6, 6)
   say
   end angles
 return

</lang>

Output:

The mean angle of 350,10 is:
       0.000000

The mean angle of 90,180,270,360 is:
     -90.000000

The mean angle of 10,20,30 is:
      20.000000

The mean angle of 370 is:
      10.000000

The mean angle of 180 is:
     180.000000

Nim

<lang nim>import math, complex

proc rect(r, phi: float): Complex = (r * cos(phi), sin(phi)) proc phase(c: Complex): float = arctan2(c.im, c.re)

proc radians(x: float): float = (x * Pi) / 180.0 proc degrees(x: float): float = (x * 180.0) / Pi

proc meanAngle(deg: openArray[float]): float =

 var c: Complex
 for d in deg:
   c += rect(1.0, radians(d))
 degrees(phase(c / float(deg.len)))

echo "The 1st mean angle is: ", meanAngle([350.0, 10.0]), " degrees" echo "The 2nd mean angle is: ", meanAngle([90.0, 180.0, 270.0, 360.0]), " degrees" echo "The 3rd mean angle is: ", meanAngle([10.0, 20.0, 30.0]), " degrees"</lang> Output:

The 1st mean angle is: -2.745176884498468e-14 degrees
The 2nd mean angle is: -90.0 degrees
The 3rd mean angle is: 20.0 degrees

Oberon-2

Works with: oo2c

<lang oberon2> MODULE MeanAngle; IMPORT

 M := LRealMath,
 Out;

CONST

 toRads = M.pi / 180;
 toDegs = 180 / M.pi;

VAR

 grades: ARRAY 64 OF LONGREAL;

PROCEDURE Mean(g: ARRAY OF LONGREAL): LONGREAL; VAR

 i: INTEGER;
 sumSin, sumCos: LONGREAL;

BEGIN

 i := 0;sumSin := 0.0;sumCos := 0.0;
 WHILE g[i] # 0.0 DO
   sumSin := sumSin + M.sin(g[i] * toRads);
   sumCos := sumCos + M.cos(g[i] * toRads);  
   INC(i)
 END;
 RETURN M.arctan2(sumSin / i,sumCos / i);

END Mean;

BEGIN

 grades[0] := 350.0;grades[1] := 10.0;
 Out.LongRealFix(Mean(grades) * toDegs,15,9);Out.Ln;
 grades[0] := 90.0;grades[1] := 180.0;grades[2] := 270.0;grades[3] := 360.0;
 Out.LongRealFix(Mean(grades) * toDegs,15,9);Out.Ln;
 grades[0] := 10.0;grades[1] := 20.0;grades[2] := 30.0;grades[3] := 0.0;
 Out.LongRealFix(Mean(grades) * toDegs,15,9);Out.Ln;

END MeanAngle. </lang>

Output:

   -0.000000000
  -90.000000000
   20.000000000

OCaml

<lang ocaml>let pi = 3.14159_26535_89793_23846_2643

let deg2rad d =

 d *. pi /. 180.0

let rad2deg r =

 r *. 180.0 /. pi

let mean_angle angles =

 let rad_angles = List.map deg2rad angles in
 let sum_sin = List.fold_left (fun sum a -> sum +. sin a) 0.0 rad_angles
 and sum_cos = List.fold_left (fun sum a -> sum +. cos a) 0.0 rad_angles
 in
 rad2deg (atan2 sum_sin sum_cos)

let test angles =

 Printf.printf "The mean angle of [%s] is: %g degrees\n"
   (String.concat "; " (List.map (Printf.sprintf "%g") angles))
   (mean_angle angles)

let () =

 test [350.0; 10.0];
 test [90.0; 180.0; 270.0; 360.0];
 test [10.0; 20.0; 30.0];

</lang>

or using the Complex module: <lang ocaml>open Complex

let mean_angle angles =

 let sum =
   List.fold_left (fun sum a -> add sum (polar 1.0 (deg2rad a))) zero angles
 in
   rad2deg (arg sum)</lang>

Output:

The mean angle of [350; 10] is: -2.74518e-14 degrees
The mean angle of [90; 180; 270; 360] is: -90 degrees
The mean angle of [10; 20; 30] is: 20 degrees

ooRexx

Translation of: REXX

<lang oorexx>/*REXX program computes the mean angle (angles expressed in degrees). */ numeric digits 50 /*use fifty digits of precision, */

      showDig=10                      /*··· but only display 10 digits.*/

xl = 350 10 ; say showit(xl, meanAngleD(xl) ) xl = 90 180 270 360 ; say showit(xl, meanAngleD(xl) ) xl = 10 20 30 ; say showit(xl, meanAngleD(xl) ) exit /*stick a fork in it, we're done.*/ /*----------------------------------MEANANGD subroutine-----------------*/ meanAngleD: procedure; parse arg xl; numeric digits digits()+digits()%4 sum.=0 n=words(xl) do j=1 to n

 sum.0sin+=rxCalcSin(word(xl,j))
 sum.0cos+=rxCalcCos(word(xl,j))
 End

If sum.0cos=0 Then

 Return sign(sum.0sin/n)*90

Else

 Return rxCalcArcTan((sum.0sin/n)/(sum.0cos/n))

showit: procedure expose showDig; numeric digits showDig; parse arg a,mA

       return left('angles='a,30)  'mean angle='  format(mA,,showDig,0)/1

requires rxMath library;</lang>

Output:

angles=350 10                  mean angle= 0
angles=90 180 270 360          mean angle= 0
angles=10 20 30                mean angle= 20

PARI/GP

<lang parigp>meanAngle(v)=atan(sum(i=1,#v,sin(v[i]))/sum(i=1,#v,cos(v[i])))%(2*Pi) meanDegrees(v)=meanAngle(v*Pi/180)*180/Pi apply(meanDegrees,[[350, 10], [90, 180, 270, 360], [10, 20, 30]])</lang>

Output:

[360.000000, 296.565051, 20.0000000]

Pascal

uses library math for sincos, a function of FPU80x87, atan2 and DegToRad. Tested with free pascal. Try to catch very small cos values and set to 0.0 degrees " Error : Not meaningful" as http://rosettacode.org/wiki/Averages/Mean_angle#Euler_Math_Toolbox complains. <lang pascal>program MeanAngle; {$IFDEF DELPHI}

 {$APPTYPE CONSOLE}

{$ENDIF} uses

 math;// sincos and atan2

type

 tAngles = array of double;

function MeanAngle(const a:tAngles;cnt:longInt):double; // calculates mean angle. // returns 0.0 if direction is not sure. const

 eps = 1e-10;

var

 i : LongInt;
 s,c,
 Sumsin,SumCos : extended;

begin

 IF cnt = 0 then
 Begin
   MeanAngle := 0.0;
   EXIT;
 end;

 SumSin:= 0;
 SumCos:= 0;
 For i := Cnt-1 downto 0 do
 Begin
   sincos(DegToRad(a[i]),s,c);
   Sumsin := sumSin+s;
   SumCos := sumCos+c;
 end;
 s := SumSin/cnt;
 c := sumCos/cnt;
 IF c > eps then
   MeanAngle := RadToDeg(arctan2(s,c))
 else
   // Not meaningful
   MeanAngle := 0.0;

end;

Procedure OutMeanAngle(const a:tAngles;cnt:longInt); var

 i : longInt;

Begin

 IF cnt > 0 then
 Begin
   write('The mean angle of [');
   For i := 0 to Cnt-2 do
     write(a[i]:0:2,',');
   write(a[Cnt-1]:0:2,'] => ');
   writeln(MeanAngle(a,cnt):0:16);
 end;

end;

var

 a:tAngles;

Begin

 setlength(a,4);

 a[0] := 350;a[1] := 10;
 OutMeanAngle(a,2);
 a[0] := 90;a[1] := 180;a[2] := 270;a[3] := 360;
 OutMeanAngle(a,4);
 a[0] := 10;a[1] := 20;a[2] := 30;
 OutMeanAngle(a,3);

 setlength(a,0);

end.</lang>

output

The mean angle of [350.00,10.00] => 0.0000000000000000
The mean angle of [90.00,180.00,270.00,360.00] => 0.0000000000000000
The mean angle of [10.00,20.00,30.00] => 20.0000000000000000

Perl

<lang perl>sub Pi () { 3.1415926535897932384626433832795028842 }

sub meanangle {

 my($x, $y) = (0,0);
 ($x,$y) = ($x + sin($_), $y + cos($_)) for @_;
 my $atan = atan2($x,$y);
 $atan += 2*Pi while $atan < 0;    # Ghetto fmod
 $atan -= 2*Pi while $atan > 2*Pi;
 $atan;

}

sub meandegrees {

 meanangle( map { $_ * Pi/180 } @_ ) * 180/Pi;

}

print "The mean angle of [@$_] is: ", meandegrees(@$_), " degrees\n"

 for ([350,10], [90,180,270,360], [10,20,30]);</lang>

Output:

The mean angle of [350 10] is: 360 degrees
The mean angle of [90 180 270 360] is: 270 degrees
The mean angle of [10 20 30] is: 20 degrees

Phix

Copied from Euphoria, and slightly improved <lang Phix>function atan2(atom y, atom x)

   return 2*arctan((sqrt(power(x,2)+power(y,2))-x)/y)

end function

function MeanAngle(sequence angles) atom x=0, y=0, ai_rad integer l=length(angles)

   for i=1 to l do
       ai_rad = angles[i]*PI/180
       x += cos(ai_rad)
       y += sin(ai_rad)
   end for
   if abs(x)<1e-16 then return "not meaningful" end if
   return sprintf("%9.5f",atan2(y,x)*180/PI)

end function

constant AngleLists = {{350,10},{90,180,270,360},{10,20,30},{180},{0,180}} sequence ai for i=1 to length(AngleLists) do

   ai = AngleLists[i]
   printf(1,"%+16s: Mean Angle is %s\n",{sprint(ai),MeanAngle(ai)})

end for {} = wait_key()</lang>

Output:

        {350,10}: Mean Angle is   0.00000
{90,180,270,360}: Mean Angle is not meaningful
      {10,20,30}: Mean Angle is  20.00000
           {180}: Mean Angle is 180.00000
         {0,180}: Mean Angle is not meaningful

PHP

Translation of: C

<lang php><?php $samples = array( '1st' => array(350, 10), '2nd' => array(90, 180, 270, 360), '3rd' => array(10, 20, 30) );

foreach($samples as $key => $sample){ echo 'Mean angle for ' . $key . ' sample: ' . meanAngle($sample) . ' degrees.' . PHP_EOL; }

function meanAngle ($angles){ $y_part = $x_part = 0; $size = count($angles); for ($i = 0; $i < $size; $i++){ $x_part += cos(deg2rad($angles[$i])); $y_part += sin(deg2rad($angles[$i])); } $x_part /= $size; $y_part /= $size; return rad2deg(atan2($y_part, $x_part)); } ?></lang>

Output:

Mean angle for 1st sample: -1.6148099320579E-15 degrees.
Mean angle for 2nd sample: -90 degrees.
Mean angle for 3rd sample: 20 degrees.

PicoLisp

<lang PicoLisp>(load "@lib/math.l")

(de meanAngle (Lst)

  (*/
     (atan2
        (sum '((A) (sin (*/ A pi 180.0))) Lst)
        (sum '((A) (cos (*/ A pi 180.0))) Lst) )
     180.0 pi ) )

(for L '((350.0 10.0) (90.0 180.0 270.0 360.0) (10.0 20.0 30.0))

  (prinl
     "The mean angle of ["
     (glue ", " (mapcar round L '(0 .)))
     "] is: " (round (meanAngle L))) )</lang>

Output:

The mean angle of [350, 10] is: 0.000
The mean angle of [90, 180, 270, 360] is: 90.000
The mean angle of [10, 20, 30] is: 20.000

PL/I

<lang PL/I>averages: procedure options (main); /* 31 August 2012 */

  declare b1(2) fixed initial (350, 10);
  declare b2(4) fixed initial (90, 180, 270, 360);
  declare b3(3) fixed initial (10, 20, 30);

  put edit ( b1) (f(7));
  put edit ( ' mean=', mean(b1) )   (a, f(7) );
  put skip edit ( b3) (f(7));
  put edit ( ' mean=', mean(b3) )   (a, f(7) );
  put skip edit ( b2) (f(7));
  put edit ( ' mean=', mean(b2) )   (a, f(7) );

mean: procedure (a) returns (fixed);

  declare a(*) float (18);
  return ( atand(sum(sind(a))/hbound(a), sum(cosd(a))/hbound(a) ) );

end mean;

end averages;</lang> Results (the final one brings up an error in inverse tangent):

    350     10 mean=      0
     10     20     30 mean=     20
     90    180    270    360 mean=
IBM0683I  ONCODE=1521  X or Y in ATAN(X,Y) or ATAND(X,Y) was invalid.
   At offset +000009CC in procedure with entry AVERAGES

PowerShell

<lang PowerShell> function Get-MeanAngle ([double[]]$Angles) {

   $x = ($Angles | ForEach-Object {[Math]::Cos($_ * [Math]::PI / 180)} | Measure-Object -Average).Average
   $y = ($Angles | ForEach-Object {[Math]::Sin($_ * [Math]::PI / 180)} | Measure-Object -Average).Average

   [Math]::Atan2($y, $x) * 180 / [Math]::PI

} </lang>

<lang PowerShell> @(350, 10), @(90, 180, 270, 360), @(10, 20, 30) | ForEach-Object {Get-MeanAngle $_} </lang>

Output:

-2.74517688449847E-14
-90
20

Python

Works with: Python version 2.6+

<lang python>>>> from cmath import rect, phase >>> from math import radians, degrees >>> def mean_angle(deg): ... return degrees(phase(sum(rect(1, radians(d)) for d in deg)/len(deg))) ... >>> for angles in [[350, 10], [90, 180, 270, 360], [10, 20, 30]]: ... print('The mean angle of', angles, 'is:', round(mean_angle(angles), 12), 'degrees') ... The mean angle of [350, 10] is: -0.0 degrees The mean angle of [90, 180, 270, 360] is: -90.0 degrees The mean angle of [10, 20, 30] is: 20.0 degrees >>> </lang>

R

<lang R> deg2rad <- function(x) {

 x * pi/180

}

rad2deg <- function(x) {

 x * 180/pi

}

deg2vec <- function(x) {

 c(sin(deg2rad(x)), cos(deg2rad(x)))

}

vec2deg <- function(x) {

 rad2deg(atan2(x[1], x[2]))

}

mean_vec <- function(x) {

 y <- lapply(x, deg2vec)
 Reduce(`+`, y)/length(y)

}

mean_deg <- function(x) {

 res <- vec2deg(mean_vec(x))
 if (res < 0) {
   360 + res
 } else {
   res
 }

}

mean_deg(c(350, 10)) mean_deg(c(90, 180, 270, 360)) mean_deg(c(10, 20, 30)) </lang>

Output:

360
270
20

Racket

The formula given above can be straightforwardly transcribed into a program: <lang racket>

lang racket

(define (mean-angle αs)

 (radians->degrees
  (mean-angle/radians
   (map degrees->radians αs))))

(define (mean-angle/radians αs)

 (define n (length αs))
 (atan (* (/ 1 n) (for/sum ([α_j αs]) (sin α_j)))
       (* (/ 1 n) (for/sum ([α_j αs]) (cos α_j)))))

(mean-angle '(350 0 10)) (mean-angle '(90 180 270 360)) (mean-angle '(10 20 30)) </lang>

Output:

-1.0710324872062297e-15
-90.0
19.999999999999996

Raku

(formerly Perl 6)

Works with: Rakudo version 2015.12

This solution refuses to return an answer when the angles cancel out to a tiny magnitude. <lang perl6># Of course, you can still use pi and 180. sub deg2rad { $^d * tau / 360 } sub rad2deg { $^r * 360 / tau }

sub phase ($c) {

   my ($mag,$ang) = $c.polar;
   return NaN if $mag < 1e-16;
   $ang;

}

sub meanAngle { rad2deg phase [+] map { cis deg2rad $_ }, @^angles }

say meanAngle($_).fmt("%.2f\tis the mean angle of "), $_ for

   [350, 10],
   [90, 180, 270, 360],
   [10, 20, 30];</lang>

Output:

-0.00	is the mean angle of 350 10
NaN	is the mean angle of 90 180 270 360
20.00	is the mean angle of 10 20 30

REXX

This REXX version uses an ATAN2 solution.

The REXX language doesn't have most of the higher mathematical functions (like sqrt), and none of the trigonometric
functions, so all of them have to be included as RYO (Roll-Your-Own).

Note that the second set of angles:    90    180   270   360

is the same as:                        90    180   -90     0
           and:                      -270   -180   -90  -360

and other combinations.

All the trigonometric functions use normalization (converting the angle to a unit circle) before computation, and most
of them use shortcuts for some exact values, so there is a minimum of errors due to rounding for near values for some
common values.

The consequence is the trig functions may return exact values such as 0 (zero) for sin(-2 $\pi$ ) instead of -8.154E-61.

This very small difference (almost inconsequential) makes a significant difference when that value is used for a parameter
for the ATAN2 function; in particular, the sign of the value.

There isn't much difference between:

           -8.154e-61   and
           +8.154e-61

in magnitude, but the ATAN2 function treats those two numbers very differently as the angle will be in different quadrants,
thereby yielding a different value.

Usually this just results in an angle of -90º instead of +270º (both angles are equivalent).

Also note that the REXX subroutines are largely not commented as they provide a support structure that's normally present
in other computer programming languages as BIFs (Built-In-Functions); to add comments and expand the REXX statements
into single lines would increase the program's bulk and detract from the main program. <lang rexx>/*REXX program computes the mean angle for a group of angles (expressed in degrees). */ call pi /*define the value of pi to some accuracy.*/ numeric digits length(pi) - 1; /*use PI width decimal digits of precision,*/

       showDig= 10                              /*only display  ten  decimal digits.        */

= 350 10 ; say show(#, meanAngleD(#)) /*display mean angle (in degrees), 1st case.*/
= 90 180 270 360 ; say show(#, meanAngleD(#)) /* " " " " " 2nd " */
= 10 20 30 ; say show(#, meanAngleD(#)) /* " " " " " 3rd " */

exit /*stick a fork in it, we're all done with it*/ /*───────────────────────────────────────────────────────────────────────────────────────────*/ .sinCos: arg z,_,i; x=x*x; do k=2 by 2 until p=z; p=z; _=-_*x/(k*(k+i)); z=z+_; end; return z $fuzz: return min(arg(1), max(1, digits() - arg(2) ) ) Acos: procedure; parse arg x; return pi() * .5 - Asin(x) Atan: parse arg x; if abs(x)=1 then return pi()*.25 * sign(x); return Asin(x/sqrt(1 + x*x)) d2d: return arg(1) // 360 d2r: return r2r(d2d(arg(1)) / 180 * pi() ) r2d: return d2d((r2r(arg(1)) / pi()) * 180) r2r: return arg(1) // (pi() * 2) pi: pi=3.1415926535897932384626433832795028841971693993751058209749445923078164062862;return pi /*───────────────────────────────────────────────────────────────────────────────────────────*/ Asin: procedure; parse arg x 1 z 1 o 1 p; a= abs(x); aa= a * a

       if a>1  then call AsinErr x                            /*argument X  is out of range.*/
       if a >= sqrt(2) * .5  then  return sign(x) * acos( sqrt(1 - aa),  '-ASIN')
         do j=2  by 2  until p=z;  p= z;  o= o * aa * (j-1) / j;   z= z +o / (j+1);   end
       return  z                                              /* [↑]  compute until no noise*/

/*───────────────────────────────────────────────────────────────────────────────────────────*/ Atan2: procedure; parse arg y,x; call pi; s= sign(y)

         select
         when x=0  then  z= s * pi * .5
         when x<0  then  if  y=0  then z= pi;  else z= s * (pi - abs( Atan(y/x) ) )
         otherwise       z= s * Atan(y / x)
         end   /*select*/;                           return z

/*───────────────────────────────────────────────────────────────────────────────────────────*/ cos: procedure; parse arg x; x= r2r(x); numeric fuzz $fuzz(6, 3)

       a= abs(x);      if a=0  then return   1;      if a=pi    then return -1
       if a=pi*.5 | a= pi*1.5  then return   0;      if a=pi/3  then return .5
                 if a= pi*2/3  then return -.5;                      return .sinCos(1, 1, -1)

/*───────────────────────────────────────────────────────────────────────────────────────────*/ meanAngleD: procedure; parse arg x; numeric digits digits() + digits() % 4

           n= words(x);                              _sin= 0;              _cos= 0
              do j=1  for n;  != d2r( word(x, j));   _sin= _sin + sin(!);  _cos= _cos + cos(!)
              end   /*j*/
           return  r2d( Atan2( _sin/n, _cos/n) )

/*───────────────────────────────────────────────────────────────────────────────────────────*/ show: parse arg a,mA; _= format(ma, , showDig, 0) / 1

       return left('angles='a, 30)   "mean angle="   right(_, max(4, length(_) ) )

/*───────────────────────────────────────────────────────────────────────────────────────────*/ sin: procedure; parse arg x; x= r2r(x); numeric fuzz $fuzz(5, 3)

       if x=pi * .5         then return 1;           if x==pi * 1.5  then return -1
       if abs(x)=pi | x=0   then return 0;           return .sinCos(x, x, +1)

/*───────────────────────────────────────────────────────────────────────────────────────────*/ sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); m.=9; numeric form; h= d+6

       numeric digits;  parse value format(x,2,1,,0) 'E0'  with  g "E" _ .;  g= g * .5'e'_ % 2
                do j=0  while h>9;       m.j= h;               h= h % 2  +  1;     end  /*j*/
                do k=j+5  to 0  by -1;   numeric digits m.k;   g= (g + x/g) * .5;  end  /*k*/
       return g</lang>

output when using the default internal inputs:

angles=350 10                  mean angle=    0
angles=90 180 270 360          mean angle=  -90
angles=10 20 30                mean angle=   20

Note that with the increase in decimal digit precision, the 2^nd mean angle changed dramatically from an earlier result.

Ring

Project : Averages/Mean angle

load "stdlib.ring" decimals(6) pi = 3.1415926535897 angles = [350,10] see meanangle(angles, len(angles)) + nl angles = [90,180,270,360] see meanangle(angles, len(angles)) + nl angles = [10,20,30] see meanangle(angles, len(angles)) + nl

func meanangle(angles, n)

       sumsin = 0
       sumcos = 0
       for i = 1 to n
            sumsin = sumsin + sin(angles[i]*pi/180)
            sumcos = sumcos + cos(angles[i]*pi/180)
       next
       return 180/pi*atan3(sumsin, sumcos)

func atan3(y,x)

       if x <= 0
          return sign(y)*pi/2
       ok
       if x>0
          return atan(y/x)
       else
          if y>0 
             return atan(y/x)+pi
          else 
             return atan(y/x)-pi
          ok
       ok

</lang> Output:

-0.000000
-90
20.000000

Ruby

<lang ruby>require 'complex' # Superfluous in Ruby >= 2.0; complex is added to core.

def deg2rad(d)

 d * Math::PI / 180

end

def rad2deg(r)

 r * 180 / Math::PI

end

def mean_angle(deg)

 rad2deg((deg.inject(0) {|z, d| z + Complex.polar(1, deg2rad(d))} / deg.length).arg)

end

[[350, 10], [90, 180, 270, 360], [10, 20, 30]].each {|angles|

 puts "The mean angle of %p is: %f degrees" % [angles, mean_angle(angles)]

}</lang>

Output:

The mean angle of [350, 10] is: -0.000000 degrees
The mean angle of [90, 180, 270, 360] is: -90.000000 degrees
The mean angle of [10, 20, 30] is: 20.000000 degrees

Rust

<lang rust> use std::f64; // the macro is from // http://stackoverflow.com/questions/30856285/assert-eq-with-floating- // point-numbers-and-delta fn mean_angle(angles: &[f64]) -> f64 {

   let length: f64 = angles.len() as f64;
   let cos_mean: f64 = angles.iter().fold(0.0, |sum, i| sum + i.to_radians().cos()) / length;
   let sin_mean: f64 = angles.iter().fold(0.0, |sum, i| sum + i.to_radians().sin()) / length;
   (sin_mean).atan2(cos_mean).to_degrees()

}

fn main() {

   let angles1 = [350.0_f64, 10.0];
   let angles2 = [90.0_f64, 180.0, 270.0, 360.0];
   let angles3 = [10.0_f64, 20.0, 30.0];
   println!("Mean Angle for {:?}  is {:.5} degrees",
            &angles1,
            mean_angle(&angles1));
   println!("Mean Angle for {:?}  is {:.5} degrees",
            &angles2,
            mean_angle(&angles2));
   println!("Mean Angle for {:?}  is {:.5} degrees",
            &angles3,
            mean_angle(&angles3));

}

macro_rules! assert_diff{

   ($x: expr,$y : expr, $diff :expr)=>{
       if ( $x - $y ).abs() > $diff {
           panic!("floating point difference is to big {}", $x - $y );
       }
   }

}

[test]

fn calculate() {

   let angles1 = [350.0_f64, 10.0];
   let angles2 = [90.0_f64, 180.0, 270.0, 360.0];
   let angles3 = [10.0_f64, 20.0, 30.0];
   assert_diff!(0.0, mean_angle(&angles1), 0.001);
   assert_diff!(-90.0, mean_angle(&angles2), 0.001);
   assert_diff!(20.0, mean_angle(&angles3), 0.001);

} </lang>

Scala

Library: Scala

<lang Scala>trait MeanAnglesComputation {

 import scala.math.{Pi, atan2, cos, sin}

 def meanAngle(angles: List[Double], convFactor: Double = 180.0 / Pi) = {
   val sums = angles.foldLeft((.0, .0))((r, c) => {
     val rads = c / convFactor
     (r._1 + sin(rads), r._2 + cos(rads))
   })
   val result = atan2(sums._1, sums._2)
   (result + (if (result.signum == -1) 2 * Pi else 0.0)) * convFactor
 }

}

object MeanAngles extends App with MeanAnglesComputation {

 assert(meanAngle(List(350, 10), 180.0 / math.Pi).round == 360, "Unexpected result with 350, 10")
 assert(meanAngle(List(90, 180, 270, 360)).round == 270, "Unexpected result with 90, 180, 270, 360")
 assert(meanAngle(List(10, 20, 30)).round == 20, "Unexpected result with 10, 20, 30")
 println("Successfully completed without errors.")

}</lang>

Scheme

Translation of: Common Lisp

<lang scheme> (import (srfi 1 lists)) ;; use 'fold' from library

(define (average l)

 (/ (fold + 0 l) (length l)))

(define pi 3.14159265358979323846264338327950288419716939937510582097)

(define (radians a)

 (* pi 1/180 a))

(define (degrees a)

 (* 180 (/ 1 pi) a))

(define (mean-angle angles)

 (let* ((angles (map radians angles))
        (cosines (map cos angles))
        (sines (map sin angles)))
   (degrees (atan (average sines) (average cosines)))))

(for-each (lambda (angles)

           (display "The mean angle of ") (display angles) 
           (display " is ") (display (mean-angle angles)) (newline))
         '((350 10) (90 180 270 360) (10 20 30)))

</lang>

The mean angle of (350 10) is -1.614809932057922E-15
The mean angle of (90 180 270 360) is -90.0
The mean angle of (10 20 30) is 19.999999999999996

Seed7

<lang seed7>$ include "seed7_05.s7i";

 include "float.s7i";
 include "math.s7i";
 include "complex.s7i";

const func float: deg2rad (in float: degree) is

 return degree * PI / 180.0;

const func float: rad2deg (in float: rad) is

 return rad * 180.0 / PI;

const func float: meanAngle (in array float: degrees) is func

 result
   var float: mean is 0.0;
 local
   var float: degree is 0.0;
   var complex: sum is complex.value;
 begin
   for degree range degrees do
     sum +:= polar(1.0, deg2rad(degree));
   end for;
   mean := rad2deg(arg(sum / complex conv length(degrees)));
 end func;

const proc: main is func

 begin
   writeln(meanAngle([] (350.0, 10.0)) digits 4);
   writeln(meanAngle([] (90.0, 180.0, 270.0, 360.0)) digits 4);
   writeln(meanAngle([] (10.0, 20.0, 30.0)) digits 4);

end func;</lang>

Output:

0.0000
90.0000
20.0000

Sidef

<lang ruby>func mean_angle(angles) {

   atan2(
       Math.avg(angles.map{ .deg2rad.sin }...),
       Math.avg(angles.map{ .deg2rad.cos }...),
   ) -> rad2deg

}

[[350,10], [90,180,270,360], [10,20,30]].each { |angles|

 say "The mean angle of #{angles.dump} is: #{ '%.2f' % mean_angle(angles)} degrees"

}</lang>

Output:

The mean angle of [350, 10] is: 0.00 degrees
The mean angle of [90, 180, 270, 360] is: -90.00 degrees
The mean angle of [10, 20, 30] is: 20.00 degrees

Stata

<lang stata>mata function meanangle(a) { return(arg(sum(exp(C(0,a))))) }

deg=pi()/180

meanangle((350,10)*deg)/deg

 -1.61481e-15

meanangle((90,180,270,360)*deg)/deg

meanangle((10,20,30)*deg)/deg

end</lang>

Swift

<lang swift>import Foundation

@inlinable public func d2r<T: FloatingPoint>(_ f: T) -> T { f * .pi / 180 } @inlinable public func r2d<T: FloatingPoint>(_ f: T) -> T { f * 180 / .pi }

public func meanOfAngles(_ angles: [Double]) -> Double {

 let cInv = 1 / Double(angles.count)
 let (s, c) =
   angles.lazy
     .map(d2r)
     .map({ (sin($0), cos($0)) })
     .reduce(into: (0.0, 0.0), { $0.0 += $1.0; $0.1 += $1.1 })

 return r2d(atan2(cInv * s, cInv * c))

}

let fmt = { String(format: "%lf", $0) }

print("Mean of angles (350, 10) => \(fmt(meanOfAngles([350, 10])))") print("Mean of angles (90, 180, 270, 360) => \(fmt(meanOfAngles([90, 180, 270, 360])))") print("Mean of angles (10, 20, 30) => \(fmt(meanOfAngles([10, 20, 30])))")</lang>

Output:

Mean of angles (350, 10) => -0.000000
Mean of angles (90, 180, 270, 360) => -90.000000
Mean of angles (10, 20, 30) => 20.000000

Tcl

<lang tcl>proc meanAngle {angles} {

   set toRadians [expr {atan2(0,-1) / 180}]
   set sumSin [set sumCos 0.0]
   foreach a $angles {

set sumSin [expr {$sumSin + sin($a * $toRadians)}] set sumCos [expr {$sumCos + cos($a * $toRadians)}]

   }
   # Don't need to divide by counts; atan2() cancels that out
   return [expr {atan2($sumSin, $sumCos) / $toRadians}]

}</lang> Demonstration code: <lang tcl># A little pretty-printer proc printMeanAngle {angles} {

   puts [format "mean angle of \[%s\] = %.2f" \

[join $angles ", "] [meanAngle $angles]] }

printMeanAngle {350 10} printMeanAngle {90 180 270 360} printMeanAngle {10 20 30}</lang>

Output:

mean angle of [350, 10] = -0.00
mean angle of [90, 180, 270, 360] = -90.00
mean angle of [10, 20, 30] = 20.00

Vala

<lang vala>double meanAngle(double[] angles) {

 double y_part = 0.0;
 double x_part = 0.0;
 
 for (int i = 0; i < angles.length; i++) {
   x_part += Math.cos(angles[i] * Math.PI / 180.0);
   y_part += Math.sin(angles[i] * Math.PI / 180.0);
 }

 return Math.atan2(y_part / angles.length, x_part / angles.length) * 180 / Math.PI;

}

void main() {

 double[] angleSet1 = {350.0, 10.0};
 double[] angleSet2 = {90.0, 180.0, 270.0, 360.0};
 double[] angleSet3 = {10.0, 20.0, 30.0};

 print("\nMean Angle for 1st set :  %lf degrees", meanAngle(angleSet1));
 print("\nMean Angle for 2nd set : %lf degrees", meanAngle(angleSet2));
 print("\nMean Angle for 3rd set :  %lf degrees\n", meanAngle(angleSet3));

}</lang>

Output:

Mean Angle for 1st set :  -0.000000 degrees
Mean Angle for 2nd set : -90.000000 degrees
Mean Angle for 3rd set :  20.000000 degrees

VBA

<lang vb>Option Base 1 Private Function mean_angle(angles As Variant) As Double

   Dim sins() As Double, coss() As Double
   ReDim sins(UBound(angles))
   ReDim coss(UBound(angles))
   For i = LBound(angles) To UBound(angles)
       sins(i) = Sin(WorksheetFunction.Radians(angles(i)))
       coss(i) = Cos(WorksheetFunction.Radians(angles(i)))
   Next i
   mean_angle = WorksheetFunction.Degrees( _
       WorksheetFunction.Atan2( _
       WorksheetFunction.sum(coss), _
       WorksheetFunction.sum(sins)))

End Function Public Sub main()

   Debug.Print Format(mean_angle([{350,10}]), "##0")
   Debug.Print Format(mean_angle([{90, 180, 270, 360}]), "##0")
   Debug.Print Format(mean_angle([{10, 20, 30}]), "##0")

End Sub</lang>

Output:

0
-90
20

Visual Basic .NET

Translation of: C#

<lang vbnet>Imports System.Math

Module Module1

   Function MeanAngle(angles As Double()) As Double
       Dim x = angles.Sum(Function(a) Cos(a * PI / 180)) / angles.Length
       Dim y = angles.Sum(Function(a) Sin(a * PI / 180)) / angles.Length
       Return Atan2(y, x) * 180 / PI
   End Function

   Sub Main()
       Dim printMean = Sub(x As Double()) Console.WriteLine("{0:0.###}", MeanAngle(x))
       printMean({350, 10})
       printMean({90, 180, 270, 360})
       printMean({10, 20, 30})
   End Sub

End Module</lang>

Output:

0
-90
20

XPL0

<lang XPL0>include c:\cxpl\codes;

def Pi = 3.14159265358979323846; def D2R = Pi/180.0; \coefficient to convert degrees to radians

func real MeanAng(A); \Return the mean of the given list of angles int A; real X, Y; int I; [X:= 0.0; Y:= 0.0; for I:= 1 to A(0) do

   [X:= X + Cos(D2R*float(A(I)));
    Y:= Y + Sin(D2R*float(A(I)));
   ];

return ATan2(Y,X)/D2R; ];

[Format(5, 15); RlOut(0, MeanAng([2, 350, 10])); CrLf(0); RlOut(0, MeanAng([4, 90, 180, 270, 360])); CrLf(0); RlOut(0, MeanAng([3, 10, 20, 30])); CrLf(0); ]</lang>

Output:

   -0.000000000000003
  -90.000000000000000
   20.000000000000000

The second example is interesting because it computes ATan2(0.,0.), which is undefined in mathematics (like division by zero), but the floating point processor in IBM-PC-type computers defines it as zero. The reason -90 gets displayed instead is due to small rounding errors (and another extra-mathematical concept, -0). In fact almost any angle can result because of slight rounding errors as Y and X both approach zero.

zkl

<lang zkl>fcn meanA(a1,a2,etc){

  as:=vm.arglist.pump(List,"toFloat","toRad");
  n:=as.len();
  (as.apply("sin").sum(0.0)/n)
  .atan2(as.apply("cos").sum(0.0)/n)
  .toDeg()

}</lang>

Output:

zkl: meanA(350,10)
-1.61481e-15
zkl: meanA(90,180,270,360)
-90
meanA(10,20,30)
20