Updated 2015-11-26 11:42:57 by suchenwi

Purpose: Help on the math function 'atan2'.

The call
    [expr {atan2($y, $x)}]

returns the inverse tangent (expressed in radians) of the quantity,
    ($y / $x)

It never divides by zero, and is careful about the quadrant of the result (which conventional atan() can't be, since the division prior to feeding the quotient into atan() would lose information about which, if any, of $x and $y are negative. In practice, this doubles the range of the result.)

RS notes (testing functional imaging): Contrary to the above claim, on Sun computers the call atan2(0,0) raises the error:
 % expr atan2(0,0)
 domain error: argument not in valid range

Fix: check that at least one coordinate is non-zero:
 proc toPolars p {
    foreach {x y} $p break
    # for Sun, we have to make sure atan2 gets no two 0's
    list [expr {hypot($x,$y)}] [expr {$x||$y? atan2($y,$x): 0}]

Arjen Markus I tested this feature in a small C program on an HPUX and a SGI machine with native compilers. The results were quite similar:

  • HPUX reports a Nan and errno = 0
  • SGI reports a value 0.0 and a Domain error.

Tcl 8.3.1 on SGI responds with a domain error as well, so that is quite consistent!

AMG: GNU (glibc) atan2() is defined [1] to return 0 when both arguments are 0.

The most common use of atan2 is in Converting between rectangular and polar co-ordinates.

atan2 is also available as a command in Tclx.

AM (28 april 2008) Interesting feature:

  • atan2(y,x) will give the mathematical angle, that is, the angle with the positive x axis, oriented counter-clockwise
  • atan2(x,y) (so the arguments in just the opposite order) will give the nautical angle, that is, the angle with the positive y-axis, oriented clockwise (or if you like: the angle on the compass)

AMG: When dealing with maps and geospatial data, I greatly prefer to orient north along the positive X axis and east along the positive Y axis. This way atan2(y,x) gives me a compass heading. Well, atan2(y,x)*180/acos(-1) does. :^) I am told this is standard GIS convention.

GWM (28 april 2008) note that Excel uses the reverse convention from C, Fortran, Tcl (and probably most other languages),
 atan2(0,1) = 0; atan2(1,0) = 1.570796 etc.

Have all other spreadsheets been infected by Excel? Does SLK insist on Excel ordered atan2? Take care when translating a formula that works in Excel to other languages (and vice versa).

AMG: OpenOffice.org Calc uses the same backwards convention. Sigh. It stinks when the de facto standard is exactly backwards from the de jure standard. (Maybe I should say "du jour standard", hehehe.)

RS 2015-11-26 I see the same behavior, so what is "backwards"?:
 $  tclsh
 % info pa
 % expr atan2(1,0)
 % expr atan2(0,1)

But while we're at it, the Tcl manpage mathfunc has a bug there:

atan arg

Returns the arc tangent of arg, in the range [-pi/2,pi/2] radians.

atan2 y x

Returns the arc tangent of y/x, in the range [-pi,pi] radians. x and y cannot both be 0. If x is greater than 0, this is equivalent to “atan [expr {y/x}]”.


To be equivalent (which I tested), the range of atan2 is equally -pi/2...pi/2.

DKF: Tcl just uses the same convention as C, because it uses the same function.

AMG: Which explains why it accepts two arguments.