aTan always positive (Bug or intended?)

[corgano]

This thread is about something odd i noticed with AutoIt's aTan function, that caused me a lot of time to track down.

Recently i was writing a program to change GPX files, splitting each set of points into smaller directions. It works by taking the current and next point in the gpx file, calculating the angle between them, and then adding points between them every x meters until it's near enough to the second point. Lather, rinse, repeat.

The problem is sometimes the angle the new points would make would go completely the wrong direction, approximately half the time. Hurm...

Anyways I eventually discovered that aTan would always return a positive result. This is fine for calculating SLOPE, but for calculating ANGLE this was very annoying. I eventually came up with a work-around that works, but i was wondering if this behavior was intended or not. Example script here, will show the buggy behavior by default. Uncomment line 28 to see the expected behavior

#include <GUIConstantsEx.au3>

$Form1 = GUICreate("Form1", 318, 288, 192, 124)
$Radio1 = GUICtrlCreateRadio("", 160, 120, 17, 17)
$Label1 = GUICtrlCreateLabel("Label1", 16, 232, 316, 77)
GUISetState(@SW_SHOW)

Local $s, $olds, $hwnd=WinGetHandle(ControlGetHandle($Form1,"",$Radio1))

While 1
    $nMsg = GUIGetMsg()
    Switch $nMsg
        Case $GUI_EVENT_CLOSE
            Exit

    EndSwitch

    $pos = WinGetPos($hwnd)
    If @error then ContinueLoop
    $pos2 = MouseGetPos()

    $dx = $pos2[0] - $pos[0]
    $dy = $pos2[1] - $pos[1]

    $angle = ATan($dy/$dx)

    ;this is the fix, uncomment to see expected (desired) behavior
;~  If $dx < 0 Then $angle -= 3.14159265359

    $s = "Pos 1 = "&$pos[0]&" , "&$pos[0]&"       Angle= "&($angle*57.295779513)+90&@CRLF&"Pos 2 = "&$pos2[0]&" , "&$pos2[1]&@CRLF&"DX = "&$dx&"         DY = "&$dy
    If $s <> $olds Then
        $olds = $s
        GUICtrlSetData($Label1, $s)
    EndIf
WEnd

 

[BrewManNH]

I don't believe Arctangent can ever return a negative number.

[jchd]

It is common to see inverse trig functions return result in [0, 2π[. Some softwares use the range ]-π/2, π/2[ for result of ArcTan but these ranges are only conventions. Angles in radian are always Mod(2π).

Yes of course ArcTan can return something in ]-π/2, π/2[ or even [-π/2, π/2] if the underlying library deals with infinities.

[BrewManNH]

That made it even less clear than before.

[jchd]

If you consider the complex plane (or the affine plane) instead of the traditionnal trigonometric circle and conventions, you can define the arctangent of a complex number (or a pair of points in the affine plane). Then the arctangent can be mapped to any range of span 2π, the origin of the interval being your choice. That's what the OP seems to expect: angles of vectors u, v, w and t wrt the x axis to be resp. > 0, > 0, < 0, < 0 by using the range [-π, π].

Trig C.pdf

Trig trad.pdf

In the picture above the traditionnal trigonometric (real) arctangent of points A and C are drawn, resp. angles β and α. (Excuse the weird naming sequence).

β and α can be anywhere in ]-π/2, π/2[ if the line (t) maps to ℝ or even in [-π/2, π/2] if the map is to ℝ ∪ {-∞, +∞}.

 

Edited December 23, 2016 by jchd
Images didn't make it