Seems I was misunderstanding the interactions between atan2's +-Pi returns and Tk::Canvas' +-360° inputs. I was trying too hard.

The results of the simplification look like this

I'll update the original post, but here is the modified code and updated test

#! perl -slw
use strict;
use Tk;

use constant PI => 3.1415926535897932384626433832795;
use constant RAD => 180 / PI;

sub curvedArrow {
    my( $cnv, $x1, $y1, $x2, $y2, $color ) = @_;
    ## set the radius to the distance between p1 & p2
    my $rad = sqrt( abs( $x1 - $x2 )**2 + abs( $y1 - $y2 )**2 );
    my $q = sqrt( ( $x2 - $x1 )**2 + ( $y2 - $y1 )**2 );
    my( $x3, $y3 ) = ( ( $x1 + $x2 ) / 2, ( $y1 + $y2 ) / 2 );

    my $xc = $x3 + sqrt( $rad**2 - ( $q / 2 )**2 ) * ( $y1 - $y2 ) / $
+q;
    my $yc = $y3 + sqrt( $rad**2 - ( $q / 2 )**2 ) * ( $x2 - $x1 ) / $
+q;

    my $a1 = atan2( ( $yc - $y1 ), -( $xc - $x1 ) ) * RAD;
    my $a2 = atan2( ( $yc - $y2 ), -( $xc - $x2 ) ) * RAD;


    $cnv->createArc(
        $xc - $rad, $yc - $rad, $xc + $rad, $yc + $rad,
        -style => 'arc', -start => $a1, -extent => -60,
        -outline=> $color
    );

    my $r2 = $rad / 15;

    $cnv->createArc(
        $x1-$r2, $y1-$r2, $x1+$r2, $y1+$r2,
        -start=>$a1-77, -extent=> -30, -fill=> $color
    );

    $cnv->createArc(
        $x2-$r2, $y2-$r2, $x2+$r2, $y2+$r2,
        -start=> ( $a2+107 ) %360, -extent=> -30, -fill=> $color
    );
    return $xc, $yc;
}

our $W //= 1000;
our $H //= 1000;

my $mw = new MainWindow(-title => 'Test');
my $canvas = $mw->Canvas(-width  => $W, -height => $H )->pack;
for my $radius ( 500, 100 ) {
    for( my $a = 0; $a < 180; $a += 10 ) {
        my( $x, $y ) = ( sin( $a / RAD )*$radius, cos( $a / RAD )*$rad
+ius );
        my( $x1, $y1, $x2, $y2 ) = ( 500-$x, 500-$y, 500+$x, 500+$y,  
+);

        $canvas->createLine( $x1-5, $y1, $x1+5, $y1, -fill => 'blue' )
+;
        $canvas->createLine( $x1, $y1-5, $x1, $y1+5, -fill => 'blue' )
+;
        $canvas->createLine( $x2-5, $y2, $x2+5, $y2, -fill => 'green' 
+);
        $canvas->createLine( $x2, $y2-5, $x2, $y2+5, -fill => 'green' 
+);

        my( $xc, $yc ) = curvedArrow( $canvas, $x1, $y1, $x2, $y2, 'bl
+ack' );
        $canvas->createLine( $xc-5, $yc, $xc+5, $yc, -fill => 'red' );
        $canvas->createLine( $xc, $yc-5, $xc, $yc+5, -fill => 'red' );

          ( $xc, $yc ) = curvedArrow( $canvas, $x2, $y2, $x1, $y1, 're
+d' );
        $canvas->createLine( $xc-5, $yc, $xc+5, $yc, -fill => 'red' );
        $canvas->createLine( $xc, $yc-5, $xc, $yc+5, -fill => 'red' );
    }
}

MainLoop;
[download]