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]

I don't know math, but you can identify the rogues with tagging -- its usually the red "one"s for me, so I didn't expand on tags two and three

#! perl -slw
use strict;
use Tk;


our $W  = our $H =  600; ## 2012-07-31-18:49:09

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

sub curvedArrow {
    my $tag = 'thetag  my( $cnv, $x1, $y1, $x2, $y2, $color ) = ( ' . 
+join(', ', @_  ).')';
    my( $cnv, $x1, $y1, $x2, $y2, $color ) = @_;
    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;
    
    my @one = (
        $xc - $rad, $yc - $rad, $xc + $rad, $yc + $rad,
        -style => 'arc', -start => $a1, -extent => $a2 - $a1,
    );
    $cnv->createArc(
        @one,
        -outline=> $color,
        -tags => ["one $tag ", "one @one"],
    );

    my $r2 = $rad / 15;
    $cnv->createArc(
        $x1-$r2, $y1-$r2, $x1+$r2, $y1+$r2,
        -start=>$a1-77, -extent=>($a2-$a1)/2, -fill=> $color,
        -tags => ["two $tag "],
    );
    $cnv->createArc(
        $x2-$r2, $y2-$r2, $x2+$r2, $y2+$r2,
        -start=> ( $a2+107 ) %360, -extent=>($a2-$a1)/2, -fill=> $colo
+r,
        -tags => ["three $tag "],
    );
    return $xc, $yc;
}

our $W //= 1000;
our $H //=  800;

my $mw = new MainWindow(-title => 'Test');
my $canvas = $mw->Canvas(-width  => $W, -height => $H )->pack;

for( 1 .. 10 ) {
    my( $x1, $y1 ) = ( int( rand( $W ) ), int( rand( $H ) ) );
    $canvas->createLine( $x1-5, $y1, $x1+5, $y1, -fill => 'blue' );
    $canvas->createLine( $x1, $y1-5, $x1, $y1+5, -fill => 'blue' );

    my( $x2, $y2 ) = ( int( rand $W ), int( rand $H ) );
    $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, 'black'
+ );
    $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, 'red' )
+;
    $canvas->createLine( $xc-5, $yc, $xc+5, $yc, -fill => 'red' );
    $canvas->createLine( $xc, $yc-5, $xc, $yc+5, -fill => 'red' );
}

#~ $canvas->bind( '<<1>>', \&tagsUnderMouse );
#~ $canvas->bind( '<Button-1>', \&tagsUnderMouse );
#~ $canvas->bind( '<ButtonPress-1>' , \&tagsUnderMouse );
$mw->bind( ref( $canvas ), '<ButtonPress-1>' , \&tagsUnderMouse );

MainLoop;

sub tagsUnderMouse {
    warn "@_";
#~     my $canvas = $Tk::event->W;
    my $x = $Tk::event->X;
    my $y = $Tk::event->Y;
    my( $canvas  ) = @_;
    use Data::Dump;
    my @closest = $canvas->find( closest => $x, $y ) ;
#~     my %ccget = map { $_ => [ $canvas->itemconfigure($_) ]  } @clos
+est;
    my %ccget = map { $_ => [ $canvas->itemcget($_, '-tags') ]  } @clo
+sest;
    dd [ "$canvas", \@closest , \%ccget ];
    
}
[download]

Thanks for the reply and code.

Even picking out which ones produced the affect didn't make things clear to me; but thinking about selection forced me to come up with a better -- ordered rather than random -- testcase. And that crystalised things a brought me to a solution. Thanks.

---

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The start of some sanity?