Here is a PDL version, that still has loops (gasp) but uses Judd Taylor's excellent PDL binding for GD. It goes about 5 times quicker than the version in the OP (edit to note: actually, for the fancy -M=4 -P=1.5 -S=200 -L=10 one, the OP takes about 70s, this PDL one takes 7s, so 10x quicker): 
#! perl -sw
use strict;
use PDL;
use PDL::Constants qw(PI DEGRAD);
use PDL::IO::GD;
use constant FNAME => 'CGScrewPDL';
use constant {
    COS30   => 0.86602540378443864676372317075294,
    TAN30   => 0.57735026918962576450914878050196,
};
use enum qw[ X Y A ];

## Construct thread profile segment
sub constructThreadProfile {
    my @points;
    my( $dia, $pitch, $yInc ) = @_;
    my $H =  $pitch * COS30;
    # draw flat crest
    my $x = $dia / 2;
    my $y = -$yInc;
    my $yLim = $pitch / 16;
    push @points, [ $x, $y += $yInc, 1 ] while $y < $yLim;

    ## draw upper 30deg flank.
    my $xd = $yInc / TAN30;
    $yLim = $y + 5 / 16 * $pitch;
    push @points, [ $x -= $xd, $y += $yInc, 0.5 ] while $y < $yLim;

    ## draw root radius
    $yLim = $y + 4 / 16 * $pitch;
    my( $cx, $cy, $r ) = ( ( $dia/2 - 7/8*$H + $H/3 ), $pitch / 2, $H 
+/ 6 );
    while( $y < $yLim ) {
        my $dy = $cy - $y;
        my $dx = sqrt( $r**2 - $dy**2 );
        push @points, [ $cx - $dx, $y, $dx / $r ];
        $y += $yInc;
    }
    $y -= $yInc;
    ## draw lower 30deg flank
    $yLim = $y + 5 / 16 * $pitch;
    push @points, [ $x += $xd, $y +=  $yInc, - 0.5 ] while $y < $yLim;

    $yLim = $pitch;
    push @points, [ $x, $y += $yInc, 1 ] while $y < $yLim;
    return pdl \@points;
}

our $M //= 10;
our $P //= 1.5;
our $L //= 2;
our $S //= 100;
my $fname = sprintf "%sM%.2fxP%.2fxL%.2fxS%d.png", FNAME, $M, $P, $L, 
+$S;

my $profilePDL = constructThreadProfile( $M, $P, 1 / ( 10* $S ) );

my( $w, $h ) = ( $M * $S + 200, int( $L / $P + 1 ) * $P * $S + 200 );
my $xc = $w / 2;

my $imPDL = zeroes(byte, $w, $h) + 128; # no RGB as all same, dummy at
+ end

my $yTrans = $P / 360;
my $maxRad = $M * $S / 2;
my $yOff = 100;
my $turns = sequence(indx, $L / $P);
for my $turn ( $turns->list ) {
    for my $p ( map $_/2, -$maxRad*2 .. $maxRad*2 ) {
        my $rot = atan2( sqrt( $maxRad**2 - $p**2 ), $p );
        my $this_profile = $profilePDL + [0, $yTrans * DEGRAD * $rot, 
+0];
        $this_profile *= [cos($rot), 1, cos( PI/2 - $rot)];
        my ($newX, $newY) = ($xc + $this_profile->slice([X,0,0]) * $S,
+ $yOff + $this_profile->slice([Y,0,0]) * $S);
        my $color = $this_profile->slice([A,0,0])->abs * 256 + (100 - 
+$newY);
        $imPDL->indexND( $newX->cat($newY)->mv(-1,0) ) .= $color;
    }
    $yOff += $P * $S;
}

my $im = PDL::IO::GD->new({pdl=> $imPDL->dummy(2,3)});
$im->String( gdFontGetSmall, 0,0, $fname, 0 );
$im->write_Png($fname);

[download]
You can see I eliminated the "Z" enum and entry in the "profile" and "xformPoints" (inlined as a couple of PDL operations here). That made the original run about 10% faster. Top tip for programmers! If you don't want to waste 45 minutes cursing your existence in confusion, if you eliminate part of a fixed-size tuple, make sure you eliminate it in all places. Unexpected results will happen otherwise!

In reply to Re: Reflections on graphic screwing! by etj
in thread Reflections on graphic screwing! by BrowserUk

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