# Copyright 2011, 2012, 2013 Kevin Ryde
# This file is part of Math-PlanePath.
#
# Math-PlanePath is free software; you can redistribute it and/or modify it
# under the terms of the GNU General Public License as published by the Free
# Software Foundation; either version 3, or (at your option) any later
# version.
#
# Math-PlanePath is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
# or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
# for more details.
#
# You should have received a copy of the GNU General Public License along
# with Math-PlanePath. If not, see .
# Also possible would be circle involute spiral, unrolling string around
# centre of circumference 1, but is only very slightly different radius from
# an Archimedean spiral.
package Math::PlanePath::ArchimedeanChords;
use 5.004;
use strict;
use Math::Libm 'hypot', 'asinh';
use POSIX 'ceil';
#use List::Util 'min', 'max';
*min = \&Math::PlanePath::_min;
*max = \&Math::PlanePath::_max;
use vars '$VERSION', '@ISA';
$VERSION = 114;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
use Math::PlanePath::Base::Generic
'is_infinite',
'round_nearest';
use Math::PlanePath::MultipleRings;
# uncomment this to run the ### lines
# use Smart::Comments;
use constant figure => 'circle';
use constant n_start => 0;
use constant gcdxy_maximum => 1;
use constant dx_minimum => -1; # infimum when straight
use constant dx_maximum => 1; # at N=0
use constant dy_minimum => -1;
use constant dy_maximum => 1;
use constant dsumxy_minimum => -sqrt(2); # supremum when diagonal
use constant dsumxy_maximum => sqrt(2);
use constant ddiffxy_minimum => -sqrt(2); # supremum when diagonal
use constant ddiffxy_maximum => sqrt(2);
#------------------------------------------------------------------------------
use constant 1.02 _PI => 2*atan2(1,0);
# Starting at polar angle position t in radians,
#
# r = t / 2pi
#
# x = r * cos(t) = t * cos(t) / 2pi
# y = r * sin(t) = t * sin(t) / 2pi
#
# Want a polar angle amount u to move by a chord distance of 1. Hypot
# square distance from t to t+u is
#
# dist(u) = ( (t+u)/2pi*cos(t+u) - t/2pi*cos(t) )^2 # X
# + ( (t+u)/2pi*sin(t+u) - t/2pi*sin(t) )^2 # Y
# = [ (t+u)^2*cos^2(t+u) - 2*(t+u)*t*cos(t+u)*cos(t) + t^2*cos^2(t)
# + (t+u)^2*sin^2(t+u) - 2*(t+u)*t*sin(t+u)*sin(t) + t^2*sin^2(t)
# ] / (4*pi^2)
#
# and from sin^2 + cos^2 = 1
# and addition cosA*cosB + sinA*sinB = cos(A-B)
#
# = [ (t+u)^2 - 2*(t+u)*t*cos((t+u)-t) + t^2 ] /4pi^2
# = [ (t+u)^2 + t^2 - 2*t*(t+u)*cos(u) ] / (4*pi^2)
#
# then double angle cos(u) = 1 - 2*sin^2(u/2) to go to the sine since if u
# is small then cos(u) near 1.0 might lose accuracy
#
# dist(u) = [(t+u)^2 + t^2 - 2*t*(t+u)*(1 - 2*sin^2(u/2))] / (4*pi^2)
# = [(t+u)^2 + t^2 - 2*t*(t+u) + 2*t*(t+u)*2*sin^2(u/2)] / (4*pi^2)
# = [((t+u) - t)^2 + 4*t*(t+u)*sin^2(u/2)] / (4*pi^2)
# = [ u^2 + 4*t*(t+u)*sin^2(u/2) ] / (4*pi^2)
#
# Seek d(u) = 1 by letting f(u)=4*pi^2*(d(u)-1) and seeking f(u)=0
#
# f(u) = u^2 + 4*t*(t+u)*sin^2(u/2) - 4*pi^2
#
# Derivative f'(u) for the slope, starting from the cosine form,
#
# f(u) = (t+u)^2 + t^2 - 2*t*(t+u)*cos(u) - 4*pi^2
#
# f'(u) = 2*(t+u) - 2*t*[ cos(u) - (t+u)*sin(u) ]
# = 2*(t+u) - 2*t*[ 1 - 2*sin^2(u/2) - (t+u)*sin(u) ]
# = 2*t + 2*u - 2*t + 2*t*2*sin^2(u/2) + 2*t*(t+u)*sin(u)
# = 2*[ u + 2*t*sin^2(u/2) + t*(t+u)*sin(u) ]
# = 2*[ u + t * [2*sin^2(u/2) + (t+u)*sin(u) ] ]
#
# Newton's method
# */ <- f(x) high
# */|
# * / |
# * / |
# ---------*------------------
# +---+ <- subtract
#
# f(x) / sub = f'(x)
# sub = f(x) / f'(x)
#
#
# _chord_angle_inc() takes $t is a polar angle around the Archimedean spiral.
# Returns an increment polar angle $u which may be added to $t to move around
# the spiral by a chord length 1 unit.
#
# The loop is Newton's method, $f=f(u), $slope=f'(u) so $u-$f/$slope is a
# better $u, ie. f($u) closer to 0. Stop when the subtract becomes small,
# usually only about 3 iterations.
#
sub _chord_angle_inc {
my ($t) = @_;
# ### _chord_angle_inc(): $t
my $u = 2*_PI/$t; # estimate
foreach (0 .. 10) {
my $shu = sin($u/2); $shu *= $shu; # sin^2(u/2)
my $tu = ($t+$u);
my $f = $u*$u + 4*$t*$tu*$shu - 4*_PI*_PI;
my $slope = 2 * ( $u + $t*(2*$shu + $tu*sin($u)));
# unsimplified versions ...
# $f = ($t+$u)**2 + $t**2 - 2*$t*($t+$u)*cos($u) - 4*_PI*_PI;
# $slope = 2*($t+$u) - 2*$t*( cos($u) - ($t+$u)*sin($u) );
my $sub = $f/$slope;
$u -= $sub;
# printf ("f=%.6f slope=%.6f sub=%.20f u=%.6f\n", $f, $slope, $sub, $u);
last if (abs($sub) < 1e-15);
}
# printf ("return u=%.6f\n", $u);
return $u;
}
use constant 1.02; # for leading underscore
use constant _SAVE => 500;
my @save_n = (1);
my @save_t = (2*_PI);
my $next_save = $save_n[0] + _SAVE;
sub new {
### ArchimedeanChords new() ...
return shift->SUPER::new (i => $save_n[0],
t => $save_t[0],
@_);
}
sub n_to_xy {
my ($self, $n) = @_;
if ($n < 0) { return; }
if (is_infinite($n)) { return ($n,$n); }
if ($n <= 1) {
return ($n, 0); # exactly Y=0
}
{
# ENHANCE-ME: look at the N+1 position for the frac directly, without
# the full call for N+1
my $int = int($n);
if ($n != $int) {
my $frac = $n - $int; # inherit possible BigFloat/BigRat
my ($x1,$y1) = $self->n_to_xy($int);
my ($x2,$y2) = $self->n_to_xy($int+1);
my $dx = $x2-$x1;
my $dy = $y2-$y1;
return ($frac*$dx + $x1, $frac*$dy + $y1);
}
}
my $i = $self->{'i'};
my $t = $self->{'t'};
if ($i > $n) {
my $pos = min ($#save_n, int (($n - $save_n[0]) / _SAVE));
$i = $save_n[$pos];
$t = $save_t[$pos];
### resume: "$i t=$t"
}
while ($i < $n) {
$t += _chord_angle_inc($t);
if (++$i == $next_save) {
push @save_n, $i;
push @save_t, $t;
$next_save += _SAVE;
}
}
$self->{'i'} = $i;
$self->{'t'} = $t;
my $r = $t * (1 / (2*_PI));
return ($r*cos($t),
$r*sin($t));
}
sub _xy_to_nearest_r {
my ($x, $y) = @_;
my $frac = Math::PlanePath::MultipleRings::_xy_to_angle_frac($x,$y);
### assert: 0 <= $frac && $frac < 1
# if $frac > 0.5 then 0.5-$frac is negative and int() rounds towards zero
# giving $r==$frac
return int(hypot($x,$y) + 0.5 - $frac) + $frac;
}
sub xy_to_n {
my ($self, $x, $y) = @_;
### ArchimedeanChords xy_to_n(): "$x, $y"
my $r = _xy_to_nearest_r($x,$y);
my $r_limit = 1.001 * $r;
### hypot: hypot($x,$y)
### $r
### $r_limit
### save_t: "end index=$#save_t save_t[0]=".($save_t[0]//'undef')
if (is_infinite($r_limit)) {
### infinite range, r inf or too big ...
return undef;
}
my $theta = 0.999 * 2*_PI*$r;
my $n_lo = 0;
foreach my $i (1 .. $#save_t) {
if ($save_t[$i] > $theta) {
$n_lo = $save_n[$i-1];
if ($n_lo == 1) { $n_lo = 0; } # for finding X=0,Y=0
last;
}
}
### $n_lo
# loop with for(;;) since $n_lo..$n_hi limited to IV range
for (my $n = $n_lo; ; $n += 1) {
my ($nx,$ny) = $self->n_to_xy($n);
# #### $n
# #### $nx
# #### $ny
# #### hypot: hypot ($x-$nx,$y-$ny)
if (hypot($x-$nx,$y-$ny) <= 0.5) {
### hypot in range ...
return $n;
}
if (hypot($nx,$ny) >= $r_limit) {
last;
}
}
### n not found ...
return undef;
}
# int (max (0, int(_PI*$r2) - 4*$r));
#
# my $r2 = $r * $r;
# my $n_lo = int (max (0, int(_PI*$r2) - 4*$r));
# my $n_hi = $n_lo + 7*$r + 2;
# ### $r2
# $n_lo == $n_lo-1 ||
# x,y has radius hypot(x,y), then want the next higher spiral arc which is r
# >= hypot(x,y)+0.5, with the 0.5 being the width of the circle figure on
# the spiral.
#
# The polar angle of x,y is a=atan2(y,x) and frac=a/2pi is the extra away
# from an integer radius for the spiral. So seek integer k with k+a/2pi >=
# h with h=hypot(x,y)+0.5.
#
# k + a/2pi >= h
# k >= h-a/2pi
# k = ceil(h-a/2pi)
# = ceil(hypot(x,y) + 0.5 - atan2(y,x)/2pi)
#
#
# circle radius i has circumference 2*pi*i and at most that many N on it
# rectangle corner at radius Rcorner = hypot(x,y)
#
# sum i=1 to i=Rlimit of 2*pi*i = 2*pi/2 * Rlimit*(Rlimit+1)
# = pi * Rlimit*(Rlimit+1)
# is an upper bound, though a fairly slack one
#
#
# cf. arc length along the spiral r=a*theta with a=1/2pi
# arclength = (1/2) * a * (theta*sqrt(1+theta^2) + asinh(theta))
# = (1/4*pi) * (theta*sqrt(1+theta^2) + asinh(theta))
# and theta = 2*pi*r
# = (1/4*pi) * (4*pi^2*r^2 * sqrt(1+1/theta^2) + asinh(theta))
# = pi * (r^2 * sqrt(1+1/r^2) + asinh(theta)/(4*pi^2))
#
# and to compare to the circles formula
#
# = pi * (r*(r+1) * r/(r+1) * sqrt(1+1/r^2)
# + asinh(theta)/(4*pi^2))
#
# so it's smaller hence better upper bound. Only a little smaller than the
# squaring once get to 100 loops or so.
#
#
# not exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
### rect_to_n_range() ...
my $rhi = 0;
foreach my $x ($x1, $x2) {
foreach my $y ($y1, $y2) {
my $frac = atan2($y,$x) / (2*_PI); # -0.5 <= $frac <= 0.5
$frac += ($frac < 0); # 0 <= $frac < 1
$rhi = max ($rhi, ceil(hypot($x,$y)+0.5 - $frac) + $frac);
}
}
### $rhi
# arc length pi * (r^2 * sqrt(1+1/r^2) + asinh(theta)/(4*pi^2))
# = pi*r^2*sqrt(1+1/r^2) + asinh(theta)/4pi
my $rhi2 = $rhi*$rhi;
return (0,
ceil (_PI * $rhi2 * sqrt(1+1/$rhi2)
+ asinh(2*_PI*$rhi) / (4*_PI)));
# # each loop begins at N = pi*k^2 - 2 or thereabouts
# return (0,
# int(_PI*$rhi*($rhi+1) + 1));
}
1;
__END__
# my $slope = 2*($t + (-$c1-$s1*$t)*cos($t) + ($c1*$t-$s1)*sin($t));
# my $dist = ( ($t*cos($t) - $c1) ** 2
# + ($t*sin($t) - $s1) ** 2
# - 4*_PI*_PI );
# my $slope = (2*($t*cos($t)-$c1)*(cos($t) - $t*sin($t))
# + 2*($t*sin($t)-$s1)*(sin($t) + $t*cos($t)));
# my $c1 = $t1 * cos($t1);
# my $s1 = $t1 * sin($t1);
# my $c1_2 = $c1*2;
# my $s1_2 = $s1*2;
# my $t = $t1 + 2*_PI/$t1; # estimate
# my $ct = cos($t);
# my $st = sin($t);
# my $dist = (($t - $ct*$c1_2 - $st*$s1_2) * $t + $t1sqm);
# my $slope = 2 * (($t*$ct - $c1) * ($ct - $t*$st)
# + ($t*$st - $s1) * ($st + $t*$ct));
#
# my $sub = $dist/$slope;
# $t -= $sub;
# use constant _A => 1 / (2*_PI);
# my @radius = (0, 1);
# # my $theta = _inverse($n);
# # my $r = _A * $theta;
# # return ($r * cos($theta),
# # $r * sin($theta));
#
#
# # $n = floor($n);
# #
# # for (my $i = scalar(@radius); $i <= $n; $i++) {
# # my $prev = $radius[$i-1];
# # # my $step = 8 * asin (.25/4 / $prev) / pi();
# # my $step = (.5 / pi()) / $prev;
# # $radius[$i] = $prev + $step;
# # }
# #
# # my $r = $radius[$n];
# # my $theta = 2 * pi() * ($r - int($r)); # radians 0 to 2*pi
# # return ($r * cos($theta),
# # $r * sin($theta));
# sub _arc_length {
# my ($theta) = @_;
# my $hyp = hypot(1,$theta);
# return 0.5 * _A * ($theta*$hyp + asinh($theta));
# }
#
# # upper bound $hyp >= $theta
# # a/2 * $theta * $theta
# # so theta = sqrt (2/_A * $length)
# #
# # lower bound $hyp <= $theta+1, log(x)<=x
# # length <= a/2 * ($theta * ($theta+1))^2
# # 2/a * length <= (2*$theta * $theta)^2
# # so theta >= sqrt (1/(2*_A) * $length)
# #
# sub _inverse {
# my ($length) = @_;
# my $lo_theta = sqrt (1/(2*_A) * $length);
# my $hi_theta = sqrt ((2/_A) * $length);
# my $lo_length = _arc_length($lo_theta);
# my $hi_length = _arc_length($hi_theta);
# #### $length
# #### $lo_theta
# #### $hi_theta
# #### $lo_length
# #### $hi_length
# die if $lo_length > $length;
# die if $hi_length < $length;
# my $m_theta;
# for (;;) {
# $m_theta = ($hi_theta + $lo_theta) / 2;
# last if ($hi_length - $lo_length) < 0.000001;
# my $m_length = _arc_length($m_theta);
# if ($m_length < $length) {
# $lo_theta = $m_theta;
# $lo_length = $m_length;
# } else {
# $hi_theta = $m_theta;
# $hi_length = $m_length;
# }
# }
# return $m_theta;
# }
=for stopwords Archimedean Ryde ie cartesian Math-PlanePath arcsin
=head1 NAME
Math::PlanePath::ArchimedeanChords -- radial spiral chords
=head1 SYNOPSIS
use Math::PlanePath::ArchimedeanChords;
my $path = Math::PlanePath::ArchimedeanChords->new;
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
This path puts points at unit chord steps along an Archimedean spiral. The
spiral goes outwards by 1 unit each revolution and the points are spaced 1
apart.
R = theta/(2*pi)
The result is roughly
31
32 30 ... 3
33 29
14
34 15 13 28 50 2
12
16 3
35 2 27 49 1
4 11
17
36 5 0 1 26 48 <- Y=0
10
18
37 6 25 47 -1
9
19 7 8 24 46
38 -2
20 23
39 21 22 45
-3
40 44
41 42 43
^
-3 -2 -1 X=0 1 2 3 4
X,Y positions returned are fractional. Each revolution is about 2*pi longer
than the previous, so the effect is a kind of 6.28 increment looping.
Because the spacing is by unit chords, adjacent unit circles centred on each
N position touch but don't overlap. The spiral spacing of 1 unit per
revolution means they don't overlap radially either.
The unit chords here are a little like the C. But the
C goes by unit steps at a fixed right-angle and
approximates an Archimedean spiral (of 3.14 radial spacing). Whereas this
C is an actual Archimedean spiral (of radial spacing 1),
with unit steps angling along that.
=head1 FUNCTIONS
See L for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::ArchimedeanChords-Enew ()>
Create and return a new path object.
=item C<($x,$y) = $path-En_to_xy ($n)>
Return the X,Y coordinates of point number C<$n> on the path.
C<$n> can be any value C<$n E= 0> and fractions give positions on the
chord between the integer points (ie. straight line between the points).
C<$n==0> is the origin 0,0.
For C<$n < 0> the return is an empty list, it being considered there are no
negative points in the spiral.
=item C<$n = $path-Exy_to_n ($x,$y)>
Return an integer point number for coordinates C<$x,$y>. Each integer N
is considered the centre of a circle of diameter 1 and an C<$x,$y> within
that circle returns N.
The unit spacing of the spiral means those circles don't overlap, but they
also don't cover the plane and if C<$x,$y> is not within one then the
return is C.
The current implementation is a bit slow.
=item C<$n = $path-En_start ()>
Return 0, the first C<$n> on the path.
=item C<$str = $path-Efigure ()>
Return "circle".
=back
=head1 FORMULAS
=head2 N to X,Y
The current code keeps a position as a polar angle t and calculates an
increment u needed to move along by a unit chord. If dist(u) is the
straight-line distance between t and t+u, then squared is the hypotenuse
dist^2(u) = ((t+u)/2pi*cos(t+u) - t/2pi*cos(t))^2 # X
+ ((t+u)/2pi*sin(t+u) - t/2pi*sin(t))^2 # Y
which simplifies to
dist^2(u) = [ (t+u)^2 + t^2 - 2*t*(t+u)*cos(u) ] / (4*pi^2)
Switch from cos to sin using the half angle cos(u) = 1 - 2*sin^2(u/2) in
case if u is small then the cos(u) near 1.0 might lose floating point
accuracy, and also as a slight simplification,
dist^2(u) = [ u^2 + 4*t*(t+u)*sin^2(u/2) ] / (4*pi^2)
Then want the u which has dist(u)=1 for a unit chord. The u*sin(u) part
probably doesn't have a good closed form inverse, so the current code is a
Newton/Raphson iteration on f(u) = dist^2(u)-1, seeking f(u)=0
f(u) = u^2 + 4*t*(t+u)*sin^2(u/2) - 4*pi^2
Derivative f'(u) for the slope from the cos form is
f'(u) = 2*(t+u) - 2*t*[ cos(u) - (t+u)*sin(u) ]
And again switching from cos to sin in case u is small,
f'(u) = 2*[ u + t*[2*sin^2(u/2) + (t+u)*sin(u)] ]
=head2 X,Y to N
A given x,y point is at a fraction of a revolution
frac = atan2(y,x) / 2pi # -.5 <= frac <= .5
frac += (frac < 0) # 0 <= frac < 1
And the nearest spiral arm measured radially from x,y is then
r = int(hypot(x,y) + .5 - frac) + frac
Perl's C is the same as the C library and gives -pi E= angle
E= pi, hence allowing for fracE0. It may also be "unspecified" for
x=0,y=0, and give +/-pi for x=negzero, which has to be a special case so 0,0
gives r=0. The C rounds towards zero, so frac>.5 ends up as r=0.
So the N point just before or after that spiral position may cover the x,y,
but how many N chords it takes to get around to there is 's not so easily
calculated.
The current code looks in saved C positions for an N below the
target, and searches up from there until past the target and thus not
covering x,y. With C points saved 500 apart this means searching
somewhere between 1 and 500 points.
One possibility for calculating a lower bound for N, instead of the saved
positions, and both for C and C, would be to
add up chords in circles. A circle of radius k fits pi/arcsin(1/2k) many
unit chords, so
k=floor(r) pi
total = sum ------------
k=0 arcsin(1/2k)
and this is less than the chords along the spiral. Is there a good
polynomial over-estimate of arcsin, to become an under-estimate total,
without giving away so much?
=head2 Rectangle to N Range
For the C upper bound, the current code takes the arc
length along with spiral with the usual formula
arc = 1/4pi * (theta*sqrt(1+theta^2) + asinh(theta))
Written in terms of the r radius (theta = 2pi*r) as calculated from the
biggest of the rectangle x,y corners,
arc = pi*r^2*sqrt(1+1/r^2) + asinh(2pi*r)/4pi
The arc length is longer than chords, so N=ceil(arc) is an upper bound for
the N range.
An upper bound can also be calculated simply from the circumferences of
circles 1 to r, since a spiral loop from radius k to k+1 is shorter than a
circle of radius k.
k=ceil(r)
total = sum 2pi*k
k=1
= pi*r*(r+1)
This is bigger than the arc length, thus a poorer upper bound, but an easier
calculation. (Incidentally, for smallish r have arc length E=
pi*(r^2+1) which is a tighter bound and an easy calculation, but it only
holds up to somewhere around r=10^7.)
=head1 SEE ALSO
L,
L,
L
=head1 HOME PAGE
L
=head1 LICENSE
Copyright 2010, 2011, 2012, 2013 Kevin Ryde
This file is part of Math-PlanePath.
Math-PlanePath is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by the Free
Software Foundation; either version 3, or (at your option) any later
version.
Math-PlanePath is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for
more details.
You should have received a copy of the GNU General Public License along with
Math-PlanePath. If not, see .
=cut