Math::PlanePath -- points on a path through the 2-D plane
use Math::PlanePath; # only a base class, see the subclasses for actual operation
This is the base class for some mathematical paths which turn an integer position $n into coordinates $x,$y. The current classes include
$n
$x,$y
SquareSpiral four-sided spiral DiamondSpiral four-sided spiral, looping faster PyramidSpiral square base pyramid TriangleSpiral equilateral triangle TriangleSpiralSkewed equilateral skewed for compactness PentSpiralSkewed five-sided spiral, compact HexSpiral six-sided spiral HexSpiralSkewed six-sided spiral skewed for compactness HeptSpiralSkewed seven-sided spiral, compact SacksSpiral quadratic on an Archimedean spiral VogelFloret seeds in a sunflower KnightSpiral an infinite knight's tour Rows fixed-width rows Columns fixed-height columns Diagonals diagonals between X and Y axes Corner expanding stripes around a corner PyramidRows expanding rows pyramid PyramidSides along the sides of a 45-degree pyramid
The paths are object oriented to allow parameters, though only a few subclasses actually have any parameters.
The classes are generally oriented towards integer $n positions and the classes designed for a square grid turn an integer $n into integer $x,$y. Usually they give in-between positions for fractional $n too. Classes not on a square grid, like SacksSpiral and VogelFloret, are scaled for a unit circle at each $n but they too can give in-between positions on request.
In general there's no parameters for scaling, or an offset for the 0,0 origin, or a reflection up or down. Those things are thought better done by a general coordinate transformer that might expand or invert for display. Even clockwise instead of counter-clockwise spiralling can be had just by negating $x (or negate $y to stay starting at the right), or a quarter turn by swapping $x and $y.
$x
$y
The paths can be characterized by how much longer each loop or repetition is than the preceding one. For example each cycle around the SquareSpiral is 8 longer than the preceding.
Step Path(s) ---- ------- 0 Rows, Columns (fixed widths) 1 Diagonals 2 SacksSpiral, PyramidSides, Corner 2 PyramidRows (default step parameter) 4 DiamondSpiral 5 PentSpiralSkewed 6 HexSpiral, HexSpiralSkewed 7 HeptSpiralSkewed 8 SquareSpiral, PyramidSpiral 9 TriangleSpiral, TriangleSpiralSkewed 32 KnightSpiral (counting the 2-wide trip as the reps)
The step determines which quadratic number sequences fall on straight lines. For example the gap between successive perfect squares increases by 2 each time (4 add 5 to get 9, then add 7 to 16, then add 9 to 25, etc), so the perfect squares make a straight line in the paths of step 2.
A factor of 4 splits a straight line into two, so for example on the SquareSpiral of step 8 the perfect squares fall on two lines to the lower left and upper right. Effectively it's one line of the even squares (2k)^2 == 4*k^2 and another of the odd squares (2k+1)^2 == 4*k^2+4*k+1. The gap between successive even squares increases by 8 each time, and likewise the odd squares.
$path = Math::PlanePath::Foo->new (key=>value, ...)
Create and return a new path object. Optional key/value parameters may control aspects of the object. Foo here is one of the various subclasses, see the list under "SEE ALSO".
Foo
($x,$y) = $path->n_to_xy ($n)
Return x,y coordinates of point $n on the path. If there's no point $n then the return is an empty list, so for example
my ($x,$y) = $path->n_to_xy (-123) or next; # likely no negatives in $path
Currently all paths start from N=1, though some will give a position for N=0 or N=0.5 too.
$n = $path->xy_to_n ($x,$y)
Return the point number for coordinates $x,$y. If there's nothing at $x,$y then return undef.
undef
my $n = $path->xy_to_n(20,20); if (! defined $n) { next; # nothing at this x,y }
$x and $y can be fractional and the path classes will give an integer $n which contains $x,$y within a unit square, circle, or intended figure centred on that $n.
For paths which completely tile the plane there's always an $n to return, but for the spread-out paths an $x,$y position may fall in between (no $n close enough).
($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)
Return a range of N values which occur in a rectangle with corners at $x1,$y1 and $x2,$y2. The range is inclusive, so for instance
$x1
$y1
$x2
$y2
my ($n_lo, $n_hi) = $path->rect_to_n_range (-5,-5, 5,5); foreach my $n ($n_lo .. $n_hi) { my ($x, $y) = $path->n_to_xy ($n) or next; print "$n $x,$y"; }
Note the return may be an over-estimate, and of course some of the points between $n_lo and $n_hi may go outside the rectangle. $n_hi is usually no more than an extra partial row or revolution. $n_lo is often just the starting point 1, which is correct if you always want the origin 0,0, but a rectangle away from the origin could start higher.
$n_lo
$n_hi
$x1,$y1 and $x2,$y2 can be fractional and if they partly overlap some N figures then those N's are included in the return. If there's no points in the rectangle then the return is a "crossed" range like $n_lo=1, $n_hi=0 (which makes a foreach do no loops).
$n_lo=1
$n_hi=0
foreach
$bool = $path->x_negative
$bool = $path->y_negative
Return true if the path extends into negative X coordinates and/or negative Y coordinates respectively.
$str = $path->figure
Return the name of the figure (shape) intended to be drawn at each $n position. Currently this is one of
square side 1 centred on $x,$y circle diameter 1 centred on $x,$y
Of course this is only a suggestion as PlanePath doesn't draw anything itself. A figure like a diamond for instance would work well too.
Math::PlanePath::SquareSpiral, Math::PlanePath::DiamondSpiral, Math::PlanePath::PentSpiralSkewed, Math::PlanePath::HexSpiral, Math::PlanePath::HexSpiralSkewed, Math::PlanePath::HeptSpiralSkewed, Math::PlanePath::SacksSpiral, Math::PlanePath::VogelFloret, Math::PlanePath::KnightSpiral
Math::PlanePath::Rows, Math::PlanePath::Columns, Math::PlanePath::Diagonals, Math::PlanePath::Corner, Math::PlanePath::PyramidRows, Math::PlanePath::PyramidSides
examples/numbers.pl in the sources to print all the paths.
http://user42.tuxfamily.org/math-planepath/index.html
Math-PlanePath is Copyright 2010 Kevin Ryde
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 <http://www.gnu.org/licenses/>.
To install Math::PlanePath, copy and paste the appropriate command in to your terminal.
cpanm
cpanm Math::PlanePath
CPAN shell
perl -MCPAN -e shell install Math::PlanePath
For more information on module installation, please visit the detailed CPAN module installation guide.