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 a base class for some mathematical paths which map an integer position $n
to and from coordinates $x,$y
in the 2D plane.
The current classes include the following. The intention is that any Math::PlanePath::Something
is a PlanePath, and supporting base classes or related things are further down like Math::PlanePath::Base::Xyzzy
.
SquareSpiral four-sided spiral PyramidSpiral square base pyramid TriangleSpiral equilateral triangle spiral TriangleSpiralSkewed equilateral skewed for compactness DiamondSpiral four-sided spiral, looping faster PentSpiral five-sided spiral PentSpiralSkewed five-sided spiral, compact HexSpiral six-sided spiral HexSpiralSkewed six-sided spiral skewed for compactness HeptSpiralSkewed seven-sided spiral, compact AnvilSpiral anvil shape OctagramSpiral eight pointed star KnightSpiral an infinite knight's tour CretanLabyrinth 7-circuit extended infinitely SquareArms four-arm square spiral DiamondArms four-arm diamond spiral AztecDiamondRings four-sided rings HexArms six-arm hexagonal spiral GreekKeySpiral square spiral with Greek key motif MPeaks "M" shape layers SacksSpiral quadratic on an Archimedean spiral VogelFloret seeds in a sunflower TheodorusSpiral unit steps at right angles ArchimedeanChords unit chords on an Archimedean spiral MultipleRings concentric circles PixelRings concentric rings of midpoint pixels FilledRings concentric rings of pixels Hypot points by distance HypotOctant first octant points by distance TriangularHypot points by triangular distance PythagoreanTree X^2+Y^2=Z^2 by trees PeanoCurve 3x3 self-similar quadrant WunderlichSerpentine transpose parts of PeanoCurve HilbertCurve 2x2 self-similar quadrant HilbertSides 2x2 self-similar quadrant segments HilbertSpiral 2x2 self-similar whole-plane ZOrderCurve replicating Z shapes GrayCode Gray code splits WunderlichMeander 3x3 "R" pattern quadrant BetaOmega 2x2 self-similar half-plane AR2W2Curve 2x2 self-similar of four parts KochelCurve 3x3 self-similar of two parts DekkingCurve 5x5 self-similar, edges DekkingCentres 5x5 self-similar, centres CincoCurve 5x5 self-similar ImaginaryBase replicate in four directions ImaginaryHalf half-plane replicate three directions CubicBase replicate in three directions SquareReplicate 3x3 replicating squares CornerReplicate 2x2 replicating "U" LTiling self-simlar L shapes DigitGroups digits grouped by zeros FibonacciWordFractal turns by Fibonacci word bits Flowsnake self-similar hexagonal tile traversal FlowsnakeCentres likewise but centres of hexagons GosperReplicate self-similar hexagonal tiling GosperIslands concentric island rings GosperSide single side or radial QuintetCurve self-similar "+" traversal QuintetCentres likewise but centres of squares QuintetReplicate self-similar "+" tiling DragonCurve paper folding DragonRounded paper folding rounded corners DragonMidpoint paper folding segment midpoints AlternatePaper alternating direction folding AlternatePaperMidpoint alternating direction folding, midpoints TerdragonCurve ternary dragon TerdragonRounded ternary dragon rounded corners TerdragonMidpoint ternary dragon segment midpoints R5DragonCurve radix-5 dragon curve R5DragonMidpoint radix-5 dragon curve midpoints CCurve "C" curve ComplexPlus base i+realpart ComplexMinus base i-realpart, including twindragon ComplexRevolving revolving base i+1 SierpinskiCurve self-similar right-triangles SierpinskiCurveStair self-similar right-triangles, stair-step HIndexing self-similar right-triangles, squared up KochCurve replicating triangular notches KochPeaks two replicating notches KochSnowflakes concentric notched 3-sided rings KochSquareflakes concentric notched 4-sided rings QuadricCurve eight segment zig-zag QuadricIslands rings of those zig-zags SierpinskiTriangle self-similar triangle by rows SierpinskiArrowhead self-similar triangle connectedly SierpinskiArrowheadCentres likewise but centres of triangles Rows fixed-width rows Columns fixed-height columns Diagonals diagonals between X and Y axes DiagonalsAlternating diagonals Y to X and back again DiagonalsOctant diagonals between Y axis and X=Y centre Staircase stairs down from the Y to X axes StaircaseAlternating stairs Y to X and back again Corner expanding stripes around a corner PyramidRows expanding stacked rows pyramid PyramidSides along the sides of a 45-degree pyramid CellularRule cellular automaton by rule number CellularRule54 cellular automaton rows pattern CellularRule57 cellular automaton (rule 99 mirror too) CellularRule190 cellular automaton (rule 246 mirror too) UlamWarburton cellular automaton diamonds UlamWarburtonQuarter cellular automaton quarter-plane DiagonalRationals rationals X/Y by diagonals FactorRationals rationals X/Y by prime factorization GcdRationals rationals X/Y by rows with GCD integer RationalsTree rationals X/Y by tree FractionsTree fractions 0<X/Y<1 by tree ChanTree rationals X/Y multi-child tree CfracDigits continued fraction 0<X/Y<1 by digits CoprimeColumns coprime X,Y DivisibleColumns X divisible by Y WythoffArray Fibonacci recurrences WythoffPreliminaryTriangle PowerArray powers in rows File points from a disk file
And in the separate Math-PlanePath-Toothpick distribution
ToothpickTree pattern of toothpicks ToothpickReplicate same by replication rather than tree ToothpickUpist toothpicks only growing upwards ToothpickSpiral toothpicks around the origin LCornerTree L-shape corner growth LCornerReplicate same by replication rather than tree OneOfEight HTree H shapes replicated
The paths are object oriented to allow parameters, though many have none. See examples/numbers.pl
in the Math-PlanePath sources for a sample printout of numbers from selected paths or all paths.
The $n
and $x,$y
parameters can be either integers or floating point. The paths are meant to do something sensible with fractions but expect round-off for big floating point exponents.
Floating point infinities (when available) give NaN or infinite returns of some kind (some unspecified kind as yet). n_to_xy()
on negative infinity is an empty return the same as other negative $n
.
Floating point NaNs (when available) give NaN, infinite, or empty/undef returns, but again of some unspecified kind as yet.
Most of the classes can operate on overloaded number types as inputs and give corresponding outputs.
Math::BigInt maybe perl 5.8 up for ** operator Math::BigRat Math::BigFloat Number::Fraction 1.14 or higher for abs()
A few classes might truncate a bignum or a fraction to a float as yet. In general the intention is to make the calculations generic enough to act on any sensible number type. Recent enough versions of the bignum modules might be required, perhaps BigInt
of Perl 5.8 or higher for **
exponentiation operator.
For reference, an undef
input as $n
, $x
, $y
, etc, is designed to provoke an uninitialized value warning when warnings are enabled. Perhaps that will change, but the warning at least prevents bad inputs going unnoticed.
In the following Foo
is one of the various subclasses, see the list above and under "SEE ALSO".
$path = Math::PlanePath::Foo->new (key=>value, ...)
Create and return a new path object. Optional key/value parameters may control aspects of the object.
($x,$y) = $path->n_to_xy ($n)
Return X,Y coordinates of point $n
on the path. If there's no point $n
then the return is an empty list. For example
my ($x,$y) = $path->n_to_xy (-123) or next; # no negatives in $path
Paths start from $path->n_start()
below, though some will give a position for N=0 or N=-0.5 too.
($dx,$dy) = $path->n_to_dxdy ($n)
Return the change in X and Y going from point $n
to point $n+1
, or for paths with multiple arms from $n
to $n+$arms_count
(thus advancing one point along the arm of $n
).
+ $n+1 == $next_x,$next_y ^ | | $dx = $next_x - $x + $n == $x,$y $dy = $next_y - $y
$n
can be fractional and in that case the dX,dY is from that fractional $n
position to $n+1
(or $n+$arms
).
frac $n+1 == $next_x,$next_y v integer *---+---- | / | / |/ $dx = $next_x - $x frac + $n == $x,$y $dy = $next_y - $y | integer *
In both cases n_to_dxdy()
is the difference $dx=$next_x-$x, $dy=$next_y-$y
. Currently for most paths it's merely two n_to_xy()
calls to calculate the two points, but some paths can calculate a dX,dY with a little less work.
$rsquared = $path->n_to_radius ($n)
$rsquared = $path->n_to_rsquared ($n)
Return the radial distance R=sqrt(X^2+Y^2) of point $n
, or the radius squared R^2=X^2+Y^2. If there's no point $n
then the return is undef
.
For a few paths these might be calculated with less work than n_to_xy()
. For example the SacksSpiral
is simply R^2=N, or the MultipleRings
path with its default step=6 has an integer radius for integer $n
whereas $x,$y
are fractional (and so inexact).
$n = $path->xy_to_n ($x,$y)
Return the N point number at coordinates $x,$y
. If there's nothing at $x,$y
then return 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 the integer $n
.
For paths which completely fill 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) and give undef
.
@n_list = $path->xy_to_n_list ($x,$y)
Return a list of N point numbers at coordinates $x,$y
. If there's nothing at $x,$y
then return an empty list.
my @n_list = $path->xy_to_n(20,20);
Most paths have just a single N for a given X,Y but some such as DragonCurve
and TerdragonCurve
have multiple N's and this method returns all of them.
$bool = $path->xy_is_visited ($x,$y)
Return true if $x,$y
is visited. This is equivalent to
defined($path->xy_to_n($x,$y))
Some paths cover the plane and for them xy_is_visited()
is always true. For others it might be less work to test a point than to calculate its $n
.
$n = $path->xyxy_to_n($x1,$y1, $x2,$y2)
$n = $path->xyxy_to_n_either($x1,$y1, $x2,$y2)
@n_list = $path->xyxy_to_n_list($x1,$y1, $x2,$y2)
@n_list = $path->xyxy_to_n_list_either($x1,$y1, $x2,$y2)
Return <$n> which goes from $x1,$y1
to $x2,$y2
. <$n> is at $x1,$y1
and $n+1
is at $x2,$y2
, or for a multi-arm path $n+$arms
so a step along the same arm. If there's no such $n
then return undef
.
The either()
forms allow <$n> in either direction, so $x1,$y1
to $x2,$y2
or the other way $x2,$y2
to $x1,$y1
.
The n_list()
forms return a list of all $n
going between $x1,$y1
and $x2,$y2
. For example in Math::PlanePath::CCurve
some segments are traversed twice, once in each direction.
The possible N values at each X,Y are determined the same way as for xy_to_n()
.
($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)
Return a range of N values covering or exceeding a rectangle with corners at $x1
,$y1
and $x2
,$y2
. The range is inclusive. For example,
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"; }
The return might be an over-estimate of the N range required to cover the rectangle. Even if the range is exact the nature of the path may mean many points between $n_lo
and $n_hi
are outside the rectangle. But the range is at least a lower and upper bound on the N values which occur in the rectangle. Classes which can guarantee an exact lo/hi range say so in their docs.
$n_hi
is usually no more than an extra partial row, revolution, or self-similar level. $n_lo
might be merely the starting $path->n_start()
, which is fine if the origin is in the desired rectangle but away from the origin might actually start higher.
$x1
,$y1
and $x2
,$y2
can be fractional. 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 can be a "crossed" range like $n_lo=1
, $n_hi=0
(which makes a foreach
do no loops). But rect_to_n_range()
may not always notice there's no points in the rectangle and might instead return some over-estimate.
$n = $path->n_start()
Return the first N in the path. The start is usually either 0 or 1 according to what is most natural for the path. Some paths have an n_start
parameter to control the numbering.
Some classes have secret dubious undocumented support for N values below this start (zero or negative), but n_start()
is the intended starting point.
$f = $path->n_frac_discontinuity()
Return the fraction of N at which there may be discontinuities in the path. For example if there's a jump in the coordinates between N=7.4999 and N=7.5 then the returned $f
is 0.5. Or $f
is 0 if there's a discontinuity between 6.999 and 7.0.
If there's no discontinuities in the path then the return is undef
. That means for example fractions between N=7 to N=8 give smooth continuous X,Y values (of some kind).
This is mainly of interest for drawing line segments between N points. If there's discontinuities then the idea is to draw from say N=7.0 to N=7.499 and then another line from N=7.5 to N=8.
$arms = $path->arms_count()
Return the number of arms in a "multi-arm" path.
For example in SquareArms
this is 4 and each arm increments in turn, so the first arm is N=1,5,9,13,etc starting from $path->n_start()
and incrementing by 4 each time.
$bool = $path->x_negative()
$bool = $path->y_negative()
Return true if the path extends into negative X coordinates and/or negative Y coordinates respectively.
$bool = Math::PlanePath::Foo->class_x_negative()
$bool = Math::PlanePath::Foo->class_y_negative()
$bool = $path->class_x_negative()
$bool = $path->class_y_negative()
Return true if any paths made by this class extend into negative X coordinates and/or negative Y coordinates, respectively.
For some classes the X or Y extent may depend on parameter values.
$n = $path->x_negative_at_n()
$n = $path->y_negative_at_n()
Return the integer N where X or Y respectively first goes negative, or return undef
if it does not go negative (x_negative()
or y_negative()
respectively is false).
$x = $path->x_minimum()
$y = $path->y_minimum()
$x = $path->x_maximum()
$y = $path->y_maximum()
Return the minimum or maximum of the X or Y coordinate reached by integer N values in the path. If there's no minimum or maximum then return undef
.
$dx = $path->dx_minimum()
$dx = $path->dx_maximum()
$dy = $path->dy_minimum()
$dy = $path->dy_maximum()
Return the minimum or maximum change dX, dY occurring in the path for integer N to N+1. For a multi-arm path the change is N to N+arms so it's the change along the same arm.
Various paths which go by rows have non-decreasing Y. For them dy_minimum()
is 0.
$adx = $path->absdx_minimum()
$adx = $path->absdx_maximum()
$ady = $path->absdy_minimum()
$ady = $path->absdy_maximum()
Return the minimum or maximum change abs(dX) or abs(dY) occurring in the path for integer N to N+1. For a multi-arm path the change is N to N+arms so it's the change along the same arm.
absdx_maximum()
is simply max(dXmax,-dXmin), the biggest change either positive or negative. absdy_maximum()
similarly.
absdx_minimum()
is 0 if dX=0 occurs anywhere in the path, which means any vertical step. If X always changes then absdx_minimum()
will be something bigger than 0. absdy_minimum()
likewise 0 if any horizontal dY=0, or bigger if Y always changes.
$sum = $path->sumxy_minimum()
$sum = $path->sumxy_maximum()
Return the minimum or maximum values taken by coordinate sum X+Y reached by integer N values in the path. If there's no minimum or maximum then return undef
.
S=X+Y is an anti-diagonal. A path which is always right and above some anti-diagonal has a minimum. Some paths might be entirely left and below and so have a maximum, though that's unusual.
\ Path always above \ | has minimum S=X+Y \| ---o---- Path always below |\ has maximum S=X+Y | \ \ S=X+Y
$sum = $path->sumabsxy_minimum()
$sum = $path->sumabsxy_maximum()
Return the minimum or maximum values taken by coordinate sum abs(X)+abs(Y) reached by integer N values in the path. A minimum always exists but if there's no maximum then return undef
.
SumAbs=abs(X)+abs(Y) is sometimes called the "taxi-cab" or "Manhattan" distance, being how far to travel through a square-grid city to get to X,Y. sumabsxy_minimum()
is then how close to the origin the path extends.
SumAbs can also be interpreted geometrically as numbering the anti-diagonals of the quadrant containing X,Y, which is equivalent to asking which diamond shape X,Y falls on. sumabsxy_minimum()
is then the smallest such diamond reached by the path.
| /|\ SumAbs = which diamond X,Y falls on / | \ / | \ -----o----- \ | / \ | / \|/ |
$diffxy = $path->diffxy_minimum()
$diffxy = $path->diffxy_maximum()
Return the minimum or maximum values taken by coordinate difference X-Y reached by integer N values in the path. If there's no minimum or maximum then return undef
.
D=X-Y is a leading diagonal. A path which is always right and below such a diagonal has a minimum, for example HypotOctant
. A path which is always left and above some diagonal has a maximum D=X-Y. For example various wedge-like paths such as PyramidRows
in its default step=2, and "upper octant" paths have a maximum.
/ D=X-Y Path always below | / has maximum D=X-Y |/ ---o---- /| / | Path always above / has minimum D=X-Y
$absdiffxy = $path->absdiffxy_minimum()
$absdiffxy = $path->absdiffxy_maximum()
Return the minimum or maximum values taken by abs(X-Y) for integer N in the path. The minimum is 0 or more. If there's maximum then return undef
.
abs(X-Y) can be interpreted geometrically as the distance away from the X=Y diagonal and measured at right-angles to that line.
d=abs(X-Y) X=Y line ^ / \ / \/ /\ / \ / \ o v / d=abs(X-Y)
Paths which visit the X=Y line (or approach it as an infimum) have absdiffxy_minimum() = 0
. Otherwise absdiffxy_minimum()
is how close they come to the line.
If the path is entirely below the X=Y line so X>=Y then X-Y>=0 and absdiffxy_minimum()
is the same as diffxy_minimum()
. If the path is entirely below the X=Y line then absdiffxy_minimum()
is - diffxy_maximum()
.
$dsumxy = $path->dsumxy_minimum()
$dsumxy = $path->dsumxy_maximum()
$ddiffxy = $path->ddiffxy_minimum()
$ddiffxy = $path->ddiffxy_maximum()
Return the minimum or maximum change dSum or dDiffXY occurring in the path for integer N to N+1. For a multi-arm path the change is N to N+arms so it's the change along the same arm.
$rsquared = $path->rsquared_minimum()
$rsquared = $path->rsquared_maximum()
Return the minimum or maximum Rsquared = X^2+Y^2 reached by integer N values in the path. If there's no minimum or maximum then return undef
.
Rsquared is always >= 0 so it always has a minimum. The minimum will be more than 0 for paths which don't include the origin X=0,Y=0.
RSquared generally has no maximum since the paths usually extend infinitely in some direction. rsquared_maximum()
returns undef
in that case.
($dx,$dy) = $path->dir_minimum_dxdy()
($dx,$dy) = $path->dir_maximum_dxdy()
Return a vector which is the minimum or maximum angle taken by a step integer N to N+1, or for a multi-arm path N to N+arms so it's the change along the same arm. Directions are reckoned anti-clockwise around from the X axis.
| * dX=2,dY=2 dX=-1,dY=1 * | / \|/ ------+----* dX=1,dY=0 | | * dX=0,dY=-1
A path which is always goes N,S,E,W such as the SquareSpiral
has minimum East dX=1,dY=0 and maximum South dX=0,dY=-1.
Paths which go diagonally may have different limits. For example the KnightSpiral
goes in 2x1 steps and so has minimum East-North-East dX=2,dY=1 and maximum East-South-East dX=2,dY=-1.
If the path has directions approaching 360 degrees then dir_maximum_dxdy()
is 0,0 which should be taken to mean a full circle as a supremum. For example MultipleRings
.
If the path only ever goes East then the maximum is East dX=1,dY=0, and the minimum the same. This isn't particularly interesting, but arises for example in the Columns
path height=0.
$bool = $path->turn_any_left()
$bool = $path->turn_any_right()
$bool = $path->turn_any_straight()
Return true if the path turns left, right, or straight (which includes 180deg reverse) at any integer N.
left ^ | N-1 ----------- N --> straight | v left
A line from N-1 to N is a current direction and the turn at N is then whether point N+1 is to the left or right of that line. Directly along the line is straight, and so is anything directly behind as a reverse. This is the turn style of Math::NumSeq::PlanePathTurn.
$str = $path->figure()
Return a string name of the figure (shape) intended to be drawn at each $n
position. This is currently either
"square" side 1 centred on $x,$y "circle" diameter 1 centred on $x,$y
Of course this is only a suggestion since PlanePath doesn't draw anything itself. A figure like a diamond for instance can look good too.
Some paths are structured like a tree where each N has a parent and possibly some children.
123 / | \ 456 999 458 / / \ 1000 1001 1005
The N numbering and any relation to X,Y positions varies among the paths. Some are numbered by rows in breadth-first style and some have children with X,Y positions adjacent to their parent, but that shouldn't be assumed, only that there's a parent-child relation down from some set of root nodes.
$bool = $path->is_tree()
Return true if $path
is a tree.
The various tree methods have empty or undef
returns on non-tree paths. Often it's enough to check for that from a desired method rather than a separate is_tree()
check.
@n_children = $path->tree_n_children($n)
Return a list of N values which are the child nodes of $n
, or return an empty list if $n
has no children.
There could be no children either because $path
is not a tree or because there's no children at a particular $n
.
$num = $path->tree_n_num_children($n)
Return the number of children of $n
, or 0 if $n
has no children, or undef
if $n < n_start()
(ie. before the start of the path).
If the tree is considered as a directed graph then this is the "out-degree" of $n
.
$n_parent = $path->tree_n_parent($n)
Return the parent node of $n
, or undef
if it has no parent.
There is no parent at the root node of the tree, or one of multiple roots, or if $path
is not a tree.
$n_root = $path->tree_n_root ($n)
Return the N which is the root node of $n
. This is the top of the tree as would be found by following tree_n_parent()
repeatedly.
The return is undef
if there's no $n
point or if $path
is not a tree.
$depth = $path->tree_n_to_depth($n)
Return the depth of node $n
, or undef
if there's no point $n
. The top of the tree is depth=0, then its children are depth=1, etc.
The depth is a count of how many parent, grandparent, etc, levels are above $n
, ie. until reaching tree_n_to_parent()
returning undef
. For non-tree paths tree_n_to_parent()
is always undef
and tree_n_to_depth()
is always 0.
$n_lo = $path->tree_depth_to_n($depth)
$n_hi = $path->tree_depth_to_n_end($depth)
($n_lo, $n_hi) = $path->tree_depth_to_n_range ($depth)
Return the first or last N, or both those N, for tree level $depth
in the path. If there's no such $depth
or if $path
is not a tree then return undef
, or for tree_depth_to_n_range()
return an empty list.
The points $n_lo
through $n_hi
might not necessarily all be at $depth
. It's possible for depths to be interleaved or intermixed in the point numbering. But many paths are breadth-wise successive rows and for them $n_lo
to $n_hi
inclusive is all $depth
.
$n_hi
can only exist if the row has a finite number of points. That's true of all current paths, but perhaps allowance ought to be made for $n_hi
as undef
or some such if there is no maximum N for some row.
$num = $path->tree_depth_to_width ($depth)
Return the number of points at $depth
in the tree. If there's no such $depth
or $path
is not a tree then return undef
.
$height = $path->tree_n_to_subheight($n)
Return the height of the sub-tree starting at $n
, or undef
if infinite. The height of a tree is the longest distance down to a leaf node. For example,
... N subheight \ --- --------- 6 7 8 0 undef \ \ / 1 undef 3 4 5 2 2 \ \ / 3 undef 1 2 4 1 \ / 5 0 0 ...
At N=0 and all of the left side the tree continues infinitely so the sub-height there is undef
for infinite. For N=2 the sub-height is 2 because the longest path down is 2 levels (to N=7 or N=8). For a leaf node such as N=5 the sub-height is 0.
$num = $path->tree_num_roots()
Return the number of root nodes in $path
. If $path
is not a tree then return 0. Many tree paths have a single root and for them the return is 1.
@n_list = $path->tree_root_n_list()
Return a list of the N values which are the root nodes in $path
. If $path
is not a tree then this is an empty list. There are tree_num_roots()
many return values.
$num = $path->tree_num_children_minimum()
$num = $path->tree_num_children_maximum()
@nums = $path->tree_num_children_list()
Return the possible number of children of the nodes of $path
, either the minimum, the maximum, or a list of all possible numbers of children.
For tree_num_children_list()
the list of values is in increasing order, so the first value is tree_num_children_minimum()
and the last is tree_num_children_maximum()
.
$bool = $path->tree_any_leaf()
Return true if there are any leaf nodes in the tree, meaning any N for which tree_n_num_children()
is 0.
This is the same as tree_num_children_minimum()==0
since if NumChildren=0 occurs then there are leaf nodes.
Some trees may have no leaf nodes, for example in the complete binary tree of RationalsTree
every node always has 2 children.
level = $path->n_to_level($n)
Return the replication level containing $n
. The first level is 0.
($n_lo,$n_hi) = $path->level_to_n_range($level)
Return the range of N values, inclusive, which comprise a self-similar replication level in $path
. If $path
has no notion of such levels then return an empty list.
my ($n_lo, $n_hi) = $path->level_to_n_range(6) or print "no levels in this path";
For example the DragonCurve
has levels running 0
to 2**$level
, or the HilbertCurve
is 0
to 4**$level - 1
. Most levels are powers like this. A power 2**$level
is a "vertex" style whereas 2**$level - 1
is a "centre" style. The difference is generally whether the X,Y points represent vertices of the object's segments as opposed to centres or midpoints.
$aref = Math::PlanePath::Foo->parameter_info_array()
@list = Math::PlanePath::Foo->parameter_info_list()
Return an arrayref of list describing the parameters taken by a given class. This meant to help making widgets etc for user interaction in a GUI. Each element is a hashref
{ name => parameter key arg for new() share_key => string, or undef description => human readable string type => string "integer","boolean","enum" etc default => value minimum => number, or undef maximum => number, or undef width => integer, suggested display size choices => for enum, an arrayref }
type
is a string, one of
"integer" "enum" "boolean" "string" "filename"
"filename" is separate from "string" since it might require subtly different handling to reach Perl as a byte string, whereas a "string" type might in principle take Perl wide chars.
For "enum" the choices
field is the possible values, such as
{ name => "flavour", type => "enum", choices => ["strawberry","chocolate"], }
minimum
and/or maximum
are omitted if there's no hard limit on the parameter.
share_key
is designed to indicate when parameters from different PlanePath
classes can done by a single control widget in a GUI etc. Normally the name
is enough, but when the same name has slightly different meanings in different classes a share_key
allows the same meanings to be matched up.
$hashref = Math::PlanePath::Foo->parameter_info_hash()
Return a hashref mapping parameter names $info->{'name'}
to their $info
records.
{ wider => { name => "wider", type => "integer", ... }, }
The classes are mostly based on integer $n
positions and those 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 but instead giving fractional X,Y such as SacksSpiral
and VogelFloret
are designed for a unit circle at each $n
but they too can give in-between positions on request.
All X,Y positions are calculated by separate n_to_xy()
calls. To follow a path use successive $n
values starting from $path->n_start()
.
foreach my $n ($path->n_start .. 100) { my ($x,$y) = $path->n_to_xy($n); print "$n $x,$y\n"; }
The separate n_to_xy()
calls were motivated by plotting just some N points of a path, such as just the primes or the perfect squares. Successive positions in paths could perhaps be done more efficiently in an iterator style. Paths with a quadratic "step" are not much worse than a sqrt()
to break N into a segment and offset, but the self-similar paths which chop N into digits of some radix could increment instead of recalculate.
If interested only in a particular rectangle or similar region then iterating has the disadvantage that it may stray outside the target region for a long time, making an iterator much less useful than it seems. For wild paths it can be better to apply xy_to_n()
by rows or similar across the desired region.
Math::NumSeq::PlanePathCoord etc offer the PlanePath coordinates, directions, turns, etc as sequences. The iterator forms there simply make repeated calls to n_to_xy()
etc.
The paths generally make a first move to the right and go anti-clockwise around from the X axis, unless there's some more natural orientation. Anti-clockwise is the usual direction for mathematical spirals.
There's no parameters for scaling, offset or reflection as those things are thought better left to a general coordinate transformer, for example to expand or invert for display. Some easy transformations can be had just from the X,Y with
-X,Y flip horizontally (mirror image) X,-Y flip vertically (across the X axis) -Y,X rotate +90 degrees (anti-clockwise) Y,-X rotate -90 degrees (clockwise) -X,-Y rotate 180 degrees
Flip vertically makes spirals go clockwise instead of anti-clockwise, or a flip horizontally the same but starting on the left at the negative X axis. See "Triangular Lattice" below for 60 degree rotations of the triangular grid paths too.
The Rows and Columns paths are exceptions to the rule of not having rotated versions of paths. They began as ways to pass in width and height as generic parameters and let the path use the one or the other.
For scaling and shifting see for example Transform::Canvas, and to rotate as well see Geometry::AffineTransform.
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 more N points than the preceding.
Step Path ---- ---- 0 Rows, Columns (fixed widths) 1 Diagonals 2/2 DiagonalsOctant (2 rows for +2) 2 SacksSpiral, PyramidSides, Corner, PyramidRows (default) 4 DiamondSpiral, AztecDiamondRings, Staircase 4/2 CellularRule54, CellularRule57, DiagonalsAlternating (2 rows for +4) 5 PentSpiral, PentSpiralSkewed 5.65 PixelRings (average about 4*sqrt(2)) 6 HexSpiral, HexSpiralSkewed, MPeaks, MultipleRings (default) 6/2 CellularRule190 (2 rows for +6) 6.28 ArchimedeanChords (approaching 2*pi), FilledRings (average 2*pi) 7 HeptSpiralSkewed 8 SquareSpiral, PyramidSpiral 16/2 StaircaseAlternating (up and back for +16) 9 TriangleSpiral, TriangleSpiralSkewed 12 AnvilSpiral 16 OctagramSpiral, ToothpickSpiral 19.74 TheodorusSpiral (approaching 2*pi^2) 32/4 KnightSpiral (4 loops 2-wide for +32) 64 DiamondArms (each arm) 72 GreekKeySpiral 128 SquareArms (each arm) 128/4 CretanLabyrinth (4 loops for +128) 216 HexArms (each arm) totient CoprimeColumns, DiagonalRationals numdivisors DivisibleColumns various CellularRule parameter MultipleRings, PyramidRows
The step determines which quadratic number sequences make straight lines. For example the gap between successive perfect squares increases by 2 each time (4 to 9 is +5, 9 to 16 is +7, 16 to 25 is +9, etc), so the perfect squares make a straight line in the paths of step 2.
In general straight lines on stepped paths are quadratics
N = a*k^2 + b*k + c where a=step/2
The polygonal numbers are like this, with the (step+2)-gonal numbers making a straight line on a "step" path. For example the 7-gonals (heptagonals) are 5/2*k^2-3/2*k and make a straight line on the step=5 PentSpiral
. Or the 8-gonal octagonal numbers 6/2*k^2-4/2*k on the step=6 HexSpiral
.
There are various interesting properties of primes in quadratic progressions. Some quadratics seem to have more primes than others. For example see "Lucky Numbers of Euler" in Math::PlanePath::PyramidSides. Many quadratics have no primes at all, or none above a certain point, either trivially if always a multiple of 2 etc, or by a more sophisticated reasoning. See "Step 3 Pentagonals" in Math::PlanePath::PyramidRows for a factorization on the roots making a no-primes gap.
A 4*step path splits a straight line in two, so for example the perfect squares are a straight line on the step=2 "Corner" path, and then on the step=8 SquareSpiral
they instead fall on two lines (lower left and upper right). In the bigger step there's one line of the even squares (2k)^2 == 4*k^2 and another of the odd squares (2k+1)^2. The gap between successive even squares increases by 8 each time and likewise between odd squares.
The self-similar patterns such as PeanoCurve
generally have a base pattern which repeats at powers N=base^level or squares N=(base*base)^level. Or some multiple or relationship to such a power for things like KochPeaks
and GosperIslands
.
Base Path ---- ---- 2 HilbertCurve, HilbertSides, HilbertSpiral, ZOrderCurve (default), GrayCode (default), BetaOmega, AR2W2Curve, HIndexing, ImaginaryBase (default), ImaginaryHalf (default), SierpinskiCurve, SierpinskiCurveStair, CubicBase (default) CornerReplicate, ComplexMinus (default), ComplexPlus (default), ComplexRevolving, DragonCurve, DragonRounded, DragonMidpoint, AlternatePaper, AlternatePaperMidpoint, CCurve, DigitGroups (default), PowerArray (default) 3 PeanoCurve (default), WunderlichSerpentine (default), WunderlichMeander, KochelCurve, GosperIslands, GosperSide SierpinskiTriangle, SierpinskiArrowhead, SierpinskiArrowheadCentres, TerdragonCurve, TerdragonRounded, TerdragonMidpoint, UlamWarburton, UlamWarburtonQuarter (each level) 4 KochCurve, KochPeaks, KochSnowflakes, KochSquareflakes, LTiling, 5 QuintetCurve, QuintetCentres, QuintetReplicate, DekkingCurve, DekkingCentres, CincoCurve, R5DragonCurve, R5DragonMidpoint 7 Flowsnake, FlowsnakeCentres, GosperReplicate 8 QuadricCurve, QuadricIslands 9 SquareReplicate Fibonacci FibonacciWordFractal, WythoffArray parameter PeanoCurve, WunderlichSerpentine, ZOrderCurve, GrayCode, ImaginaryBase, ImaginaryHalf, CubicBase, ComplexPlus, ComplexMinus, DigitGroups, PowerArray
Many number sequences plotted on these self-similar paths tend to be fairly random, or merely show the tiling or path layout rather than much about the number sequence. Sequences related to the base can make holes or patterns picking out parts of the path. For example numbers without a particular digit (or digits) in the relevant base show up as holes. See for example "Power of 2 Values" in Math::PlanePath::ZOrderCurve.
Some paths are on triangular or "A2" lattice points like
*---*---*---*---*---* / \ / \ / \ / \ / \ / *---*---*---*---*---* \ / \ / \ / \ / \ / \ *---*---*---*---*---* / \ / \ / \ / \ / \ / *---*---*---*---*---* \ / \ / \ / \ / \ / \ *---*---*---*---*---* / \ / \ / \ / \ / \ / *---*---*---*---*---*
This is done in integer X,Y on a square grid by using every second square and offsetting alternate rows. This means sum X+Y even, ie. X,Y either both even or both odd, not of opposite parity.
. * . * . * . * . * . * * . * . * . * . * . * . . * . * . * . * . * . * * . * . * . * . * . * . . * . * . * . * . * . * * . * . * . * . * . * .
The X axis the and diagonals X=Y and X=-Y divide the plane into six equal parts in this grid.
X=-Y X=Y \ / \ / \ / ----------------- X=0 / \ / \ / \
The diagonal X=3*Y is the middle of the first sixth, representing a twelfth of the plane.
The resulting triangles are flatter than they should be. The triangle base is width=2 and top is height=1, whereas it would be height=sqrt(3) for an equilateral triangle. That sqrt(3) factor can be applied if desired,
X, Y*sqrt(3) side length 2 X/2, Y*sqrt(3)/2 side length 1
Integer Y values have the advantage of fitting pixels on the usual kind of raster computer screen, and not losing precision in floating point results.
If doing a general-purpose coordinate rotation then be sure to apply the sqrt(3) scale factor before rotating or the result will be skewed. 60 degree rotations can be made within the integer X,Y coordinates directly as follows, all giving integer X,Y results.
(X-3Y)/2, (Y+X)/2 rotate +60 (anti-clockwise) (X+3Y)/2, (Y-X)/2 rotate -60 (clockwise) -(X+3Y)/2, (X-Y)/2 rotate +120 (3Y-X)/2, -(X+Y)/2 rotate -120 -X,-Y rotate 180 (X+3Y)/2, (X-Y)/2 mirror across the X=3*Y twelfth line
The sqrt(3) factor can be worked into a hypotenuse radial distance calculation as follows if comparing distances from the origin.
hypot = sqrt(X*X + 3*Y*Y)
See for instance TriangularHypot
which is triangular points ordered by this radial distance.
The formulas section in the POD of each class describes some of the calculations. This might be of interest even if the code is not.
For a triangular lattice the rotation formulas above allow calculations to be done in the rectangular X,Y coordinates which are the inputs and outputs of the PlanePath functions. Another way is to number vertically on a 60 degree angle with coordinates i,j,
... * * * 2 * * * 1 * * * j=0 i=0 1 2
These coordinates are sometimes used for hexagonal grids in board games etc. Using this internally can simplify rotations a little,
-j, i+j rotate +60 (anti-clockwise) i+j, -i rotate -60 (clockwise) -i-j, i rotate +120 j, -i-j rotate -120 -i, -j rotate 180
Conversions between i,j and the rectangular X,Y are
X = 2*i + j i = (X-Y)/2 Y = j j = Y
A third coordinate k at a +120 degrees angle can be used too,
k=0 k=1 k=2 * * * * * * * * * 0 1 2
This is redundant in that it doesn't number anything i,j alone can't already, but it has the advantage of turning rotations into just sign changes and swaps,
-k, i, j rotate +60 j, k, -i rotate -60 -j, -k, i rotate +120 k, -i, -j rotate -120 -i, -j, -k rotate 180
The conversions between i,j,k and the rectangular X,Y are like the i,j above but with k worked in too.
X = 2i + j - k i = (X-Y)/2 i = (X+Y)/2 Y = j + k j = Y or j = 0 k = 0 k = Y
n_to_dxdy()
is the change from N to N+1, and is designed both for integer N and fractional N. For fractional N it can be convenient to calculate a dX,dY at floor(N) and at floor(N)+1 and then combine the two in proportion to frac(N).
int+2 | | N+1 \ /| | / | | / | | frac / | | / | | / | / int-----N------int+1 this_dX dX,dY next_dX this_dY next_dY |-------|------| frac 1-frac int = int(N) frac = N - int 0 <= frac < 1 this_dX,this_dY at int next_dX,next_dY at int+1 at fractional N dX = this_dX * (1-frac) + next_dX * frac dY = this_dY * (1-frac) + next_dY * frac
This is combination of this_dX,this_dY and next_dX,next_dY in proportion to the distances from positions N to int+1 and from int+1 to N+1.
The formulas can be rearranged to
dX = this_dX + frac*(next_dX - this_dX) dY = this_dY + frac*(next_dY - this_dY)
which is like dX,dY at the integer position plus fractional part of a turn or change to the next dX,dY.
For most of the self-similar paths such as HilbertCurve
the change dX,dY is determined by following the state table transitions down through either all digits of N, or to the last non-9 digit, ie. drop any low digits equal to radix-1.
Generally paths which are the edges of some tiling use all digits, and those which are the centres of a tiling stop at the lowest non-9. This can be seen for example in the DekkingCurve
using all digits, whereas its DekkingCentres
variant stops at the lowest non-24.
Perhaps this all-digits vs low-non-9 even characterizes path style as edges or centres of a tiling, when a path is specified in some way that a tiling is not quite obvious.
The mandatory methods for a PlanePath subclass are
n_to_xy() xy_to_n() xy_to_n_list() if multiple N's map to an X,Y rect_to_n_range()
It sometimes happens that one of n_to_xy()
or xy_to_n()
is easier than the other but both should be implemented.
n_to_xy()
should do something sensible on fractional N. The suggestion is to make it an X,Y proportionally between integer N positions. It can be along a straight line or an arc as best suits the path. A straight line can be done simply by two calculations of the surrounding integer points, until it's clear how to work the fraction into the code directly.
xy_to_n_list()
has a base implementation calling plain xy_to_n()
to give a single N at X,Y. If a path has multiple Ns at an X,Y (eg. DragonCurve
) then it should implement xy_to_n_list()
to return all those Ns and also implement a plain xy_to_n()
returning the first of them.
rect_to_n_range()
can initially be any convenient over-estimate. It should give N big enough that from there onwards all points are sure to be beyond the given X,Y rectangle.
The following descriptive methods have base implementations
n_start() 1 class_x_negative() \ 1, so whole plane class_y_negative() / x_negative() calls class_x_negative() y_negative() calls class_x_negative() x_negative_at_n() undef \ as for no negatives y_negative_at_n() undef /
The base n_start()
starts at N=1. Paths which treat N as digits of some radix or where there's self-similar replication are often best started from N=0 instead since doing so puts nice powers-of-2 etc on the axes or diagonals.
use constant n_start => 0; # digit or replication style
Paths which use only parts of the plane should define class_x_negative()
and/or class_y_negative()
to false. For example if only the first quadrant X>=0,Y>=0 then
use constant class_x_negative => 0; use constant class_y_negative => 0;
If negativeness varies with path parameters then x_negative()
and/or y_negative()
follow those parameters and the class_()
forms are whether any set of parameters ever gives negative.
The following methods have base implementations calling n_to_xy()
. A subclass can implement them directly if they can be done more efficiently.
n_to_dxdy() calls n_to_xy() twice n_to_rsquared() calls n_to_xy() n_to_radius() sqrt of n_to_rsquared()
SacksSpiral
is an example of an easy n_to_rsquared()
. TheodorusSpiral
is only slightly trickier. Unless a path has some sort of easy X^2+Y^2 then it might as well let the base implementation call n_to_xy()
.
The way n_to_dxdy()
supports fractional N can be a little tricky. One way is to calculate dX,dY on the integer N below and above and combine as described in "N to dX,dY -- Fractional". For some paths the calculation of turn or direction at ceil(N) can be worked into a calculation of the direction at floor(N) so not much more work.
The following methods have base implementations calling xy_to_n()
. A subclass might implement them directly if it can be done more efficiently.
xy_is_visited() defined(xy_to_n($x,$y)) xyxy_to_n() \ xyxy_to_n_either() | calling xy_to_n_list() xyxy_to_n_list() | xyxy_to_n_list_either() /
Paths such as SquareSpiral
which fill the plane have xy_is_visited()
always true, so for them
use constant xy_is_visited => 1;
For a tree path the following methods are mandatory
tree_n_parent() tree_n_children() tree_n_to_depth() tree_depth_to_n() tree_num_children_list() tree_n_to_subheight()
The other tree methods have base implementations,
is_tree()
Checks for n_start()
having non-zero tree_n_to_num_children()
. Usually this suffices, expecting n_start()
to be a root node and to have some children.
tree_n_num_children()
Calls tree_n_children()
and counts the number of return values. Many trees can count the children with less work than calculating outright, for example RationalsTree
is simply always 2 for N>=Nstart.
tree_depth_to_n_end()
Calls tree_depth_to_n($depth+1)-1
. This assumes that the depth level ends where the next begins. This is true for the various breadth-wise tree traversals, but anything interleaved etc will need its own implementation.
tree_depth_to_n_range()
Calls tree_depth_to_n()
and tree_depth_to_n_end()
. For some paths the row start and end, or start and width, might be calculated together more efficiently.
tree_depth_to_width()
Returns tree_depth_to_n_end() - tree_depth_to_n() + 1
. This suits breadth-wise style paths where all points at $depth
are in a contiguous block. Any path not like that will need its own tree_depth_to_width()
.
tree_num_children_minimum()
, tree_num_children_maximum()
Return the first and last values of tree_num_children_list()
as the minimum and maximum.
tree_any_leaf()
Calls tree_num_children_minimum()
. If the minimum num_children
is 0 then there's leaf nodes.
Math::PlanePath::SquareSpiral, Math::PlanePath::PyramidSpiral, Math::PlanePath::TriangleSpiral, Math::PlanePath::TriangleSpiralSkewed, Math::PlanePath::DiamondSpiral, Math::PlanePath::PentSpiral, Math::PlanePath::PentSpiralSkewed, Math::PlanePath::HexSpiral, Math::PlanePath::HexSpiralSkewed, Math::PlanePath::HeptSpiralSkewed, Math::PlanePath::AnvilSpiral, Math::PlanePath::OctagramSpiral, Math::PlanePath::KnightSpiral, Math::PlanePath::CretanLabyrinth
Math::PlanePath::HexArms, Math::PlanePath::SquareArms, Math::PlanePath::DiamondArms, Math::PlanePath::AztecDiamondRings, Math::PlanePath::GreekKeySpiral, Math::PlanePath::MPeaks
Math::PlanePath::SacksSpiral, Math::PlanePath::VogelFloret, Math::PlanePath::TheodorusSpiral, Math::PlanePath::ArchimedeanChords, Math::PlanePath::MultipleRings, Math::PlanePath::PixelRings, Math::PlanePath::FilledRings, Math::PlanePath::Hypot, Math::PlanePath::HypotOctant, Math::PlanePath::TriangularHypot, Math::PlanePath::PythagoreanTree
Math::PlanePath::PeanoCurve, Math::PlanePath::WunderlichSerpentine, Math::PlanePath::WunderlichMeander, Math::PlanePath::HilbertCurve, Math::PlanePath::HilbertSides, Math::PlanePath::HilbertSpiral, Math::PlanePath::ZOrderCurve, Math::PlanePath::GrayCode, Math::PlanePath::AR2W2Curve, Math::PlanePath::BetaOmega, Math::PlanePath::KochelCurve, Math::PlanePath::DekkingCurve, Math::PlanePath::DekkingCentres, Math::PlanePath::CincoCurve
Math::PlanePath::ImaginaryBase, Math::PlanePath::ImaginaryHalf, Math::PlanePath::CubicBase, Math::PlanePath::SquareReplicate, Math::PlanePath::CornerReplicate, Math::PlanePath::LTiling, Math::PlanePath::DigitGroups, Math::PlanePath::FibonacciWordFractal
Math::PlanePath::Flowsnake, Math::PlanePath::FlowsnakeCentres, Math::PlanePath::GosperReplicate, Math::PlanePath::GosperIslands, Math::PlanePath::GosperSide
Math::PlanePath::QuintetCurve, Math::PlanePath::QuintetCentres, Math::PlanePath::QuintetReplicate
Math::PlanePath::KochCurve, Math::PlanePath::KochPeaks, Math::PlanePath::KochSnowflakes, Math::PlanePath::KochSquareflakes
Math::PlanePath::QuadricCurve, Math::PlanePath::QuadricIslands
Math::PlanePath::SierpinskiCurve, Math::PlanePath::SierpinskiCurveStair, Math::PlanePath::HIndexing
Math::PlanePath::SierpinskiTriangle, Math::PlanePath::SierpinskiArrowhead, Math::PlanePath::SierpinskiArrowheadCentres
Math::PlanePath::DragonCurve, Math::PlanePath::DragonRounded, Math::PlanePath::DragonMidpoint, Math::PlanePath::AlternatePaper, Math::PlanePath::AlternatePaperMidpoint, Math::PlanePath::TerdragonCurve, Math::PlanePath::TerdragonRounded, Math::PlanePath::TerdragonMidpoint, Math::PlanePath::R5DragonCurve, Math::PlanePath::R5DragonMidpoint, Math::PlanePath::CCurve
Math::PlanePath::ComplexPlus, Math::PlanePath::ComplexMinus, Math::PlanePath::ComplexRevolving
Math::PlanePath::Rows, Math::PlanePath::Columns, Math::PlanePath::Diagonals, Math::PlanePath::DiagonalsAlternating, Math::PlanePath::DiagonalsOctant, Math::PlanePath::Staircase, Math::PlanePath::StaircaseAlternating, Math::PlanePath::Corner
Math::PlanePath::PyramidRows, Math::PlanePath::PyramidSides, Math::PlanePath::CellularRule, Math::PlanePath::CellularRule54, Math::PlanePath::CellularRule57, Math::PlanePath::CellularRule190, Math::PlanePath::UlamWarburton, Math::PlanePath::UlamWarburtonQuarter
Math::PlanePath::DiagonalRationals, Math::PlanePath::FactorRationals, Math::PlanePath::GcdRationals, Math::PlanePath::RationalsTree, Math::PlanePath::FractionsTree, Math::PlanePath::ChanTree, Math::PlanePath::CfracDigits, Math::PlanePath::CoprimeColumns, Math::PlanePath::DivisibleColumns, Math::PlanePath::WythoffArray, Math::PlanePath::WythoffPreliminaryTriangle, Math::PlanePath::PowerArray, Math::PlanePath::File
Math::PlanePath::LCornerTree, Math::PlanePath::LCornerReplicate, Math::PlanePath::ToothpickTree, Math::PlanePath::ToothpickReplicate, Math::PlanePath::ToothpickUpist, Math::PlanePath::ToothpickSpiral, Math::PlanePath::OneOfEight, Math::PlanePath::HTree
Math::NumSeq::PlanePathCoord, Math::NumSeq::PlanePathDelta, Math::NumSeq::PlanePathTurn, Math::NumSeq::PlanePathN
math-image, displaying various sequences on these paths.
examples/numbers.pl in the Math-PlanePath source code, to print all the paths.
Math::Fractal::Curve, Math::Curve::Hilbert, Algorithm::SpatialIndex::Strategy::QuadTree
PerlMagick (module Image::Magick) demo scripts lsys.pl and tree.pl
http://user42.tuxfamily.org/math-planepath/index.html
http://user42.tuxfamily.org/math-planepath/gallery.html
Copyright 2010, 2011, 2012, 2013, 2014 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 <http://www.gnu.org/licenses/>.