Math::PlanePath::TriangleSpiralSkewed -- integer points drawn around a skewed equilateral triangle
use Math::PlanePath::TriangleSpiralSkewed; my $path = Math::PlanePath::TriangleSpiralSkewed->new; my ($x, $y) = $path->n_to_xy (123);
This path makes an spiral shaped as an equilateral triangle (each side the same length), but skewed to the left to fit on a square grid,
16 4 |\ 17 15 3 | \ 18 4 14 2 | |\ \ 19 5 3 13 1 | | \ \ 20 6 1--2 12 ... <- Y=0 | | \ \ 21 7--8--9-10-11 30 -1 | \ 22-23-24-25-26-27-28-29 -2 ^ -2 -1 X=0 1 2 3 4 5
The properties are the same as the spread-out TriangleSpiral
. The triangle numbers fall on straight lines as the do in the TriangleSpiral
but the skew means the top corner goes up at an angle to the vertical and the left and right downwards are different angles plotted (but are symmetric by N count).
Option skew => 'right'
directs the skew towards the right, giving
4 16 skew="right" / | 3 17 15 / | 2 18 4 14 / / | | 1 ... 5 3 13 / | | Y=0 -> 6 1--2 12 / | -1 7--8--9-10-11 ^ -2 -1 X=0 1 2
This is a shear "X -> X+Y" of the default skew="left" shown above. The coordinates are related by
Xright = Xleft + Yleft Xleft = Xright - Yright Yright = Yleft Yleft = Yright
2 16-15-14-13-12-11 skew="up" | / 1 17 4--3--2 10 | | / / Y=0 -> 18 5 1 9 | | / -1 ... 6 8 |/ -2 7 ^ -2 -1 X=0 1 2
This is a shear "Y -> X+Y" of the default skew="left" shown above. The coordinates are related by
Xup = Xleft Xleft = Xup Yup = Yleft + Xleft Yleft = Yup - Xup
2 ..-18-17-16 skew="down" | 1 7--6--5--4 15 \ | | Y=0 -> 8 1 3 14 \ \ | | -1 9 2 13 \ | -2 10 12 \ | 11 ^ -2 -1 X=0 1 2
This is a rotate by -90 degrees of the skew="up" above. The coordinates are related
Xdown = Yup Xup = - Ydown Ydown = - Xup Yup = Xdown
Or related to the default skew="left" by
Xdown = Yleft + Xleft Xleft = - Ydown Ydown = - Xleft Yleft = Xdown + Ydown
The default is to number points starting N=1 as shown above. An optional n_start
can give a different start, with the same shape etc. For example to start at 0,
15 n_start => 0 |\ 16 14 | \ 17 3 13 ... | |\ \ \ 18 4 2 12 31 | | \ \ \ 19 5 0--1 11 30 | | \ \ 20 6--7--8--9-10 29 | \ 21-22-23-24-25-26-27-28
With this adjustment for example the X axis N=0,1,11,30,etc is (9X-7)*X/2, the hendecagonal numbers (11-gonals). And South-East N=0,8,25,etc is the hendecagonals of the second kind, (9Y-7)*Y/2 with Y negative.
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::TriangleSpiralSkewed->new ()
$path = Math::PlanePath::TriangleSpiralSkewed->new (skew => $str, n_start => $n)
Create and return a new skewed triangle spiral object. The skew
parameter can be
"left" (the default) "right" "up" "down"
$n = $path->xy_to_n ($x,$y)
Return the point number for coordinates $x,$y
. $x
and $y
are each rounded to the nearest integer, which has the effect of treating each N in the path as centred in a square of side 1, so the entire plane is covered.
Within each row there's a minimum N and the N values then increase monotonically away from that minimum point. Likewise in each column. This means in a rectangle the maximum N is at one of the four corners of the rectangle.
| x1,y2 M---|----M x2,y2 maximum N at one of | | | the four corners -------O--------- of the rectangle | | | | | | x1,y1 M---|----M x1,y1 |
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A117625 (etc)
n_start=1, skew="left" (the defaults) A204439 abs(dX) A204437 abs(dY) A010054 turn 1=left,0=straight, extra initial 1 A117625 N on X axis A064226 N on Y axis, but without initial value=1 A006137 N on X negative A064225 N on Y negative A081589 N on X=Y leading diagonal A038764 N on X=Y negative South-West diagonal A081267 N on X=-Y negative South-East diagonal A060544 N on ESE slope dX=+2,dY=-1 A081272 N on SSE slope dX=+1,dY=-2 A217010 permutation N values of points in SquareSpiral order A217291 inverse A214230 sum of 8 surrounding N A214231 sum of 4 surrounding N n_start=0 A051682 N on X axis (11-gonal numbers) A081268 N on X=1 vertical (next to Y axis) A062708 N on Y axis A062725 N on Y negative axis A081275 N on X=Y+1 North-East diagonal A062728 N on South-East diagonal (11-gonal second kind) A081266 N on X=Y negative South-West diagonal A081270 N on X=1-Y North-West diagonal, starting N=3 A081271 N on dX=-1,dY=2 NNW slope up from N=1 at X=1,Y=0 n_start=-1 A023531 turn 1=left,0=straight, being 1 at N=k*(k+3)/2 A023532 turn 1=straight,0=left n_start=1, skew="right" A204435 abs(dX) A204437 abs(dY) A217011 permutation N values of points in SquareSpiral order but with 90-degree rotation A217292 inverse A214251 sum of 8 surrounding N n_start=1, skew="up" A204439 abs(dX) A204435 abs(dY) A217012 permutation N values of points in SquareSpiral order but with 90-degree rotation A217293 inverse A214252 sum of 8 surrounding N n_start=1, skew="down" A204435 abs(dX) A204439 abs(dY)
The square spiral order in A217011,A217012 and their inverses has first step at 90-degrees to the first step of the triangle spiral, hence the rotation by 90 degrees when relating to the SquareSpiral
path. A217010 on the other hand has no such rotation since it reckons the square and triangle spirals starting in the same direction.
Math::PlanePath, Math::PlanePath::TriangleSpiral, Math::PlanePath::PyramidSpiral, Math::PlanePath::SquareSpiral
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016 Kevin Ryde
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