| Math::PlanePath::TriangleSpiralSkewed(3pm) | User Contributed Perl Documentation | Math::PlanePath::TriangleSpiralSkewed(3pm) |
NAME¶
Math::PlanePath::TriangleSpiralSkewed -- integer points drawn around a skewed equilateral triangle
SYNOPSIS¶
use Math::PlanePath::TriangleSpiralSkewed; my $path = Math::PlanePath::TriangleSpiralSkewed->new; my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION¶
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 they 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).
Skew Right¶
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
Skew Up¶
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
Skew Down¶
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
N Start¶
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.
FUNCTIONS¶
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.
FORMULAS¶
Rectangle to N Range¶
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
|
OEIS¶
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
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.
SEE ALSO¶
Math::PlanePath, Math::PlanePath::TriangleSpiral, Math::PlanePath::PyramidSpiral, Math::PlanePath::SquareSpiral
HOME PAGE¶
LICENSE¶
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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/>.
| 2021-01-23 | perl v5.32.0 |