/ 
Math-PlanePath-76
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::PeanoHalf

NAME

Math::PlanePath::PeanoHalf -- 9-segment self-similar spiral

SYNOPSIS

use Math::PlanePath::PeanoHalf;
my $path = Math::PlanePath::PeanoHalf->new;
my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This is an integer version of a 9-segment self-similar curve by ...

     4

     3

     2

     1

<- Y=0

    -1

    -2

    -3

    -4

   -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8  9 10 11 12

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Arms

The optional arms => 2 parameter can give a second copy of the spiral rotated 180 degrees. With two arms all points of the plane are covered.

 93--91  81--79--77--75  57--55  45--43--41--39 122-124  ..
  |   |   |           |   |   |   |           |   |   |   |
 95  89  83  69--71--73  59  53  47  33--35--37 120 126 132 
  |   |   |   |           |   |   |   |           |   |   | 
 97  87--85  67--65--63--61  51--49  31--29--27 118 128-130 
  |                                           |   |
 99-101-103  22--20  10-- 8-- 6-- 4  13--15  25 116-114-112 
          |   |   |   |           |   |   |   |           | 
109-107-105  24  18  12   1   0-- 2  11  17  23 106-108-110 
  |           |   |   |   |           |   |   |   |         
111-113-115  26  16--14   3-- 5-- 7-- 9  19--21 104-102-100 
          |   |                                           | 
129-127 117  28--30--32  50--52  62--64--66--68  86--88  98 
  |   |   |           |   |   |   |           |   |   |   |
131 125 119  38--36--34  48  54  60  74--72--70  84  90  96 
  |   |   |   |           |   |   |   |           |   |   | 
 .. 123-121  40--42--44--46  56--58  76--78--80--82  92--94

The first arm is the even numbers N=0,2,4,etc and the second arm is the odd numbers N=1,3,5,etc.

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.

$path = Math::PlanePath::PeanoHalf->new ()

Create and return a new path object.

($x,$y) = $path->n_to_xy ($n)

Return the X,Y coordinates of point number $n on the path. Points begin at 0 and if $n < 0 then the return is an empty list.

FORMULAS

X,Y to N

The correspondence to Wunderlich's 3x3 serpentine curve can be used to turn X,Y coordinates in base 3 into an N. Reckoning the innermost 3x3 as level=1 then the smallest abs(X) or abs(Y) in a level is

Xlevelmin = (3^level + 1) / 2
eg. level=2 Xlevelmin=5

which can be reversed as

level = log3floor( max(abs(X),abs(Y)) * 2 - 1 )
eg. X=7 level=log3floor(2*7-1)=2

An offset can be applied to put X,Y in the range 0 to 3^level-1,

offset = (3^level-1)/2
eg. level=2 offset=4

Then a table can give the N base-9 digit corresponding to X,Y digits

Y=2   4   3   2      N digit
Y=1  -1   0   1
Y=0  -2  -3  -4
     X=0 X=1 X=2

A current rotation maintains the "S" part directions and is updated by a table

Y=2   0  +3   0     rotation when descending
Y=1  +1  +2  +1     into sub-part
Y=0   0  +3   0
     X=0 X=1 X=2

The negative digits of N represent backing up a little in some higher part. If N goes negative at any state then X,Y was off the main curve and instead on the second arm. If the second arm is not of interest the calculation can stop at that stage.

It no doubt would also work to take take X,Y as balanced ternary digits 1,0,-1, but it's not clear that would be any faster or easier to calculate.

SEE ALSO

Math::PlanePath, Math::PlanePath::PeanoCurve

HOME PAGE

http://user42.tuxfamily.org/math-planepath/index.html

LICENSE

Copyright 2011, 2012 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/>.