Kevin Ryde > Math-PlanePath > Math::PlanePath::QuadricIslands

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Module Version: 116   Source  

NAME ^

Math::PlanePath::QuadricIslands -- quadric curve rings

SYNOPSIS ^

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

DESCRIPTION ^

This is concentric islands made from four sides of the QuadricCurve,

                                  27--26                     3
                                   |   |
                              29--28  25  22--21             2
                               |       |   |   |
                              30--31  24--23  20--19         1
                                   | 4--3          |
                          34--33--32    | 16--17--18     <- Y=0
                           |         1--2  |
                          35--36   7---8  15--14            -1
                                   |   |       |
                               5---6   9  12--13            -2
                                       |   |
                          55--56      10--11                -3
                           |   |
      ...             53--54  57  60--61                    -4
                       |       |   |   |
                      52--51  58--59  62--63                -5
                           |               |
                  48--49--50      66--65--64                -6
                   |               |
          39--40  47--46          67--68                    -7
           |   |       |               |
      37--38  41  44--45              69                    -8
               |   |                   |
              42--43                  70--71                -9
                                           |
                                  74--73--72               -10
                                   |
                                  75--76  79--80      ...  -11
                                       |   |   |       |
                                      77--78  81  84--85   -12
                                               |   |
                                              82--83       -13

                                       ^
      -8  -7  -6  -5  -4  -3  -2  -1  X=0  1   2   3   4

The initial figure is the square N=1,2,3,4 then for the next level each straight side expands to 4x longer and a zigzag like N=5 through N=13 and the further sides to N=36.

                                *---*
                                |   |
      *---*     becomes     *---*   *   *---*
                                    |   |
                                    *---*

Level Ranges

Counting the innermost square as level 0, each ring is

    length = 4 * 8^level     many points
    Nstart = 1 + length[0] + ... + length[level-1]
           = (4*8^level + 3)/7
    Xstart = - 4^level / 2
    Ystart = - 4^level / 2

For example the lower partial ring shown above is level 2 starting N=(4*8^2+3)/7=37 at X=-(4^2)/2=-8,Y=-8.

The innermost square N=1,2,3,4 is on 0.5 coordinates, for example N=1 at X=-0.5,Y=-0.5. This is centred on the origin and consistent with the (4^level)/2. Points from N=5 onwards are integer X,Y.

       4-------3    Y=+1/2
       |       |
       |   o   |
               |
       1-------2    Y=-1/2

    X=-1/2   X=+1/2

FUNCTIONS ^

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

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

Create and return a new path object.

SEE ALSO ^

Math::PlanePath, Math::PlanePath::QuadricCurve, Math::PlanePath::KochSnowflakes, Math::PlanePath::GosperIslands

HOME PAGE ^

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

LICENSE ^

Copyright 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/>.

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