NAME

Math::PlanePath::TheodorusSpiral -- right-angle unit step spiral

SYNOPSIS

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

DESCRIPTION

This path puts points on the spiral of Theodorus, also called the square root spiral.

                                   61                 6
                                     60
               27 26 25 24                            5
            28            23           59
          29                 22          58           4

       30                      21         57          3
      31                         20
                   4                       56         2
     32          5    3          19
               6         2                 55         1
    33                            18
              7       0  1                 54    <- y=0
    34                           17
              8                            53        -1
    35                          16
               9                          52         -2
     36                       15
                 10         14           51          -3
      37           11 12 13            50
                                                     -4
        38                           49
          39                       48                -5
            40                  47
               41             46                     -6
                  42 43 44 45


                      ^
   -6 -5 -4 -3 -2 -1 x=0 1  2  3  4  5  6  7

Each step is a unit distance at right angles to the previous radial. So for example,

         3        -- y=1+1/sqrt(2)
          \
           \
             2       y=1
             |
             |
       0-----1    <- y=0

       ^
      x=0   x=1

1 to 2 is a unit step at right angles to the 0 to 1 radial. Then 2 to 3 steps at a right angle to radial 0 to 2 (menaing 45 degrees up to the left), etc. The distance 0 to 2 is sqrt(2), the distance 0 to 3 is sqrt(3), and in general r(N) = sqrt(N) since each step is a right triangle, r(N+1)^2 = r(N)^2 + 1. The resulting shape is very close to an Archimedean spiral with successive loops increasing in radius by pi (3.14159) or thereabouts each time.

X,Y positions returned are fractional and each integer N position is exactly 1 away from the previous. Fractional N gives positions on the straight line between the integer points.

Each loop is just under 2*pi^2 (19.7392) longer than the previous. This means quadratic points 9.8696*k^2 for integer k form an almost straight line. Quadratics close to 9.87 or a square multiple of it nearly line up. For example the 22-polygonal numbers have 10*k^2 and at low values nearly line up but then spiral away.

FUNCTIONS

The code is currently implemented by adding up unit steps in X,Y coordinates, so it's not particularly fast. The last X,Y is saved in the object anticipating successively higher N (not necessarily consecutive), plus past positions 1000 apart ready for re-use or to go back to.

$path = Math::PlanePath::TheodorusSpiral->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.

$n can be any value $n >= 0 and fractions give positions on the spiral in between the integer points.

For $n < 0 the return is an empty list, it being currently considered there are no negative points in the spiral. (The analytic continuation by Davis would be a possibility, though the resulting "inner spiral" makes positive and negative points overlap a bit. A spiral starting at x=-1 would fit in between the positive points.)

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

Return an integer point number for coordinates $x,$y. Each integer N is considered the centre of a circle of diameter 1 and an $x,$y within that circle returns N.

The unit steps of the spiral means those unit circles don't overlap, but the loops are 3.14 apart so there's gaps in between. If $x,$y is not within one of the unit circles then the return is undef.

SEE ALSO

Math::PlanePath, Math::PlanePath::SacksSpiral, Math::PlanePath::MultipleRings

HOME PAGE

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

LICENSE

Math-PlanePath is Copyright 2010 Kevin Ryde

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/>.