SIMPLE SOLUTIONS

# MATH::PLANEPATH::THEODORUSSPIRAL(3PM) - man page online | library functions

Chapter
2016-01-11
```Math::PlanePath::TheodorusSpiraUsermContributed Perl DocumenMath::PlanePath::TheodorusSpiral(3pm)

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 spoke.  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 which is 45 degrees, etc.

The radial distance 0 to 2 is sqrt(2), 0 to 3 is sqrt(3), and in general

R = sqrt(N)

because each step is a right triangle with radius(N+1)^2 = radius(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 values give positions on the straight line between the integer
points.  (An analytic continuation for a rounded curve between points is possible, but not
currently implemented.)

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

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

The code is currently implemented by adding 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), and previous positions 1000 apart are saved for re-use or
to go back.

"\$path = Math::PlanePath::TheodorusSpiral->new ()"
Create and return a new Theodorus spiral 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.)

"\$rsquared = \$path->n_to_rsquared (\$n)"
Return the radial distance R^2 of point \$n, or "undef" if \$n is negative.  For integer
\$n this is simply \$n itself.

"\$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
roughly 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".

"\$str = \$path->figure ()"
Return string "circle".

FORMULAS
N to RSquared
For integer N the spiral has radius R=sqrt(N) and the square is simply RSquared=R^2=N.
For fractional N the point is on a straight line at right angles to the integer position,
so

R = hypot(sqrt(Ninteger), Nfrac)
RSquared = (sqrt(Ninteger))^2 + Nfrac^2
= Ninteger + Nfrac^2

X,Y to N
For a given X,Y the radius R=hypot(X,Y) determines the N position as N=R^2.  An N point up
to 0.5 away radially might cover X,Y, so the range of N to consider is

Nlo = (R-.5)^2
Nhi = (R+.5)^2

A simple search is made through those N's seeking which, if any, covers X,Y.  The number
of N's searched is Nhi-Nlo = 2*R+1 which is about 1/3 of a loop around the spiral
(2*R/2*pi*R ~= 1/3).  Actually 0.51 is used to guard against floating point round-off,
which is then about 4*.51 = 2.04*R many points.

The angle of the X,Y position determines which part of the spiral is intersected, but
using that doesn't seem particularly easy.  The angle for a given N is an arctan sum and
there doesn't seem to be a good closed-form or converging series to invert, or apply some
Newton's method, or whatever.

Rectangle to N Range
For "rect_to_n_range()" the corner furthest from the origin determines the high N.  For
that corner

Rhi = hypot(xhi,yhi)
Nhi = (Rhi+.5)^2

The extra .5 is since a unit circle figure centred as much as .5 further out might
intersect the xhi,yhi.  The square root hypot() can be avoided by the following over-
estimate, and ceil can keep it in integers for integer Nhi.

Nhi = Rhi^2 + Rhi + 1/4
<= Xhi^2+Yhi^2 + Xhi+Yhi + 1      # since Rhi<=Xhi+Yhi
= Xhi*(Xhi+1) + Yhi*(Yhi+1) + 1
<= ceilXhi*(ceilXhi+1) + ceilYhi*(ceilYhi+1) + 1

With either formula the worst case is when Nhi doesn't intersect the xhi,yhi corner but is
just before it, anti-clockwise.  Nhi is then a full revolution bigger than it need be,
depending where the other corners fall.

Similarly for the corner or axis crossing nearest the origin (when the origin itself isn't
covered by the rectangle),

Rlo = hypot(Xlo,Ylo)
Nlo = (Rlo-.5)^2, or 0 if origin covered by rectangle

And again in integers without a square root if desired,

Nlo = Rlo^2 - Rlo + 1/4
>= Xlo^2+Ylo^2 - (Xlo+Ylo)        # since Xlo+Ylo>=Rlo
= Xlo*(Xlo-1) + Ylo*(Ylo-1)
>= floorXlo*(floorXlo-1) + floorYlo(floorYlo-1)

The worst case is when this Nlo doesn't intersect the xlo,ylo corner but is just after it
anti-clockwise, so Nlo is a full revolution smaller than it need be.

OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

<http://oeis.org/A072895> (etc)

A072895    N just below X axis
A137515    N-1 just below X axis
counting num points for n revolutions
A172164    loop length increases

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

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

Copyright 2010, 2011, 2012, 2013, 2014, 2015 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