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Math::PlanePath::TheodorusSpiral -- right-angle unit step spiral
use Math::PlanePath::TheodorusSpiral; my $path = Math::PlanePath::TheodorusSpiral->new; my ($x, $y) = $path->n_to_xy (123);
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.
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
$non the path.
$ncan be any value
$n >= 0and fractions give positions on the spiral in between the integer points.
$n < 0the 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
$nis negative. For integer
$nthis is simply
$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
$x,$ywithin 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
$x,$yis not within one of the unit circles then the return is
$str = $path->figure ()
Return string "circle".
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
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 so as 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 don't currently have a good closed-form or converging series to invert, or apply some Newton's method, or whatever.
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.
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
A072895 N just below X axis A137515 N-1 just below X axis counting num points for n revolutions A172164 loop length increases A164102 2*pi^2
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
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