Math::PlanePath::ArchimedeanChords(3) radial spiral chords


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


This path puts points at unit chord steps along an Archimedean spiral. The spiral goes outwards by 1 unit each revolution and the points are spaced 1 apart.

    R = theta/(2*pi)

The result is roughly

                   32          30         ...                3
             33                   29
       34       15          13          28    50             2
             16        3
    35                       2             27    49          1
                    4                11
    36           5        0     1          26    48     <- Y=0
    37              6                      25    47         -1
             19        7     8          24    46
       38                                                   -2
                20                23
          39          21    22             45
                40                   44
                   41    42    43
       -3    -2    -1    X=0    1     2     3     4

X,Y positions returned are fractional. Each revolution is about 2*pi longer than the previous, so the effect is a kind of 6.28 increment looping.

Because the spacing is by unit chords, adjacent unit circles centred on each N position touch but don't overlap. The spiral spacing of 1 unit per revolution means they don't overlap radially either.

The unit chords here are a little like the "TheodorusSpiral". But the "TheodorusSpiral" goes by unit steps at a fixed right-angle and approximates an Archimedean spiral (of 3.14 radial spacing). Whereas this "ArchimedeanChords" is an actual Archimedean spiral (of radial spacing 1), with unit steps angling along that.


See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::ArchimedeanChords->new ()"
Create and return a new path object.
"($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 chord between the integer points (ie. straight line between the points). "$n==0" is the origin 0,0.

For "$n < 0" the return is an empty list, it being considered there are no negative points in the spiral.

"$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 spacing of the spiral means those circles don't overlap, but they also don't cover the plane and if "$x,$y" is not within one then the return is "undef".

The current implementation is a bit slow.

"$n = $path->n_start ()"
Return 0, the first $n on the path.
"$str = $path->figure ()"
Return ``circle''.


N to X,Y

The current code keeps a position as a polar angle t and calculates an increment u needed to move along by a unit chord. If dist(u) is the straight-line distance between t and t+u, then squared is the hypotenuse

    dist^2(u) =   ((t+u)/2pi*cos(t+u) - t/2pi*cos(t))^2     # X
                + ((t+u)/2pi*sin(t+u) - t/2pi*sin(t))^2     # Y

which simplifies to

    dist^2(u) = [ (t+u)^2 + t^2 - 2*t*(t+u)*cos(u) ] / (4*pi^2)

Switch from cos to sin using the half angle cos(u) = 1 - 2*sin^2(u/2) in case if u is small then the cos(u) near 1.0 might lose floating point accuracy, and also as a slight simplification,

    dist^2(u) = [ u^2 + 4*t*(t+u)*sin^2(u/2) ] / (4*pi^2)

Then want the u which has dist(u)=1 for a unit chord. The u*sin(u) part probably doesn't have a good closed form inverse, so the current code is a Newton/Raphson iteration on f(u) = dist^2(u)-1, seeking f(u)=0

    f(u) = u^2 + 4*t*(t+u)*sin^2(u/2) - 4*pi^2

Derivative f'(u) for the slope from the cos form is

    f'(u) = 2*(t+u) - 2*t*[ cos(u) - (t+u)*sin(u) ]

And again switching from cos to sin in case u is small,

    f'(u) = 2*[ u + t*[2*sin^2(u/2) + (t+u)*sin(u)] ]

X,Y to N

A given x,y point is at a fraction of a revolution

    frac = atan2(y,x) / 2pi     # -.5 <= frac <= .5
    frac += (frac < 0)          # 0 <= frac < 1

And the nearest spiral arm measured radially from x,y is then

    r = int(hypot(x,y) + .5 - frac) + frac

Perl's "atan2" is the same as the C library and gives -pi <= angle <= pi, hence allowing for frac<0. It may also be ``unspecified'' for x=0,y=0, and give +/-pi for x=negzero, which has to be a special case so 0,0 gives r=0. The "int" rounds towards zero, so frac>.5 ends up as r=0.

So the N point just before or after that spiral position may cover the x,y, but how many N chords it takes to get around to there is 's not so easily calculated.

The current code looks in saved "n_to_xy()" positions for an N below the target, and searches up from there until past the target and thus not covering x,y. With "n_to_xy()" points saved 500 apart this means searching somewhere between 1 and 500 points.

One possibility for calculating a lower bound for N, instead of the saved positions, and both for "xy_to_n()" and "rect_to_n_range()", would be to add up chords in circles. A circle of radius k fits pi/arcsin(1/2k) many unit chords, so

             k=floor(r)     pi
    total = sum         ------------
             k=0        arcsin(1/2k)

and this is less than the chords along the spiral. Is there a good polynomial over-estimate of arcsin, to become an under-estimate total, without giving away so much?

Rectangle to N Range

For the "rect_to_n_range()" upper bound, the current code takes the arc length along with spiral with the usual formula

    arc = 1/4pi * (theta*sqrt(1+theta^2) + asinh(theta))

Written in terms of the r radius (theta = 2pi*r) as calculated from the biggest of the rectangle x,y corners,

    arc = pi*r^2*sqrt(1+1/r^2) + asinh(2pi*r)/4pi

The arc length is longer than chords, so N=ceil(arc) is an upper bound for the N range.

An upper bound can also be calculated simply from the circumferences of circles 1 to r, since a spiral loop from radius k to k+1 is shorter than a circle of radius k.

    total = sum         2pi*k
          = pi*r*(r+1)

This is bigger than the arc length, thus a poorer upper bound, but an easier calculation. (Incidentally, for smallish r have arc length <= pi*(r^2+1) which is a tighter bound and an easy calculation, but it only holds up to somewhere around r=10^7.)


Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016 Kevin Ryde

This file is part of Math-PlanePath.

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