[math-fun] http://mathworld.wolfram.com/SphericalSpiral.html still disputes my old rhumb (= loxodrome) math-fun
Despite the constant-heading definition, the graph (http://mathworld.wolfram.com/images/eps-gif/SphericalSpiral_700.gif) clearly shows a course approximating due west near the poles, but about 30º north of west near the equator. Here is the old(!) mail minus a long list of ~b (BCC), some deceased. math-fun@optima.cs.arizona.edu rhumb lines ~c [Eric Weisstein]@wolfram.com Date: (unfortunately missing) Under "Spherical Spiral", Eric's Stupendium (at least the versions I can access) gives "The path taken by a ship which travels from the south pole to the north pole of a Sphere while keeping a fixed (but not Right) Angle with respect to the meridians. The curve has an infinite number of loops since the separation of consecutive revolutions gets smaller and smaller near the poles. It is given by the parametric equations [...]" which are equivalent to [x,y,z]= cos(t) sin(t) a t [---------------, ---------------, - ---------------]. 2 2 2 2 2 2 sqrt(a t + 1) sqrt(a t + 1) sqrt(a t + 1) He also gives [the above-mentioned] 3D plot for some unspecified value of a, which presumably depends on the ship's heading. Leaving aside the problems of navigability at the south pole, and of having any initial heading other than due north, this still can't be right. The coils of a rhumb line do not, in fact, appear to bunch up at the poles because the curve approaches a logarithmic spiral, and has finite total arc length. It indeed has infinite winding number, but due to their exponential shrinkage, only a few coils are visually discernable. E.g., for a northwest heading, each orbit of the pole gets you a factor of e^(2 pi) ~ 535.5 closer, so you don't even see a full turn. As a further check, I get as the unit due north (tangent to the unit sphere) vector from the above curve: a t cos(t) a t sin(t) 1 [---------------, ---------------, ---------------], 2 2 2 2 2 2 sqrt(a t + 1) sqrt(a t + 1) sqrt(a t + 1) which dotted with the unit tangent to the curve gave me a - --------------------, 2 2 2 sqrt(a t + a + 1) which should not depend on t. For a heading of a radians west of north, I propose instead the formula t t [sin(t) cos(tan(a) log(tan(-))), sin(t) sin(tan(a) log(tan(-))), - cos(t)], 2 2 which runs from south pole to north as 0 < t < pi. This paramaterization has the virtue that arc length s is merely t sec a. [Now that there's Mathematica, Out[398]= {Cos[Log[Tan[t/2]] Tan[a]] Sin[t], Sin[t] Sin[Log[Tan[t/2]] Tan[a]], -Cos[t]} In[399]:= ArcLength[%, {t, 0, t}] Out[399]= t Sqrt[Sec[a]^2] ] Thus sailing northwest [a = -π/4, gosper.org/NW rhumb.png] will get you to the pole only sqrt 2 times slower than sailing due north. Unfortunately, you are killed by centrifugal whiplash [assuming you personally have a positive radius] before being able to contemplate the miracle of having sailed northwest to a point not northwest of anywhere. [ --rwg]
Nobody told me Mathematica now has extensive support for rhumb GeoGraphics! Here are six of the claimed 529(!) projections offered: GeoGraphics[{{Blue, GeoPath[{GeoPosition[{45, 90}], GeoPosition[{-45, -90}]}, "Rhumb"]}}, GeoRange -> "World", GeoProjection -> #, Epilog -> Inset[#]] & /@ {"Bonne", "Equirectangular", "Gnomonic", "Mercator", "Orthographic", "TissotConicEqualArea"} gosper.org/6rhumbs.png . Several of the 529 (notably Mercator) evidently portray rhumbs as straight line segments, suggesting an alternative definition of rhumb. GeoPath seems to automatically choose the heading with shortest rhumb, so it is not possible to spiral even once around a Pole, but these pictures pretty strongly contradict the one in MathWorld (that Mercator couldn't possibly straighten). --rwg On Sat, Mar 18, 2017 at 12:59 PM, Bill Gosper <billgosper@gmail.com> wrote:
Despite the constant-heading definition, the graph (http://mathworld.wolfram.com/images/eps-gif/SphericalSpiral_700.gif) clearly shows a course approximating due west near the poles, but about 30º north of west near the equator.
Here is the old(!) mail minus a long list of ~b (BCC), some deceased.
math-fun@optima.cs.arizona.edu rhumb lines ~c [Eric Weisstein]@wolfram.com Date: (unfortunately missing)
Under "Spherical Spiral", Eric's Stupendium (at least the versions I can access) gives "The path taken by a ship which travels from the south pole to the north pole of a Sphere while keeping a fixed (but not Right) Angle with respect to the meridians. The curve has an infinite number of loops since the separation of consecutive revolutions gets smaller and smaller near the poles. It is given by the parametric equations [...]"
which are equivalent to [x,y,z]=
cos(t) sin(t) a t [---------------, ---------------, - ---------------]. 2 2 2 2 2 2 sqrt(a t + 1) sqrt(a t + 1) sqrt(a t + 1)
He also gives [the above-mentioned] 3D plot for some unspecified value of a, which presumably depends on the ship's heading.
Leaving aside the problems of navigability at the south pole, and of having any initial heading other than due north, this still can't be right. The coils of a rhumb line do not, in fact, appear to bunch up at the poles because the curve approaches a logarithmic spiral, and has finite total arc length. It indeed has infinite winding number, but due to their exponential shrinkage, only a few coils are visually discernable. E.g., for a northwest heading, each orbit of the pole gets you a factor of e^(2 pi) ~ 535.5 closer, so you don't even see a full turn.
As a further check, I get as the unit due north (tangent to the unit sphere) vector from the above curve:
a t cos(t) a t sin(t) 1 [---------------, ---------------, ---------------], 2 2 2 2 2 2 sqrt(a t + 1) sqrt(a t + 1) sqrt(a t + 1)
which dotted with the unit tangent to the curve gave me
a - --------------------, 2 2 2 sqrt(a t + a + 1)
which should not depend on t.
For a heading of a radians west of north, I propose instead the formula
t t [sin(t) cos(tan(a) log(tan(-))), sin(t) sin(tan(a) log(tan(-))), - cos(t)], 2 2
which runs from south pole to north as 0 < t < pi. This paramaterization has the virtue that arc length s is merely t sec a.
[Now that there's Mathematica, Out[398]= {Cos[Log[Tan[t/2]] Tan[a]] Sin[t], Sin[t] Sin[Log[Tan[t/2]] Tan[a]], -Cos[t]}
In[399]:= ArcLength[%, {t, 0, t}]
Out[399]= t Sqrt[Sec[a]^2] ] Thus sailing northwest [a = -π/4, gosper.org/NW rhumb.png <http://gosper.org/NW%20rhumb.png>]
will get you to the pole only sqrt 2 times slower than sailing due north. Unfortunately, you are killed by centrifugal whiplash [assuming you personally have a positive radius] before being able to contemplate the miracle of having sailed northwest to a point not northwest of anywhere. [ --rwg]
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Bill Gosper