Wikipedia:Reference desk/Archives/Mathematics/2023 November 23

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November 23

Radii of Craig retroazimuthal projection

CC @Strebe:

Craig retroazimuthal projection centered on Mecca

The Craig retroazimuthal projection is overly stretched in the extremities, making it difficult to use it there and also cutting off outlying regions.

It occurred to me that if finding direction (instead of distance) is the requirement, one could scale the radius appropriately as in a fisheye lens without affecting bearing, perhaps like so:

${\begin{aligned}x&=\lambda -\lambda _{0}\\y&={\frac {\lambda -\lambda _{0}}{\sin \left(\lambda -\lambda _{0}\right)}}{\Big (}\sin \varphi \cos \left(\lambda -\lambda _{0}\right)-\tan \varphi _{0}\cos \varphi {\Big )}\\r&={\sqrt {x^{2}+y^{2}}}\\\theta &={\text{arctan2}}(y,x)\\x'&=r^{k}\cos \theta \\y'&=r^{k}\sin \theta \end{aligned}}$

for some constant 0 < k < 1. Is there a reason this isn't done?

Thanks,
cmɢʟee⎆τaʟκ 07:39, 23 November 2023 (UTC)

The main reason scaling the radius is not a common thing may be that the projection is mainly of historical interest. It is not hard to create an algorithm to compute the bearing from a source to a target, given their geographic coordinates, which can be further refined by modelling the shape of the Earth as a spheroid.

There are three hidden parameters in Craig's projection that can be used to reduce local distortions. If the preservation of bearing with respect to a target is the main objective, there is no specific reason to use the conventional latitude–longitude grid in which the North and South poles have latitude 90° and Greenwich has longitude 0°. By applying a 3D rotation to the grid before applying the projection, distortion in an area of interest can be diminished. One may furthermore replace the target by its antipodal point, in which the bearing to the target on the projection is given by the opposite direction of the bearing to the antitarget. (This will be expedient if one wants to found a mosque near Tematagi :).)

BTW, the modification you proposed introduces strange deformations near the target. In using r′ = f(r), instead of using f(r) = r^k, you should use a function f with a tame derivative near the origin.  --Lambiam 09:03, 23 November 2023 (UTC)

Thanks, @Lambiam: Good to know. Would you be able to suggest a projection which has the 3D rotation you described?

You're right that k → 0, r^k → 1. Cheers, cmɢʟee⎆τaʟκ 23:51, 23 November 2023 (UTC)

Here is a possible approach, totally original research. Let $T$ stand for the coordinate transformation, expressed in the geographical coordinate system of latitude and longitude:

$T(\varphi ,\lambda )=(T_{\varphi }(\varphi ,\lambda ),T_{\lambda }(\varphi ,\lambda ))=(\varphi ',\lambda ').$

Rather in general, we want to keep the absolute values of the 2D $x$ and $y$ (as calculated using the formulas of Craig's projection) small in the area of interest, say centred around a "source" with coordinates $(\varphi _{1},\lambda _{1}).$

The first factor of the formula for $y$ is the inverse of ${\text{sinc}}(x),$ which has an asymptote when $x=\lambda -\lambda _{0}\to \pi ,$ so keeping $x$ in check is particularly important. This suggests the constraint $\lambda _{1}'-\lambda _{0}'=0.$ In words, the rotation should bring source and target to the same meridian.

Another potential source of wild behaviour is the occurrence of $\tan \varphi _{0},$ so we add the constraint $\varphi _{0}'=0,$ that is,

$T_{\varphi }(\varphi _{0},\lambda _{0})=0.$

At this point Craig's formulas have become simplified to

$(x,y)=(0,\sin \varphi ).$

Since ${\text{SO}}(3)$ has 3 degrees of freedom, we have room for one more constraint, for which we might be tempted to pick $\varphi _{1}'=0.$ However, together with the first constraint this would imply $T(\varphi _{0},\lambda _{0})=T(\varphi _{1},\lambda _{1}),$ which is not possible unless the source coincides with the target. Instead, just for definiteness, we can pick $\lambda _{1}'=0,$ which can be combined with the first constraint $\lambda _{1}'-\lambda _{0}'=0$ to give:

$T_{\lambda }(\varphi _{1},\lambda _{1})=T_{\lambda }(\varphi _{0},\lambda _{0})=0.$

So the target should be brought to Null Island $(0,0)$ and the source to the prime meridian through the new target. Alternatively, one might consider bringing the point halfway between source and target to $(0,0)$, so that source and target are on the prime meridian at equal distances from the new equator.  --Lambiam 07:47, 24 November 2023 (UTC)

Great work, thanks very much! cmɢʟee⎆τaʟκ 07:40, 26 November 2023 (UTC)

Another reason why there may be little interest in modifications of the Craig retroazimuthal projection is probably that its purpose – giving the bearing form any source location to a fixed given target location – can be achieved in a much simpler way by any of several azimuthal projections, putting the zenith directly above the target. The azimuthal conformal projection keeps local distortions small in any modestly large area of interest.  --Lambiam 14:34, 24 November 2023 (UTC)

Good thinking though I think that would require the user to rotate the map: if north is upwards at the zenith, it isn't so away from the meridian it is on up to the north pole. cmɢʟee⎆τaʟκ 07:42, 26 November 2023 (UTC)

The original projection invented by Craig does indeed have the property that all meridians are parallel and equally spaced, a property it shares with the common cylindrical projections. This property is not preserved when $(x,y)$ is replaced by $(r'\cos \theta ,r'\sin \theta )=(\rho \,x,\rho \,y)$ where $\rho =r'/r,$ so I did not assume it was important.  --Lambiam 22:27, 26 November 2023 (UTC)