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In addition to solving the geodesic problem for the triaxial ellipsoid, Jacobi (1839) briefly mentions the problem of the conformal projection of ellipsoid. He covers this in greater detail in Vorlesungen über Dynamik, §28, which is now available in an English translation: Lectures on Dynamics (errata).
Jacobi gives the following conformal mapping of the triaxial ellipsoid onto a plane
\[\begin{align} x(\omega) &= b \int \frac {\sqrt{1 + e^2 k'^2 \sin^2\omega}} {\sqrt{k^2 + k'^2\sin^2\omega}}\, d\omega, \\ y(\beta) &= b \int \frac {\sqrt{1 - e^2 k^2 \cos^2\beta}} {\sqrt{k'^2 + k^2 \cos^2\beta}}\, d\beta. \end{align} \]
The scale of the projection is
\[m(\beta, \omega) = \frac1 {\sqrt{k^2 \cos^2\beta + k'^2 \sin^2\omega}}. \]
I have scaled the Jacobi's projection by a constant factor,
\[\frac{\sqrt{a^2-c^2}}2 = \frac{eb}2, \]
so that it reduces to the familiar formulas in the case of an oblate ellipsoid. This mapping is an immediate consequence of the special form of the expression for \(ds^2\),
\[\begin{align} \frac{ds^2}{k^2\cos^2\beta + k'^2\sin^2\omega} &= b^2\frac{1 + e^2 k'^2 \sin^2\omega}{k^2 + k'^2\sin^2\omega} d\omega^2 \\ &\qquad+b^2\frac{1 - e^2 k^2 \cos^2\beta}{k'^2 + k^2\cos^2\beta} d\beta^2. \end{align} \]
The projection may be expressed in terms of elliptic integrals,
\[\begin{align} x(\omega)&=\frac{a^2}b \Pi(\omega',-e^2k'^2, \cos\nu),\\ y(\beta)&=\frac{c^2}b \Pi(\beta' , e^2k^2, \sin\nu), \end{align} \]
where
\[\begin{align} \tan\omega' &= \frac ba \tan(\omega - \pi/2), \\ \tan\beta' &= \frac bc \tan\beta, \\ \sin\nu &= k \sqrt{1 + e^2k'^2}, \\ \cos\nu &= k' \sqrt{1 - e^2k^2}, \end{align} \]
\(\nu\) is the geographic latitude of the umbilic point at \(\beta = \frac12\pi\), \(\omega = 0\) (the angle a normal at the umbilic point makes with the equatorial plane), and \(\Pi(\phi,\alpha^2,k)\) is the incomplete elliptic integral of the third kind, https://dlmf.nist.gov/19.2.E7. This allows the projection to be numerically computed using EllipticFunction::Pi(real phi) const. The origin of the projection is \(\beta = 0\), \(\omega = \frac12\pi\) (where the scale \(m = 1\)); as a result, the projection simply reduces to the Mercator projection in both the oblate and prolate limits (see Limiting cases). The transformation involving tangents, that connects \(\omega'\) to \(\omega\) and \(\beta'\) to \(\beta\), should be understood to preserve the quadrant of the angles though multiple revolutions. As a result, the projection can be "unrolled" arbitrarily far in \(\omega\) and \(\beta\). However, in the oblate (resp. prolate) limit, we require \(\beta \in [-\tfrac12\pi,\tfrac12\pi]\) (resp. \(\omega \in [0,\pi]\)).
Nyrtsov, et al.,
also expressed the projection in terms of elliptic integrals. However the equations are more complex involving elliptic integrals of the first and third kinds. More seriously, the origin of their elliptic integrals is taken to be the coordinates of the umbilical point. As a result, one of the projection equations diverges in the biaxial limit. The relations https://dlmf.nist.gov/19.7.E5 can be used to put their results in the simpler form given here.
\(x\) (resp. \(y\)) depends on \(\omega\) (resp. \(\beta\)) alone, so that latitude-longitude grid maps to straight lines in the projection. In this sense, the Jacobi projection is the natural generalization of the Mercator projection for the triaxial ellipsoid.
In the general case (all the axes are different), the scale diverges only at the umbilic points. The behavior of these singularities is illustrated by the complex function
\[f(z;e) = \cosh^{-1}(z/e) - \log(2/e). \]
For \(e > 0\), this function has two square root singularities at \(\pm e\), corresponding to the two northern umbilic points. Plotting contours of its real (resp. imaginary) part gives the behavior of the lines of constant latitude (resp. longitude) near the north pole in Fig. 1. If we pass to the limit \(e\rightarrow 0\), then \( f(z;e)\rightarrow\log z\), and the two branch points merge yielding a stronger (logarithmic) singularity at \(z = 0\), concentric circles of latitude, and radial lines of longitude.
Again in the general case, each octant of the ellipsoid maps to a finite rectangle \( x_0 \times y_0\), where
\[\begin{align} x_0 = \frac{x(\pi) - x(0)}2 &=\frac{a^2}b \Pi(-e^2k'^2, \cos\nu),\\ y_0 = \frac{y(\tfrac12\pi) - y(-\tfrac12\pi)}2 &=\frac{c^2}b \Pi( e^2k^2 , \sin\nu), \end{align} \]
where \(\Pi(\alpha^2,k)\) is the complete elliptic integral of the third kind, https://dlmf.nist.gov/19.2.E8.
In particular, if we substitute values appropriate for the earth,
\[\begin{align} a&=(6378137+35)\,\mathrm m,\\ b&=(6378137-35)\,\mathrm m,\\ c&=6356752\,\mathrm m,\\ \end{align} \]
we have
\[\begin{align} x_0 &= 1.5720928 \times b,\\ y_0 &= 4.2465810 \times b.\\ \end{align} \]
The projection may be inverted (to give \(\omega\) in terms of \(x\) and \(\beta\) in terms of \(y\)) by using Newton's method to invert \(x(\omega)\) and \(y(\beta)\). The derivative of the elliptic integral is, of course, just given by its defining relation.
If rhumb lines are defined as curves with a constant bearing relative to the ellipsoid coordinates, then these are straight lines in the Jacobi projection. A rhumb line which passes over an umbilic point immediately retraces its path. A rhumb line which crosses the line joining the two northerly umbilic points starts traveling south with a reversed heading (e.g., a NE heading becomes a SW heading). This behavior is preserved in the limit \(a\rightarrow b\) (although the longitude becomes indeterminate in this limit).
Oblate ellipsoid, \(a\rightarrow b\). The coordinate system is
\[\begin{align} X &= b \cos\beta \cos\omega, \\ Y &= b \cos\beta \sin\omega, \\ Z &= c \sin\beta. \end{align} \]
Thus \(\beta\) (resp. \(\beta'\)) is the parametric (resp. geographic) latitude and \(\omega=\omega'\) is the longitude; the quantity \(e\) is the eccentricity of the ellipsoid. Using https://dlmf.nist.gov/19.6.E12 and https://dlmf.nist.gov/19.2.E19 the projection reduces to the normal Mercator projection for an oblate ellipsoid,
\[\begin{align} x(\omega) &= b\omega,\\ y(\beta) &= b\bigl(\sinh^{-1}\tan\beta' - e \tanh^{-1}(e\sin\beta')\bigr),\\ m(\beta, \omega) &= \frac1{\cos\beta}, \\ x_0 &= \tfrac12\pi b, \\ y_0 &\rightarrow \infty. \end{align} \]
Prolate ellipsoid, \(c\rightarrow b\). The coordinate system is
\[\begin{align} X &= a \cos\omega, \\ Y &= b \sin\omega \cos\beta, \\ Z &= b \sin\omega \sin\beta. \end{align} \]
Thus \(\omega\) (resp. \(\omega'\)) now plays the role of the parametric colatitude (resp. geographic latitude) and while \(\beta=\beta'\) is the longitude. Using https://dlmf.nist.gov/19.6.E12 and https://dlmf.nist.gov/19.2.E18 the projection reduces to similar expressions with the roles of \(\beta\) and \(\omega\) switched,
\[\begin{align} x(\omega) &= b\bigl(\sinh^{-1}\tan\omega' + e \tan^{-1}(e\sin\omega')\bigr),\\ y(\beta) &= b\beta,\\ m(\beta, \omega) &= \frac1{\cos\omega} \\ x_0 &\rightarrow \infty, \\ y_0 &= \tfrac12\pi b. \end{align} \]
Sphere, \(a\rightarrow b\) and \(c\rightarrow b\). This is a non-uniform limit depending on the parameter \(k\),
\[\begin{align} X &= b \sin\omega \sqrt{1 - k^2\sin^2\beta}, \\ Y &= b \cos\omega \cos\beta, \\ Z &= b \sin\beta \sqrt{1 - k'^2\sin^2\omega}. \end{align} \]
Using https://dlmf.nist.gov/19.6.E13 the projection can be expressed as
\[\begin{align} x(\omega) &= bF(\omega - \pi/2, k'), \\ y(\beta) &= bF(\beta, k), \\ m(\beta, \omega) &= \frac1{\sqrt{k'^2\cos^2\omega + k^2\cos^2\beta}}, \\ x_0 &= bK(k'), \\ y_0 &= bK(k), \end{align} \]
where \(F(\phi,k)\) and \(K(k)\) are incomplete and complete elliptic integral of the first kind, https://dlmf.nist.gov/19.2.E4 and https://dlmf.nist.gov/19.2.E8. Obtaining the limit of a sphere via an oblate (resp. prolate) ellipsoid corresponds to setting \(k = 1\) (resp. \(k =0\)). In these limits, the elliptic integral reduces to elementary functions https://dlmf.nist.gov/19.6.E7 and https://dlmf.nist.gov/19.6.E8
\[\begin{align} F(\phi, 0) &= \phi, \\ F(\phi, 1) &= \mathop{\mathrm{gd}}\nolimits^{-1}\phi = \sinh^{-1}\tan\phi. \end{align} \]
This spherical limit gives the projection found by É. Guyou in
who apparently derived it without realizing that it is just a special case of the projection Jacobi had given some 40 years earlier. Guyou's name is usually associated with the particular choice, \(\nu=\frac14\pi\) or \(k = k' = 1/\sqrt2\), in which case each octant of the ellipoid is mapped into a square. However, by varying \(\nu\in[0,\frac12\pi]\) the hemisphere can be mapped into a rectangle with any aspect ratio, \(K(k') : K(k)\).
An essential tool in deriving conformal projections of an ellipsoid of revolution is the conformal mapping of the ellipsoid onto a sphere. This allows conformal projections of the sphere to be generalized to the case of an ellipsoid of revolution. This conformal mapping is obtained by using the ellipsoidal Mercator projection to map the ellipsoid to the plane and then using the spherical Mercator projection to map the plane onto the sphere.
A similar construction is possible for a triaxial ellipsoid. Map each octant of the ellipsoid onto a rectangle using the Jacobi projection. The size of this rectangle is \( x_0 \times y_0 \). Find the values of \(b_s\), \(e_s^2 = 0\), \(k_s^2\), and \(k_s'^2 = 1 - k_s^2\) for a sphere such that
\[x_0 = b_s K(k_s'), \quad y_0 = b_s K(k_s). \]
Map the rectangle for the triaxial ellipsoid onto the equivalent octant of the sphere using Guyou's projection with parameter \(k = k_s\). This reduces to the standard construction in terms of the conformal latitude in the limit of an ellipsoid of revolution.
The Triaxial::Conformal3 class provides an implementation of the Jacobi conformal projection is given here. The Conformal3Proj utility provides a command-line interface to this projections.
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