Conformal latitude — where it appears
Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.
The curvature of the Earth is not one number
An ellipsoid's Gaussian curvature varies by 1.35% from equator to pole, so the sphere is not developable onto the ellipsoid any more than the plane is onto the sphere. The spherical approximation is a projection with an irreducible cost, and the cost is 6,739 parts per million.
Where the series stops being the map
The transverse Mercator has no closed form on an ellipsoid, so every national grid computes a truncated series. Asking how many terms it needs has an answer everywhere; asking whether more terms always help has an answer only within 83.81° of the central meridian, and the boundary is computed rather than assumed.
The bound on the body the country is on
The best conformal grid a country could have had was computed against a bound solved on a sphere, with a note saying the flattening was second order and unquantified. It is second order for a named projection — every family's best candidate moves by under 0.7 per cent — and it is 22 per cent for the bound, because an optimal map has already cancelled its own variation and has nothing left to hide a new one in.
A bearing on a sphere is decided by its latitude, not its radius
A sphere that stands in for the ellipsoid cannot be wrong about a direction by being the wrong size, because enlarging a sphere moves no angle. What it can be wrong by is the latitude put on it. Geodetic latitude turns every bearing at 45° by up to 347 arcseconds before anything has moved; the conformal latitude starts exact and drifts 54 arcseconds in a thousand kilometres; and only Bessel's sphere, which changes the longitude too, is exact at every distance.
An angle is a difference, and the difference doubles the error
Substituting a sphere for the ellipsoid turns every direction at a point, and a surveyor measures angles rather than directions — so the obvious hope is that a turn common to both directions cancels in their difference. It does not cancel, because the turn is not common: it runs as the sine of twice the azimuth, and the largest angle error is exactly twice the largest direction error at a well-shaped corner. A first-order triangulation computed on the geodetic-latitude sphere carries angles nearly nine minutes of arc wrong, and its closure check passes perfectly.
The correction is the smaller of the two corrections
Legendre's theorem subtracts a third of the spherical excess from each angle of a triangle and solves the rest as a plane figure. It is the correction every nineteenth-century computer applied and the one every textbook explains. Beside the error the choice of substituted sphere leaves in the same triangle it is small: on the thirty-kilometre triangle a first-order chain is made of, omitting it costs 166 millimetres in a computed side and using the geodetic-latitude sphere costs 43 metres. The two terms scale differently, so each sphere has a size at which the correction overtakes it — and for the two spheres a careful computer would have chosen, that size falls inside a real chain.
A strip of triangles carries the sphere's error to the next base
One triangle cannot tell a substituted sphere from a bad base: both leave the computed side a few parts per million long. A triangulation is not one triangle but a strip, each computed side the next one's base, and along it the two come apart completely. A bad base is one factor, the same at every triangle. A sphere's error is made afresh in each triangle and the strip adds it up, and what it adds up to at the far end is exactly the change in that sphere's own meridian scale between the two ends — 158 parts per million for the conformal sphere over twenty triangles north from 45°, predicted to half a part. Along a parallel it is nothing; at a check base three triangles on it is already three times the noise of one-second angles.
An adjustment hides the sphere only in a strip of equal triangles
A strip computed on a substituted sphere drifts by the sphere's own stretch, 315 parts per million over forty triangles north from 45°, and an adjustment that closes it on a second base takes almost all of that away: in a strip of equal triangles what is left is the stretch less a straight line, 0.63 parts per million, a third-order bow sixty times below the noise of the old angles. A real chain is not made of equal triangles. The adjustment spreads the misclosure by how uncertain each triangle's angles are, the sphere put it in by how far north each triangle reaches, and the difference is a first-order residual that never clears the noise at any one side and that the old network's own geometry predicts exactly — so a re-observation matched against it reads the sphere at seven standard deviations.
Named alongside it
The objects these essays reach for when they reach for this one.
Auxiliary latitudeSpherical approximationGeodetic latitudeSurvey networkTriangulationVerificationEllipsoidFlatteningGeodesicScale factorSpherical excessAzimuth