A projection written as a condition

Every projection in this site’s library is a formula. Several of them were derived from a requirement — Mercator’s from the demand that a compass course be straight, Lambert’s azimuthal from the demand that area be preserved — but what the library holds is the answer, and the question survives only as a comment above it.

There is a third way to specify a map, and it produces objects the ordinary library has no member like. Write down what must be exactly true, and take the map to be whatever satisfies it.

The construction is two circles and no arithmetic #

Given the two centres $A$ and $B$, plot them a true distance apart. For any other place $P$, compute its true spherical distance $a$ from $A$ and $b$ from $B$. Then the image of $P$ is the point that is $a$ from the image of $A$ and $b$ from the image of $B$ — the intersection of two circles.

That is the whole projection. There is no series, no transcendental function of latitude, no parameter to fit. The condition determines the map up to one binary choice: two circles meet in two places, and the map takes the one on the side of the great circle $AB$ that the point is really on.

The exactness is worth stating as a measurement rather than as a consequence of the construction, because a construction can be right and its implementation wrong. Over 760 sampled points, the worst relative error in the distance from either centre is 5.1 × 10⁻¹⁴, which is the arithmetic and not the map.

What is bought, and what is not #

An exact property is a strong thing to have. It is also narrow, and the second measurement is what stops the first from being read as a claim about distances in general.

The condition names two places. It does not name a metric, and nothing in the construction constrains the distance between two ordinary points at all. A reader who takes the phrase equidistant projection to mean that distances are right has been misled by a word, which is the site’s standing complaint about the whole vocabulary.

The same narrowness shows up in the first-order quantities, which the condition also says nothing about:

Measured at five sample points: 0.2° of angular deformation near London, 6.2° on the equator south of the centres, 15.8° over the western Atlantic, 37.8° over central Asia, and a maximum of 164° in the antipodal region. The areal factor over the same points runs from 0.90 to 1.41.

The construction cannot fail, and the reason is not in the construction #

Two circles in a plane do not always meet. If the sum of the radii is less than the separation of the centres, or if the radii differ by more than it, there is no intersection and no image — so the natural expectation is that this projection has a boundary, a region where the condition is unsatisfiable and the map simply stops.

It has none. Sampled over the whole sphere at equal-area spacing, the construction places every point, for centres 10° apart and for centres 170° apart alike.

That is a theorem about the sphere doing work in a construction that never mentions it. The two radii and the separation are $a$, $b$ and $c$ of a spherical triangle; $|a - b| \le c \le a + b$ holds for every spherical triangle; and those are precisely the two conditions under which the plane circles meet.

The distinction that matters here is between a construction that cannot fail and a solver that cannot report failure. The site’s rule is that an assertion which has never rejected anything proves nothing, so the same circle-intersection routine is fed a pair of radii that no plane point satisfies — two circles of radius 0.1 with their centres 2.09 radians apart — and is required to return nothing. It does. The total coverage above is therefore a fact about the sphere rather than a bug.

What the map really is, said in one sentence #

There is a compact way to describe the construction that makes its behaviour predictable rather than surprising.

The pair of numbers $(a, b)$ — distance from the first centre, distance from the second — is a coordinate system on the sphere. It is not the graticule; it is a bipolar system, whose coordinate curves are the two families of distance circles about the two centres, and it labels every point of the sphere except for the ambiguity between mirror images. The plane has a bipolar coordinate system of its own, built from circles about two points the same distance apart.

The two-point equidistant projection is the map that sends one to the other, coordinate for coordinate. Everything else follows from that. The distances from the centres are exact because they are the coordinates. The distances between other points are unconstrained because nothing was said about them. And the map’s worst regions are where the two coordinate systems’ geometries differ most, which is far from both centres, where the sphere’s distance circles are shrinking toward a point and the plane’s are still growing.

This is also why the projection has no free parameter beyond the two centres. A coordinate change matched term for term has nothing left to choose.

Which side, and the one arbitrary decision #

The construction has exactly one free choice and it is worth naming, because it is the only place where the map is not determined by its condition.

Two circles meet twice, symmetrically about the line joining the centres. The two solutions correspond to the two points of the sphere that are the same distance from both centres — mirror images in the great circle through $A$ and $B$ — and the map has to pick one. Picking by the point’s actual side keeps the map continuous and injective; picking arbitrarily would fold the sphere onto half a map.

That choice is what makes this projection a projection rather than a two-valued relation. Nothing in the condition requires it, and the third rung of this ladder is about a conditioned map where the analogous repair is not available.

A second pair of centres, and what changes #

The centres are the projection’s parameters, and there are only two of them. Everything about the map’s behaviour follows from where they are.

A pair of centres close together gives a map that is nearly the azimuthal equidistant about their midpoint, which is the degenerate case of the same condition with one place instead of two. A pair far apart gives a map whose useful region is the band between them.

The line where the map is flat, and the point where it is not a map at all #

Two special sets are worth naming because they are where a reader’s intuition about the picture goes wrong.

Points lying on the great circle through the two centres have $y = 0$: they all fall on the straight line joining the two images, in true distance order along it. That is the one line of the map along which every distance is correct, including distances between two ordinary points, and it is exactly one line.

The antipodes of the centres are the other special case. The antipode of the first centre is at distance $\pi$ from it and $\pi - c$ from the second, and both of those are extreme values, so the construction places it at the far end of the map with distance circles crowding around it. That is where the 164° of angular deformation lives, and it is the geometric reason a two-point equidistant world map is unreadable in one particular region rather than uniformly poor.

Where it is actually used #

Hans Maurer described the construction in 1919 and Charles Close arrived at it independently in 1921, and its practical career has been narrow and real. The two-point equidistant is the projection used when the question genuinely has two fixed places in it: the distance from a pair of transmitters, the flight distance from either of two airfields, or the geometry of a route that has to be planned relative to two bases.

It has also been used for world maps, which is the use its own numbers argue against. The American Geographical Society published one, and it is a curiosity rather than a working map, because a projection with 164° of angular deformation somewhere is not a projection anybody should be measuring shapes on.

The general point is the one every projection minimises something makes from the other side. A projection built to an objective is a good map for the objective and nothing else, and the honest way to present one is with the objective attached — which for a conditioned projection is trivially easy, because the objective is the definition.

One condition, and the map is already in the library #

The obvious question, once a two-place condition has a map, is what a one-place condition gives. The answer is a projection this collection has had from the start.

The distance from one named place must be exactly right is satisfied by drawing every point at its true distance from the centre, in any direction that keeps the map continuous — which is the azimuthal equidistant projection, and it is the map the route with no shortest path and a scale bar is right in one place both use for its exact radial scale.

One condition leaves a free function — the choice of azimuth, which the ordinary azimuthal equidistant fixes by preserving direction from the centre. Two conditions leave a free binary choice, which is the side. The pattern is the one any over- and under-determined problem shows, and the count of conditions against the count of degrees of freedom is what the next rung is entirely about.

What the second derivative says about it #

The two quantities from the flexion ladder apply unchanged, because the construction produces an ordinary differentiable map away from the centres, and they are worth computing here for one reason: the condition is about distances, and flexion is about how badly a straight line on the map represents a route.

The answer for the two-point map is that it bends routes rather more than an ordinary compromise map does. Exactness in the distance from two points does not buy straightness of anything: the two properties are independent, which is obvious once stated and was not obvious before, because the words equidistant and straight keep company in ordinary speech.

A reader who wants both has to ask for both, and no map gives both — the distance from a point is exact on the azimuthal equidistant, routes are straight on the gnomonic, and the gnomonic companion is the essay about using two maps at once because one will not do.

What the map is exactly right about, listed #

It is worth collecting the exact set, because it is smaller than the projection’s reputation and its shape explains the 876 per cent.

Two one-parameter families of curves. Every distance circle about either centre is drawn as a true circle of the true radius. That is two families, each parameterised by one number, so together a set of measure zero on the sphere.

One line. Points on the great circle through the two centres are drawn on the straight segment joining their images, in true order and at true separation — so along that single line, every distance is right, including between two ordinary points.

And two points, trivially. The centres themselves.

Nothing else. A projection whose name contains the word equidistant is exactly right about a set of curves and one line, and about no region at all — which is the same shape of claim the azimuthal equidistant’s own scale bar makes with one centre instead of two.

The 876 per cent is a sample, not a maximum #

The figure reporting a worst pair drawn 876 per cent too long invites the reading that the error is bounded by something near nine. It is not bounded at all, and the reason is worth stating because it decides where the map must not be used.

Approach the antipode of one of the centres. Distance circles about that centre are shrinking towards a point on the sphere — a circle of radius π − ε has circumference 2π sin ε, which goes to zero — while their images in the plane are circles of radius π − ε, whose circumference is going to 2π². So the scale in the direction around that centre grows without bound, as 1/sin ε.

Two points separated only in that direction, a fixed ground distance apart, are therefore drawn a distance apart that diverges as the antipode is approached. There is no worst case; there is a region in which the ratio exceeds any stated number, and 876 per cent is simply the largest the two thousand sampled pairs happened to land on.

That also explains the 164° of angular deformation quoted in the same neighbourhood, and it puts the two numbers in the right relation: they are the same divergence read through two instruments, one of which saturates at 180° and one of which does not. The angular deformation is bounded because an angle is, and the distance error is not.

The practical rule that follows is sharper than the map is poor far from its centres. It is that the two antipodal regions are places where the projection has no scale at all in one direction, so any measurement taken across them is meaningless rather than merely inaccurate — and a working two-point map is one whose sheet has been cut to exclude them, which is what every published use of the construction does without saying so.

Where the model stops #

Three limits, and the first is the one that decides whether the projection is usable.

The centres must be distinct, and the construction degenerates as they approach each other: the plane separation $c$ appears in a denominator, so a pair of centres a metre apart produces a map dominated by rounding. The library asserts a minimum separation rather than dividing and hoping.

The exactness is exact on the sphere, and the Earth is not one. Recomputing the distances on the ellipsoid would move each of them by up to about a third of a per cent, which is thousands of times the 5 × 10⁻¹⁴ the construction achieves against its own metric. That gap is not a failure of the projection; it is the difference between the model’s arithmetic and the model’s applicability, and the Earth is a sphere, and when it is not is the essay about which of the two a given job cares about.

The condition is checked where it is claimed and nowhere else. The 5 × 10⁻¹⁴ above is the error in distances from the centres, over the region the map covers. It is not a bound on anything else, and the essay’s second figure exists so that the exact number and the 876 per cent cannot be read as belonging to the same claim.

Nothing here is optimised. The map is not the best two-point map by any criterion; it is the only map satisfying the condition, up to the side choice. That is what distinguishes a conditioned projection from the optimised ones the rest of this field is about — Chebyshev’s criterion picks a member out of a family by minimising something, and a condition picks a member by leaving nothing else to pick.

Where the ladder goes next #

A projection can satisfy one distance condition — that is the azimuthal equidistant, and it has been in the library from the start. It can satisfy two, which is this essay.

Three is a different matter. Three distances in a plane over-determine a position: two circles already fix a point and the third has no freedom left. What Chamberlin’s trimetric construction does about that, and how large the leftover is, is the next rung — and the leftover turns out to grow as the cube of the size of the region, which is a signature worth recognising.