The lines where the bending vanishes

Nine rungs of this anchor have measured the second derivative — how much a geodesic bows on the page, how the bow ranks projections, how it behaves over a region, why it is not an invariant, and why the best compromise for angle is not the best for bending.

What none of them has asked is where it is zero.

The places where a map is exactly right asks that question of the first derivative and finds the standard parallels — the curve where the scale is one, which every projection prints in its margin and which is the whole of what a standard parallel buys is about. The second derivative has a zero set too, it is a different curve, and it is printed nowhere.

The two lines #

Both columns are measured. The first is found by solving for a parallel scale of exactly one along a meridian, not read from a definition — the site’s rule is that a property is computed from the projection’s own derivatives, and a standard parallel quoted from a table is a name. The second is found by minimising the root-mean-square flexion over all directions.

The rule is tangency #

The pattern is exact and it has a one-line explanation.

A tangent construction touches the sphere along one line. Its standard parallel is that line, and so is the line about which the whole construction is symmetric — so the flexion, which is an odd-order quantity in the displacement from the axis of symmetry, vanishes there. Both lines are the line of contact, and they coincide.

A secant construction is the same map scaled. Making a cylinder secant at ±45° instead of tangent at 0° multiplies the whole projection by $\cos 45°$ and divides the vertical coordinate by it. That moves the true-scale parallels, because the scale is a first-order quantity that the multiplication changes. It does not move the flexion zero, because a uniform scaling of the whole page cannot move a curve on which a second-order quantity vanishes: scaling multiplies the flexion by a constant and a constant times zero is still zero.

So secancy is a first-order device with no second-order effect whatsoever, and the two special lines of a secant projection are as far apart as the secancy is deep.

The conics, which are neither #

The conics are the interesting case, because their flexion zero is at neither standard parallel and at no obvious place.

Albers with standard parallels at 20° and 60° has its flexion zero at 37.67°. The Lambert conformal conic with the same two parallels has its at 41.06°. Both are strictly between their standard parallels and neither is at their mean of 40° — and the two differ from each other by three and a half degrees, on projections with identical standard parallels.

The reason the two differ is that a conic’s flexion is a balance between two curvatures — the parallels are drawn as circular arcs of a radius that varies, and the meridians as straight lines converging on the apex — and how the radius varies is what separates the equal-area conic from the conformal one. So the flexion zero is a property of the whole radial function rather than of where the cone touches, which is why it lands somewhere that has no simple name.

That figure carries one more finding. The secant conic’s flexion does not reach zero at all, it merely reaches a minimum. A tangent conic has a genuine zero at the point of contact; the secant construction, which is a tangent one composed with a scaling and a shift of the apex, has no line along which the bending vanishes exactly.

Mollweide, which is neither cylindrical nor conic #

The pseudocylindricals do not fit the tangent-or-secant story cleanly and are worth a paragraph for that reason.

Mollweide’s scale along the parallel is one at ±40.74°, which is the latitude every reference quotes for it, and its flexion is zero on the equator — a separation of nearly forty-one degrees, comparable with Gall–Peters’s. But Mollweide is not a scaled version of anything: its parallels are spaced by an elliptic construction and its meridians are half-ellipses, so the “secancy” explanation does not apply.

What applies instead is the symmetry. The flexion is odd in the displacement from the equator for the same reason it is on a cylindrical — the construction is symmetric about the equator and the second-order term changes sign across it — so a zero there is forced by the symmetry rather than by the tangency. The tangent-or-secant rule is a special case of a symmetry argument, and the general statement is that the flexion vanishes on any line the construction is symmetric about, whether or not that line has anything to do with scale.

That also explains the sinusoidal, whose parallel scale is one everywhere and whose flexion is still zero only on the equator: true scale is a property the construction has along every parallel and the symmetry is a property it has about one.

What was computed, and how #

The flexion is secondOrder’s own quantity, unchanged from the rung that introduced it: the second derivative of a geodesic’s image, measured over thirty-six directions and reduced to a root-mean-square, so the zero being sought is a zero in every direction rather than in one.

The zeros are found by scanning latitude at one degree and refining each local minimum by ternary search to forty iterations. The true-scale parallels are found by bisection on $k - 1$ to sixty iterations, with a projection whose parallel scale is one everywhere — the sinusoidal — reported as such rather than as a list of roots of arithmetic noise.

Two assertions carry the rung and they reject in opposite directions.

Mercator’s two lines must coincide to within a twentieth of a degree, and at least three projections must separate them by more than five. The first is what says the search is finding lines rather than artefacts; without it, a bug that returned zero everywhere would look like a finding. Gall–Peters’s separation is additionally pinned at forty-five degrees to within one and a half.

And a cylindrical projection’s flexion at the equator must be below $10^{-8}$ while its flexion at fifteen degrees is above 0.1, for all six cylindrical members. That is the statement “the zero is a line and not a region”, and it fails on any projection whose flexion is merely small near the equator rather than zero on it.

What it means for a reader with a straight-edge #

The practical content is the difference between two things a reader does with a map, and the fact that the map tells them about only one.

A scale bar is honest along the true-scale parallel. A distance measured with dividers is correct there and progressively wrong elsewhere, which is the first-order story and the one every margin prints.

A straight-edge is honest along the flexion zero. A great-circle route drawn as a straight line on the page departs from the true route by an amount whose leading term is the flexion, so along the flexion zero a ruled line is the route to second order. Everywhere else the ruled line bows away from it, and the line drawn straight on the page is a route prices what that does. Bending and stretching are one failure is the reason the two cannot simply be added up.

On a tangent projection the two coincide and a reader has one line to remember. On Gall–Peters they are forty-five degrees apart, so the latitude at which distances are right is the latitude at which straight lines are worst, and the latitude at which straight lines are right is where distances are off by 41 per cent.

Nothing in the margin of a Gall–Peters map says either of those things.

What a margin could say instead #

The repair is a line of print and it is worth writing down, because this collection’s rungs usually end in a recommendation nobody can act on and this one does not.

A map’s margin already carries its projection, its standard parallels and a scale bar. Adding “straight lines are true routes near 0°” costs eleven words and gives a reader the second line. On a tangent projection it is the same line as the first and the note is redundant, which is itself informative.

The stronger version is to state the rate. The flexion at the true-scale parallel is a number — 1.12 for Gall–Peters, 0.71 for Mollweide — and it says how badly a ruled line misbehaves exactly where a reader is most likely to trust the map, because that is where the scale bar told them to.

Where the model stops #

Along one meridian. The flexion zero of a cylindrical projection is a full parallel, by symmetry, and the same is true of a conic. For a pseudocylindrical it is not: the sinusoidal’s flexion is zero on the equator only at the central meridian, and grows along the equator as the meridians curve away. The scan here runs down 5° east and reports what it finds there; the general zero set is a curve rather than a parallel and is not drawn.

The flexion is a root-mean-square over directions. A projection could have zero flexion in one direction and not in others, and the quantity used here would not report a zero. Every zero found here is a zero in all thirty-six sampled directions, checked, so nothing is being hidden — but the possibility is real for an oblique aspect and is not measured.

And Robinson has no zero at all. Its flexion is 1.58 at the equator and 29.6 at 75°, because a projection defined by a table has an interpolation in it and an interpolated second derivative is a spike at every table entry. It is left out of the comparison for that reason rather than by accident.

The generalisation #

The rule is about what a construction’s parameters can and cannot reach.

A parameter that acts by scaling moves every first-order quantity and no zero of a second-order one. Tissot stops at the first derivative is why nobody would have noticed: the whole classical apparatus is first order, so a parameter tuned against it is tuned against everything the apparatus can see. Secancy is exactly such a parameter: it multiplies a projection by a constant to trade scale error at the centre against scale error at the edge. It is the single most common adjustment in the subject — every national grid uses it, every conic in an atlas uses it — and it is invisible to the whole of this anchor.

That is a genuinely useful thing to know when choosing. A projection is usually selected by its first-order behaviour and then tuned by secancy, and the tuning is free with respect to bending. Conversely, a map whose second-order behaviour matters cannot be repaired by the parameter everybody reaches for, and has to be repaired by changing the construction — a different family, a different aspect, or a different point of tangency.

The best compromise for angle is not the best for bending established that the two orders rank projections differently. This rung says something sharper about the same gap: the standard tool for improving the first-order ranking does not move the second-order one at all.

What a chooser should take from it #

Two practical consequences, and they point in opposite directions.

If the map is for measuring, the standard parallel is the line that matters and secancy is exactly the right tool: it halves the worst scale error across the sheet and the choice of where to put the two lines is the whole of the design.

If the map is for drawing routes on, the standard parallel is irrelevant and the parameter to move is the one that changes the construction’s point of tangency — the aspect, the cone constant, the family. Secancy will not help however far it is pushed, and pushing it makes the first-order behaviour better while leaving the second-order behaviour exactly where it was.

Very few maps are for only one of those, which is why the two lines being different matters at all.

Who found it, and when #

The standard parallel is as old as the conic projection and its arithmetic is in Ptolemy. Secancy as a device for halving the maximum scale error is nineteenth-century practice and is the reason every national grid has a scale factor at its central meridian of slightly less than one, which is the scale factor was chosen’s subject.

The flexion is Goldberg and Gott’s, from 2007, and their paper introduces it as a measure for comparing world maps rather than as a field with a zero set. As far as this collection can find, nobody has located that zero set for any projection, and nobody has observed that it is invariant under the one adjustment everybody makes.

The reason is probably that the two quantities belong to different jobs. The standard parallel matters to somebody measuring on the map, and the flexion matters to somebody drawing a route on it, and those two people have never been the same person for long enough to notice they were given two different lines.

Why the invariance is structural rather than lucky #

The zero set surviving the adjustment everybody makes is the rung’s most useful practical fact, and it is worth saying why it holds, because the reason means it will keep holding.

Secancy is a scaling. Making a projection secant multiplies the whole map by a constant factor chosen so that the scale error at the extremes matches the error at the centre. Every coordinate is multiplied by the same number; nothing about the shape of the map changes.

Flexion is a second-derivative quantity of a normalised kind. It measures how much a geodesic’s image bends relative to its own progress along the page — a curvature per unit length — and both halves of that ratio scale identically under a uniform magnification. So the quantity is unchanged, point by point, and its zero set is unchanged with it.

That is a stronger statement than the measurement alone. A measured invariance under one adjustment might be a coincidence of the projections tested; an invariance that follows from the quantity’s own homogeneity holds for every projection, every secancy constant, and every future member of the family.

And it has a practical consequence worth stating plainly. The flexion-zero lines need computing once per projection, not once per configuration. A national mapping agency that adjusts its grid’s scale factor — as they periodically do — moves its standard parallels and does not move these lines at all, so a margin diagram carrying both would need reprinting for one of them and not the other.

That is unusual enough to be worth dwelling on for a sentence.

Which sharpens the closing observation about the two readers. They are not merely asking different questions; they are asking questions with different stability. The measurer’s line moves whenever the grid is retuned, and the route-drawer’s does not — so of the two quantities, the one nobody prints is the one that would stay true.

Where the ladder goes next #

Ten rungs have taken the second derivative from its definition to its zero set. Every one of them has measured the bending of a geodesic, which is the curve a reader is trying to draw when they use a straight-edge. A map is also read the other way round — a straight line drawn on the page and then asked what it is on the ground — and the second-order answer to that question is a different quantity with a different zero set, which this anchor has assumed is the same one.