The second derivative cannot classify

Every essay on this ladder has used the second derivative as a magnitude. Flexion and skewness turned out to be one vector; the second-order ranking disagrees with the first-order one; flexion survives a rotation and a magnification and nothing else; it arrives at a smaller figure size than anyone expects. Seven rungs, and every one of them asks how much.

There is another thing a second derivative is famously for, and this ladder has never asked whether it works. It is asked to say what kind. A critical point of a field — a place where the gradient vanishes — is named a maximum, a minimum or a saddle by the sign of the determinant of the Hessian, and that test is in every textbook that has a chapter on optimisation.

On a sphere it does not work, and the way it fails is worth the whole rung: it is wrong by a factor of three on a field this site already draws, its wrongness is confined to exactly the fields anybody would choose to study, and the measurement that is right instead never differentiates twice at all.

The test, and what it assumes #

At a critical point of a smooth field, the second derivatives form a symmetric matrix. If its determinant is positive the point is a peak or a pit; if negative, a saddle; and if zero the test says nothing at all, which every statement of it admits in a sentence and every use of it then forgets.

The zero case is usually dismissed as a measure-zero curiosity. That dismissal is correct as a statement about randomly chosen fields and is exactly wrong as a statement about the fields anybody studies, because the fields anybody studies are the symmetric ones — and symmetry is what produces degeneracy.

A field with eight critical points #

Take Y₃₃ — the degree-three sectoral harmonic, x³ − 3xy² in the components of the unit vector. It is analytic, this site differentiates it in closed form, and it has six lobes round the equator: three ridges and three valleys, alternating.

Its critical points are found rather than detected — the gradient is exact, so a zero of it is solved for — and there are eight. Six sit on the equator, alternating peak and pit at sixty-degree intervals. The other two are at the poles, and they are where the three ridges and the three valleys all meet.

A point where three ways up and three ways down meet is a monkey saddle. It is a saddle with room for a tail.

Counting it two ways #

There are two ways to give a critical point a number, and this collection has both.

The index by winding. Walk a small circle round the point and watch the direction of steepest ascent. Count how many times it turns, with sign. A peak or a pit returns +1; an ordinary saddle returns −1; the monkey saddle returns −2, because the direction turns twice the wrong way. This measurement uses the gradient and nothing else: it is a first-derivative quantity.

The index by determinant. Take the sign of the Hessian’s determinant: +1 for a peak or a pit, −1 for a saddle. This is a second-derivative quantity, and at the monkey saddle the determinant is zero, so the sign is whatever the arithmetic noise happens to produce.

Poincaré–Hopf says the indices of the critical points of any field with isolated zeros sum to the Euler characteristic of the surface, which for a sphere is 2. That is the same 2 that Gauss–Bonnet reaches by integrating curvature, reached here with no metric anywhere in the calculation — and it is what makes north impossible to have everywhere.

Summed over the eight points of the sectoral field:

The first is the theorem. The second is a factor of three out, and the reason is that each pole contributes −2 to the truth and ±1 to the determinant’s version — and the two poles get different signs from the determinant, +1 at one and −1 at the other, on a field whose two poles are related by a symmetry.

Where the six comes from #

It is worth taking the six apart, because a wrong total is much less interesting than a wrong total whose parts are known.

The six equatorial points are ordinary. Three are peaks and three are pits, each of index +1 by both measurements, and they contribute +6 to both totals. Everything the determinant test is for, it does correctly there.

The two poles are where the totals part company. Each is a monkey saddle of index −2, so the truth is 6 − 4 = 2. The determinant contributes +1 at one pole and −1 at the other, giving 6 + 1 − 1 = 6.

The detail worth stopping on is that the determinant gives the two poles different answers. They are related by the symmetry z → −z, under which x³ − 3xy² is unchanged, so whatever is true at one pole is true at the other. Two identical places, one field, two different classifications — which is as clean a demonstration as this collection has that the quantity being computed there is not a property of the place at all. It is the sign of a number that ought to be zero, and the sign of a number that ought to be zero is decided by the last bit of the arithmetic.

Two of the five fields in the figure are not counted at all. A field that depends on latitude alone is critical along a whole parallel, and a field built from bumps on an exactly flat background is critical everywhere the bumps have died away. Poincaré–Hopf requires isolated zeros, and neither has them — so the theorem is silent rather than wrong, and the check that it is silent is part of the measurement.

The failure is exactly where the interest is #

Now nudge the field. Add ε of the degree-two tesseral harmonic — x² − y², a smooth term whose own two-fold symmetry shares no axis with the sectoral’s three-fold one — and watch both counts.

The determinant test is right on every perturbation and wrong on the field they all converge to.

That is the sentence worth keeping, and it is the reason the failure is hard to find by experiment. Any numerical study that started from a field with a rounding error in it, or a field fitted to data, or a field with any asymmetry at all, would get the right total by the determinant test and would take that as evidence the test works. The test fails on a set of measure zero and that set is the set of fields anybody constructs deliberately.

What survives the split, and what does not #

The mechanism is visible in the hero figure and is worth stating in the vocabulary this ladder uses.

The monkey saddle does not survive the nudge. It splits into two ordinary saddles, each of index −1, and −1 plus −1 is the −2 the single point had. So:

• the index is conserved across the split — it is a topological quantity, and it cannot change under a continuous deformation;
• the classification is not conserved — the kind of point changes from one the Hessian cannot name to two it can;
• the count is not conserved either: eight critical points become ten.

The index is the invariant. That is a familiar shape of statement on this site — the first-order ladder spent an essay on which quantities survive a change of coordinates, and the second-order one found that flexion survives very little. Here the surviving quantity is the one computed from the first derivative, and the quantity computed from the second is the one that moves.

Why the symmetric case is the interesting case #

A degeneracy is a coincidence between second derivatives, and the general position argument says coincidences do not happen. That argument is right about a field drawn from a hat and useless about a field written down.

Symmetry is what forces it. The sectoral harmonic has a three-fold rotational symmetry about its polar axis, and a symmetric matrix that commutes with a three-fold rotation of the plane is a multiple of the identity — so at a fixed point of that symmetry, the Hessian’s two eigenvalues are equal by force. The critical point at the pole is a fixed point of the symmetry. There is nothing accidental about the coincidence; it is a consequence of a property that was put in on purpose.

Every field a person constructs to reason with has that character. A harmonic, a polyhedral arrangement, a regular tiling, a projection with a standard parallel, a body of revolution: they are chosen because their symmetry makes them tractable, and the symmetry then puts degeneracies at exactly the special places anybody is going to look at. The set on which the determinant test fails has measure zero and contains almost every example.

That is not a licence to distrust second derivatives generally. It is a statement about which of two questions is being asked. Ask a second derivative for a size — a curvature, a flexion, a coefficient in a Taylor expansion — and symmetry is harmless. Ask it to sort a point into a category, and symmetry is the case that breaks the sorting, and it is the case that will come up.

The split is proportional, so there is no safe scale #

How far apart do the two halves land? Measured over more than a decade of perturbation:

The fitted exponent is 1.0055 — the separation is exactly proportional to the nudge, at 38.2 degrees per unit, and it reaches zero only when the nudge does.

There is a temptation, on meeting a degeneracy, to treat it as a limiting case that a small amount of realism removes. This measurement says what that costs. The two saddles are always there for any nonzero perturbation, however small; they are simply too close together to see. A degeneracy is not something a field nearly has. Every field near this one has two ordinary zeros and the classification is fine; the field itself has one degenerate zero and the classification is wrong; and no amount of shrinking ε moves the boundary between those two situations.

The instrument has a resolution too #

The winding number is not immune to any of this, and pretending it were would be the same overreach in the other direction.

A winding number is taken round a circle of a stated radius. At ε = 0.005 the two saddles are 0.191° apart. Measure each one’s index with a circle of:

A circle wider than the separation encloses both zeros and returns their sum, which is the index of the degenerate point they came from. With that radius, the total over the sphere reads −2 rather than 2, and a reader watching only the total would report a violation of Poincaré–Hopf.

Nothing about the field changed between the two measurements. What changed was the instrument’s aperture, and the step in the table falls exactly at the separation. This is the same discipline the site applies to every numerical derivative — the gradient ladder found that refining a grid makes a measured slope worse past a point, and the cure there was to fit the exponent rather than to trust two points. The cure here is to know the separation before choosing the radius, which means the measurement has to be made twice.

Where this bites in the map machinery #

None of that is about a projection yet, so it is fair to ask what a cartographic ladder is doing with it. Three answers, in increasing order of directness.

The site already runs this test on maps. The singular points of the north-direction field are counted this way, and a projection whose north field has a degenerate zero would be miscounted by any determinant test. The index there is measured by winding for exactly this reason, and until this rung the reason was a paragraph.

Every second-order classification on this ladder is a magnitude, and that is not an accident. Flexion is reported as a size and a direction, never as a type. That looked like modesty and is a requirement: the moment a second derivative is asked to name a kind rather than measure a size, it inherits this failure. The one place the collection came close is the indicatrix at a point that has none, where the gnomonic’s horizon and a polyhedral net’s corners are points at which the derivative itself stops existing — a stronger failure than degeneracy, and reached from the same direction.

A local polynomial model has the same blind spot. Fitting a polynomial between two coordinate systems buys an order of convergence, measured at 2, 3 and 4 for the three degrees. What it does not buy is any statement about the kind of behaviour at a point where the linear term vanishes — and a georeferencing fit whose second-order term is small is exactly the case where the classification would be asked for and would be unreliable.

An instrument that has an aperture should report it #

The honest procedure named above — measure, halve the radius, measure again — is a specific instance of something this collection keeps arriving at, and it is worth stating in general because it costs one extra evaluation.

Every numerical instrument has a scale below which it cannot resolve. A winding radius, a differencing step, a lattice spacing, a quadrature interval, a smoothing window: each of them sets a length below which structure is averaged rather than seen, and each of them returns a perfectly finite answer when structure is present below it.

The answer does not say which regime it is in. A winding number computed with too wide a radius reports one zero with a combined index, and that output is indistinguishable from a genuine single zero of that index. Nothing about the number is marked as an aggregate.

Halving the parameter is the test, and it is one-sided in the useful direction. A result that changes when the aperture halves was aggregating; a result that does not has resolved whatever is there at that scale, though it cannot rule out structure below the new one. That is the same one-sided reading as every other test in this collection, and it is the reason to run it twice rather than once.

And the separation is not knowable in advance, which is what makes the procedure necessary rather than merely careful. If the spacing of the features were known, the aperture could be chosen; the spacing is what the measurement is for.

So the reportable output is a pair: the value, and the aperture it was obtained at, with the note that halving the aperture did not change it. A single number with no aperture attached is a measurement whose resolution is unstated, and on a field with structure at several scales that is not enough to reconstruct what was seen.

What the rung establishes #

A second derivative measures well and classifies badly. On the sphere, on a field this collection already draws, the classical determinant test totals six where the truth is two, and it fails at precisely the symmetric fields anyone would choose to work with. The count that is right instead is a winding number, which uses the first derivative only.

The degeneracy is not approximate. It splits under any perturbation, in exact proportion to it, at a fitted exponent of 1.0055 — so a degenerate field cannot be treated as a limit of well-behaved ones for the purpose of classification, though it can be for the purpose of the total.

And the right measurement has a resolution. A winding radius wider than the separation of two zeros reports one zero with their combined index, which looks like the theorem failing and is the aperture. That number — the separation — is not available before the measurement, so the honest procedure is to measure, halve the radius, and measure again.

The ladder’s seven previous rungs measured how much a map bends. This one is about what a second derivative can be asked to say, and the answer is narrower than the ladder had been assuming: it can be asked for a size, and it cannot be asked what kind of place it is looking at.