The indicatrix at a point that has none

Tissot’s indicatrix is the image of an infinitesimal circle. That word is doing work, and rung three of this ladder did the work: the indicatrix is a limit, the departure of a real circle’s image from the ellipse falls as the square of the radius, and the limit is what the derivatives compute.

Every figure on this site since has assumed the limit exists. It usually does. This essay is about the points where it does not, which turn out to be neither rare nor pathological: one of them is the edge of a projection this collection uses in half a dozen essays, and the others are the corners of every polyhedral map ever folded.

What “the limit exists” is doing #

A projection has an indicatrix at a point when it is differentiable there — when there is a linear map that approximates it well enough that the error is o(ρ). Everything the distortion machinery computes is a property of that linear map: the two principal scale factors, the areal factor, the maximum angular deformation, the invariants that survive a change of coordinates.

There are two ways for a point to fail to have one, and they are genuinely different.

The first is that the map does not extend to the point at all — the point is on the boundary of the domain, or beyond it. Nothing is wrong with the interior; the trouble is the behaviour of the interior’s indicatrices as the boundary is approached, and whether they tend to anything.

The second is that the map extends, and is even continuous, and is still not differentiable. This is the interesting case, and it is the one polyhedral maps produce by construction rather than by accident.

The horizon, approached #

The gnomonic projection maps the hemisphere centred on its tangency point to the whole plane. A point 89.9° from the centre has a perfectly good indicatrix: a = sec²c along the radial direction, b = sec c across it, both finite, both computable, both correct.

So the indicatrix exists at every point of the domain and has no limit as the domain’s edge is approached. The distinction matters because it is the one a reader of a distortion map is most likely to blur: a map with a hole in it is not a map with a singular point in it, and a quantity that is finite everywhere and unbounded on approach is not the same as a quantity that is undefined somewhere.

Practically it says something about every gnomonic figure in this collection: any statement about “the worst distortion” on such a map is a statement about the clip, not about the projection. Choose a different latitude limit and the worst point changes, without anything about the projection changing at all. The same applies to Mercator at the pole and to every azimuthal projection whose domain is a proper subset of the sphere.

The corner, where the map is continuous and has no derivative #

Now the case that is not about a boundary. A polyhedral map is a set of projections, one per face, each of them gnomonic onto its own plane and each of them perfectly differentiable. The faces are then unfolded into a net.

Consider a point on the sphere directly above one of the solid’s vertices. It has faces all round it. The map is continuous there — walk across the vertex and the image walks across the corresponding corner of the net, without jumping. And it has no derivative, for a reason that is a theorem rather than an artefact.

A linear map takes a full turn to a full turn. It cannot take 360° to 270°, which is what the cube’s corner does, or to 180°, which is what the tetrahedron’s does. So no linear map approximates the projection at that point, to any order, and there is nothing for an indicatrix to be.

The two sums in the figure are computed by different routes and neither is looked up. The spherical one takes the tangent directions from the vertex towards each of its neighbours and adds the angles between consecutive pairs; the planar one gnomonically projects the same three points onto each face’s own plane and measures the corner angle there. Their difference agrees with the deficit that Descartes’ theorem computes from the face angles alone, to nine decimal places, and the two calculations share no line of code.

Three tests a point can fail, and what each one costs #

The three failures this essay separates are worth stating as tests, because a reader with a projection in hand can apply them.

Does the map reach the point? If not, every distortion statement about the map is a statement about where it was clipped. The gnomonic beyond 90°, the stereographic at the antipode, Mercator at either pole. The cost is that “the worst distortion on this map” stops being a property of the projection.

Does a small circle’s image settle on an ellipse? Take a circle of radius ρ, halve ρ, and watch the measured axis ratio. If it settles, the derivatives exist and the indicatrix is what they compute. If it does not settle, nothing the derivative machinery says applies — and the sequence can fail to settle either by running away, as at the gnomonic horizon, or by depending on the direction of approach, as at a cone point.

Does a full turn stay a full turn? This is the corner test, and it is the only one of the three that a picture answers directly: walk round the point in the image and add the angles. Anything other than 360° is a cone point, and the amount by which it differs is the curvature that has been concentrated there.

The first two are questions about a single projection and the third is a question about a construction. A projection given by a formula on an open set passes the third automatically, which is why it took a polyhedral map to make the case interesting — and why every distortion figure in this collection, all of which are computed on formula-given maps over open sets, has been entitled to its assumption.

What was computed, and how #

The measured axis ratio does not come from the derivatives. A circle of angular radius ρ is mapped point by point, its image’s centroid is found by the shoelace formula, and the greatest and least distances from that centroid are divided. That is a measurement of the picture, and it is what makes the convergence in the first figure evidence rather than restatement: the derivatives predict a ratio and an independent construction returns it.

The convergence is second order and the figure shows it as such. The departure from the predicted ratio falls by a factor of about four for each halving of the test radius, down to a few parts in ten thousand at ρ = 0.006 radians, where the arithmetic’s own noise begins to matter more than the geometry’s.

The runaway is checked against a closed form rather than merely observed to grow. For the gnomonic, a = sec²c and b = sec c are elementary; the measurement agrees with sec c to 3 × 10⁻⁶ at 89.9°, and the small departure there is the numerical Jacobian’s finite-difference step rather than the projection — the function being differenced changes by four orders of magnitude across that step.

Where the deficit goes when the solid gets finer #

There is an obvious escape and it does not work, which is worth showing because it is the standard response to the corner problem.

Subdivide the solid. A geodesic polyhedron with hundreds of faces has hundreds of vertices, and each vertex’s deficit is correspondingly smaller — so the map is closer to differentiable everywhere, and in the limit of infinite subdivision it is a map of the sphere with no corners at all.

The deficit at each vertex does shrink as the faces multiply. Descartes’ theorem fixes the total at exactly $4\pi$ for any convex polyhedron, so subdividing redistributes the failure and never removes any of it: twelve vertices of 60° become 162 vertices of a few degrees, and the sum is the same 720°.

The trade the polyhedral family makes across the five solids is usually stated as one thing: more faces buy less distortion inside a face and more cutting round the edges. What has not been said is that the cutting is not the only thing that grows. The number of corners grows with it, and each corner is a point where no indicatrix exists — four such points on a tetrahedron and twenty on a dodecahedron.

The total angle deficit of a convex polyhedron is 4π steradians, always. So subdivision spreads the non-differentiability more thinly and conserves it exactly, which is Gauss–Bonnet in its most elementary clothing: the sphere’s total curvature is 4π and no polyhedral approximation can be flat everywhere.

That is why the escape fails, and why it is the same escape as the one the impossibility ladder rejects at its base. It is also why the count matters: a finer solid has more points with no indicatrix, not fewer, and the reason the finer map looks better is that each of them is individually weaker. A reader looking at a subdivided icosahedral map is looking at a surface that is flat almost everywhere and carries the sphere’s entire curvature at 162 pinpoints, and the map’s own distortion figures — computed on the faces, where the derivatives exist — cannot mention any of them. A polyhedral map does not avoid the trade-off; it concentrates the whole of it into a finite set of points and reports zero distortion in between.

Where the model stops #

Two kinds of failure, not a classification. A boundary the map does not reach and a cone point the map reaches without a derivative are the two cases this essay measures. There are others — a fold, where the map is not injective; a cusp, where the derivative exists and is singular; the pole of a pseudocylindrical projection, where a whole edge maps to a point. The last of these is drawn in the collection’s essay on where a pseudocylindrical puts its error and its indicatrix degenerates rather than failing to exist.

The corner argument is for convex solids. Descartes’ theorem is a statement about convex polyhedra, and the deficit at a vertex of a non-convex one can be negative — a saddle corner, where the page turns through more than 360°. The same conclusion follows, since no linear map does that either, but the arithmetic of the total is different.

The corner measurement is of the net, not of a particular polyhedral projection. Attaching a gnomonic map to each face is what the classical polyhedral projections do, and the corner angles are then the face angles. A projection that uses a conformal map onto each face instead changes those angles — the corner exponent is exactly what that essay computes — and the deficit still cannot be removed, because the deficit is a property of the solid.

And nothing here measures what a reader loses. A corner where the map is not differentiable is a corner where a small circle is not drawn as an ellipse. Whether that is visible, and at what size, is a question about the drawing rather than about the map, and it is left alone.

The generalisation #

A derivative is a claim that a map is well approximated by a linear one, and the claim can fail while the map stays perfectly continuous. The failure is not a defect in the map; on a polyhedral projection it is where the map’s whole error has been put on purpose.

The pattern is the one that runs through this collection’s impossibility argument. A quantity that cannot be zero everywhere can be made zero almost everywhere, by concentrating it — and the concentration is a design decision with a cost that is invisible to any measurement made in the interior. Every measure of distortion this site computes is an interior measure. All of them read zero on a polyhedral face, and none of them can see the corner.

Outside cartography the same shape appears wherever a piecewise-smooth approximation replaces a smooth object: a finite-element mesh’s element interiors are exact and its element boundaries carry the error, and a spline’s segments are exact and its knots carry it. In each case the diagnostic that measures the interiors passes, and the quantity that has been concentrated is the one the diagnostic was built to find.

Who found it, and when #

Descartes’ theorem on the total angular defect is in De solidorum elementis, a manuscript found among his papers after his death and published in 1860; Leibniz had copied it in 1676. It predates Euler’s formula for polyhedra, which it implies, by most of a century — and it is the discrete Gauss–Bonnet theorem two hundred years before Gauss and Bonnet, which is why it is the natural tool here.

Tissot’s own 1881 Mémoire is careful about the limit in a way that its reproductions are not. He defines the indicatrix through the differential of the projection and states the conditions under which it exists; the century of atlases that drew scattered circles on maps dropped both the definition and the conditions.

The polyhedral projections that make the corner unavoidable are largely twentieth century — Cahill’s butterfly of 1909, Fisher’s icosahedral of 1943, Fuller’s Dymaxion of 1954 — and their literature discusses the cuts at length and the corners hardly at all. The corners are where the impossibility lives, and the maps are sold on how little distortion they have in the middle of a face.

What a numerical differentiator does at a corner #

The three tests are stated as properties of the map, and they have a consequence for every piece of software that computes distortion, because almost all of it differentiates numerically.

A finite difference does not detect a corner. It averages across it. Evaluate the projection at two points either side of a seam, divide by the separation, and the result is a perfectly finite matrix — a weighted mean of the derivative on one side and the derivative on the other, weighted by how the sample straddled the edge. Nothing throws, nothing warns, and the number has the units and the shape of a Jacobian.

So the failure is silent and the output is plausible. A distortion field computed on a lattice over a polyhedral map returns finite values at every sample, including the ones on the seams, and those values are not the distortion at those points because there is no distortion at those points. They are an artefact of the step size, and they change when the step size changes — which is the one diagnostic available, and it requires somebody to have suspected the problem.

The step-size dependence is the test. Halve the differencing interval and recompute. At an ordinary point the Jacobian is stable to the differencing error; at a corner it moves, because the two sides are being reweighted. That is a cheap sweep over a lattice that has already been computed once, and it flags every point where the derivative does not exist without needing to know where the seams are.

The remedy is not to refuse an answer there but to say that none exists: a distortion field over a polyhedral map should carry holes along its seams, and a field with no holes in it has been computed by a method that cannot tell.

And it matters more than the count of affected points suggests. The seams are a set of measure zero, so a random lattice hits few of them — but a lattice aligned to the graticule or to the net, which is what a plotting routine naturally uses, hits them systematically. A field drawn at the same spacing as the solid’s own structure samples the corners far more often than chance, and those are exactly the samples a reader will look at, because they are where the map does something visible.

Where the ladder goes next #

This rung finds the points where the first derivative does not exist. The flexion ladder next door has spent six essays on the second derivative, at points where both exist, and has already found that it is not an invariant. What neither has asked is what happens when a projection’s derivatives exist and are computed by a truncated series rather than a formula — which is where the ellipsoid ladder goes, and where a map turns out to be two maps that are not each other’s inverse.