Two surfaces with the same curvature

Six rungs of this ladder rest on one sentence: Gaussian curvature can be computed from inside a surface, a sphere has some, a plane has none, and so no map of one onto the other can preserve every distance.

That sentence says curvature is an obstruction. It does not say curvature is the only one, and the gap between those two claims has been quietly load-bearing the whole time. If some second invariant also had to match, then two surfaces could share a curvature and still be un-mappable onto each other, and every argument here that runs the curvature is the same, therefore the map exists would be missing a step.

It is possible to build the surfaces and measure. Conformal and equal-area at once forces an isometry, and an isometry is exactly what this rung is about the existence of.

Building a surface from its curvature instead of from its shape #

Parameterise the meridian of a surface of revolution by its own arclength s, so the profile is a radius r(s) and a height z(s) with r′² + z′² = 1. The metric is then

ds² + r(s)² dθ²

and the Gaussian curvature of that metric is exactly −r″/r. There is no approximation in that: it is the first fundamental form with E = 1, F = 0, G = r², put through Gauss’s own expression.

Read backwards it is a construction. Demanding a constant curvature K turns the surface into the differential equation

r″ + K r = 0

whose general solution for K = 1 is r(s) = c sin(s + s₀), a one-parameter family once the phase is fixed by requiring the surface to be symmetric. Every c gives a surface with curvature 1 everywhere, and c = 1 gives sin s, which is the sphere.

The height follows from the arclength condition: z′ = √(1 − c² cos² s), integrated. That square root decides the shape, and it decides it in three cases:

• c < 1 — the radicand is positive everywhere, the meridian reaches the axis with slope c < 1, and the surface closes to a point with a corner. A spindle, or lemon.
• c = 1 — the meridian meets the axis at a right angle and the surface closes smoothly. The sphere.
• c > 1 — the radicand goes negative near the axis, so the surface exists only in a band of latitudes and terminates in two circular edges. A bulge, or barrel.

The curvature is measured, not assumed #

Building r = c sin s and then asserting that −r″/r equals 1 proves nothing except that the algebra was copied correctly. What is worth doing is differencing the sampled profile, so that an error in the construction, in the arclength parameterisation, or in the sampling shows up as a curvature that is not 1.

The control is what makes the measurement mean anything. A routine that returned 1 for everything would pass the first three curves and fail the fourth, and the paraboloid is chosen deliberately as the hardest case — a surface that genuinely has the sphere’s curvature, at one point, and nowhere else.

The isometry, and what it agrees on #

The map between two members of the family is s ↦ s, θ ↦ (c_A/c_B) θ. It is trivial to write down; the question is whether it preserves lengths, and the honest way to ask is to measure something neither surface’s parameterisation can influence.

A geodesic circle is such a thing, and measuring one from inside is how this ladder established that curvature is intrinsic in the first place. Fire geodesics in ninety-six directions from a point, stop them all at the same arclength ρ, and measure the length of the curve their endpoints trace.

One part in 10¹⁴. The sphere and a surface that is visibly not a sphere return the same answer to a measurement made entirely from inside. This is Minding’s theorem, and this is what verifying it looks like rather than citing it.

The geodesics themselves carry their own check: Clairaut’s relation says r sin ψ is constant along any geodesic of a surface of revolution, and the integrator reports the drift in that quantity rather than assuming it, which is what makes a fourth-order Runge–Kutta trustworthy at radius 0.8 as well as at 0.1 — the same habit a direction carried round a loop needed for its holonomy.

What the isometry cannot carry #

If nothing local separates them, something global must, and the obvious candidate turns out to separate only half the cases.

That the bulges have exactly the sphere’s area is the finding this figure was not built to produce. It was built expecting area to be the separating quantity, on the reasonable grounds that total curvature ∫K dA equals the area when K = 1, and that the total curvature of a region is fixed by what it covers. Both bulges came back at 12.5664, to a part in a million.

Which is Gauss–Bonnet doing exactly what it says. A bulge is topologically an annulus, not a sphere: its Euler characteristic is 0 rather than 2, so the theorem’s statement about it involves the geodesic curvature of its two boundary circles as well as the integral over its interior. The area is free to be 4π, and the boundary term takes up the difference.

So: the sphere and a bulge agree on curvature at every point, on every length that can be measured inside them, and on total area — and are not the same surface. What separates them is that one has a boundary and the other does not, which is not a measurement at all. It is a count.

The word locally, priced #

The last column of that table is the whole content of the qualifier. Going from the sphere to the c = 1.6 bulge, a full revolution of the bulge uses 62.5 per cent of a revolution of the sphere: the isometry runs out of bulge before the sphere has finished turning. Going the other way, to the c = 0.6 spindle, it needs 166.7 per cent — the map wraps past its own start and stops being one-to-one.

Neither failure is a failure of the metric. Every distance is still right; what is wrong is that the map is not a bijection of the whole surface. That is the same distinction a cylinder and a plane already carry — both have zero curvature, both are locally isometric, and no isometry of the whole cylinder onto the whole plane exists — arriving now with a positive curvature and a measurement.

What this means for a map #

Three consequences, and the third is the reason this rung is in a collection about map projections rather than in a differential geometry text.

A projection is a map of a metric and not of a shape. Every formula in this collection takes a latitude and a longitude and returns a point on a page, and every distortion number it produces depends only on the first fundamental form of the surface those coordinates parameterise. So every projection here is equally a projection of the spindle and of the bulge, with the same indicatrix, the same areal factor, the same maximum angular deformation, at every corresponding point — and the same projection on a different body turns out to have been understating the case. The pictures would be identical and the captions would be true of all three.

Distortion measurement cannot identify the body. Identifying a projection from its graticule recovers a projection because different projections have different metrics on the page. No amount of measuring can recover which constant-curvature surface a map was made of, because the three have the same metric. The body is not underdetermined by a little; it is underdetermined by a whole one-parameter family.

And the impossibility argument is safe. Curvature is the only local obstruction, so the curvatures differ, therefore no isometry exists is a complete argument and not a partial one. No map is faithful needs Minding’s theorem to be the sharp statement it claims to be, and it has never had it.

The converse also holds and is worth having explicitly: a sphere and a plane have curvatures 1 and 0, they differ, and there is no isometry — but a cone and a plane have curvature 0 and 0, and there is one. The reason the site’s second rung could talk about unrolling a cylinder without proving anything is that Minding had already done it in 1839.

What was computed, and how #

The surfaces are built by solving r″ + Kr = 0, which for constant K has a closed form, and integrating z′ = √(1 − c² cos² s) by Simpson’s rule at four thousand steps.

The curvature is a central second difference of r with respect to arclength. Where the profile comes from a table rather than a formula — which is the paraboloid’s case — the table is interpolated with a cubic Hermite and read at a step forty times the table’s own spacing. A first version used linear interpolation and a step finer than the table, and reported the paraboloid’s vertex curvature as 1.79 against a true 1.00: the second difference of a piecewise-linear interpolant is zero inside a segment and a spike at every knot. That is the third time this collection has differenced a table at a step it was not built at, and the rule is now written where the interpolation is.

The paraboloid also has a second, independent route to its curvature — the extrinsic expression z′(r′z″ − z′r″)/(r(r′² + z′²)²), evaluated in the profile’s own parameter — and the two agree at the vertex to three decimal places, which is what makes the intrinsic figure worth reading.

Geodesics are integrated as s″ = r r′ θ′², θ″ = −2(r′/r)s′θ′ by fourth-order Runge–Kutta at fifteen hundred steps, with r′ taken numerically so that the integrator never sees a closed form.

Geodesic circles centred on the axis need no integration at all: every meridian is a geodesic through the vertex, so the circle of radius ρ about it is exactly the parallel at arclength ρ, and its circumference is 2πr(ρ). That closed form is what the paraboloid comparison uses, and it removes the polar singularity that an integrated version would have to pass through.

The assertions, and what they refuse #

Every member of the family must measure K = 1 to a part in a million; the paraboloid must measure 1 at its vertex and must not measure it at s = 1. Two surfaces of the family must agree on every geodesic circle to a part in a billion; the paraboloid must not agree with the sphere, and the disagreement must grow with the radius — a discrepancy that did not grow would be a constant offset in the measurement rather than a difference between the surfaces.

Where the model stops #

Everything here is a surface of revolution, which makes the differential equation an ordinary one. Minding’s theorem is not restricted to surfaces of revolution — any two surfaces of the same constant curvature are locally isometric — but constructing the isometry between two general surfaces means solving a partial differential equation, and this rung does not.

The theorem is also about constant curvature, and that restriction is real. Two surfaces whose curvature varies identically as a function of position are not necessarily isometric, because “as a function of position” needs a correspondence and the correspondence is what is being sought. The correct general statement involves the curvature and its derivatives along the surface’s own geodesics, and it is a great deal weaker than the constant case. Curvature that varies from place to place is where that difficulty lives, and it is not resolved here.

And the whole treatment is about intrinsic geometry. Two surfaces can be isometric and sit in space quite differently — that is what bending a sheet of paper does — and nothing measured here says anything about how a surface is embedded.

Who found it, and when #

Ferdinand Minding proved it in 1839, in Crelle’s Journal, six years before he gave the pseudosphere its constant negative curvature and eight years before Bonnet’s half of Gauss–Bonnet. Gauss’s Disquisitiones generales circa superficies curvas of 1827 had established that curvature is intrinsic; Minding established that it is all that is intrinsic, locally, which is the sharper and less quoted half.

The spindle and the bulge are usually attributed to Minding as well and are still called Minding’s surfaces. They are drawn in every differential geometry course as a curiosity — surfaces of constant positive curvature that are not spheres — and they are rarely drawn as what they also are: three bodies a cartographer could not distinguish with any instrument.

Three bodies a cartographer could not tell apart #

The remark that Minding’s surfaces are rarely drawn as what they also are is worth taking seriously, because it names precisely what this ladder’s subject is.

A cartographer’s instruments are intrinsic. Every measurement a survey can make — a distance along the ground, an angle between two lines on it, the excess of a triangle, the circumference of a circle of stated radius — is a measurement of the metric. Nothing available to a surveyor reaches outside the surface.

So an isometry is an equivalence for exactly this subject. A sphere, a spindle and a bulge of the same constant curvature return identical answers to every one of those measurements, locally, which means no amount of surveying separates them. They are not similar bodies; for the purposes of the whole apparatus this site is about, they are the same body seen three ways.

Which is a stronger statement than the textbooks’ framing. Surfaces of constant positive curvature that are not spheres presents them as a caution about a classification — a reminder that constant curvature does not imply sphere. Read the other way, they are a demonstration that the classification is the right one: what a map cares about is the metric, and the metric does not distinguish them.

It also says what a map of one of them would look like. Since the three are locally isometric, a projection of a patch of the spindle with a stated distortion has a counterpart on the sphere with exactly the same distortion — the same scale factors, the same angular deformation, the same everything the indicatrix holds. So every measurement this collection makes about mapping a sphere transfers unchanged to Minding’s surfaces, patch by patch, without any of the work being redone.

Which is the practical form of curvature is all there is. A result about mapping is a result about the metric, so it belongs to an equivalence class of surfaces rather than to a shape, and the class is labelled by the curvature. That is why this site’s arithmetic works on an ellipsoid, on a triaxial body and on an irregular one with the same machinery: nothing in it ever needed the shape, only the metric.

And the word locally is where the equivalence ends, which is why this rung prices it. Globally the three differ — a sphere closes up and Minding’s surfaces do not — and global topology is exactly the half of the subject the impossibility field’s other anchor is about. Curvature says what a map costs; topology says what a map cannot be; and Minding’s theorem is the statement that the first of those has nothing else in it.

Where the ladder goes next #

Seven rungs have established that curvature is the obstruction, that it is the only local obstruction, and that it varies over the bodies this site actually maps. What none of them has done is ask what happens when the surface is not smooth at all — when the curvature is not a function but a set of point masses, which is what a polyhedron is, and which is where the globe on a solid has been waiting for this ladder to arrive.