Angles survive and areas do not

The first rung on this ladder established that a sphere with one point removed is a plane, and left the projection as a correspondence between points. That is the weakest possible statement about it. What makes stereographic projection the map that gets used, rather than one of infinitely many bijections between a punctured sphere and a plane, is two properties that no part of the construction announces.

Every circle on the sphere becomes a circle or a line. And every angle is preserved exactly.

The formulas #

Everything below is one pair of formulas, so they are worth writing once. Put the unit sphere with its north pole at $(0,0,1)$ and project from the pole onto the plane $z = 0$. A point $(x, y, z)$ of the sphere goes to

$$\left(\frac{x}{1-z}, \frac{y}{1-z}\right),$$

and a point $(u, v)$ of the plane comes back to

$$\left(\frac{2u}{u^2+v^2+1}, \frac{2v}{u^2+v^2+1}, \frac{u^2+v^2-1}{u^2+v^2+1}\right).$$

Both are rational: no square roots, no trigonometry. That is already a hint that the map is well behaved, because a rational map of degree one takes algebraic curves to algebraic curves of the same degree, and a circle is a curve of degree two.

It is worth pausing on how much that rationality gives away. A map whose coordinate functions are ratios of polynomials of degree one in the sphere’s coordinates cannot turn a conic into anything worse than a conic, and a circle on the sphere is cut out by one linear equation. Everything in the next section is that observation carried out in symbols; the substitution has no cleverness in it, and the result is a theorem because the substitution is exact.

Circles to circles, proved #

A circle on the sphere is the intersection of the sphere with a plane, so it satisfies two equations: $x^2 + y^2 + z^2 = 1$ and $ax + by + cz = d$.

Substitute the inverse formulas into the plane’s equation. Writing $s = u^2 + v^2$, the substitution gives

$$\frac{2au + 2bv + c(s-1)}{s+1} = d,$$

and multiplying up,

$$(c - d)(u^2 + v^2) + 2au + 2bv - (c + d) = 0.$$

That is the general equation of a circle in the plane when $c \neq d$, and of a line when $c = d$. And $c = d$ says exactly that the plane passes through the north pole, since substituting $(0,0,1)$ into $ax+by+cz = d$ gives $c = d$.

So the two cases of the theorem are one algebraic condition, and the exception is not an exception at all: a line is the circle through the point that has been removed, which is what the one-point compactification says a line is.

Angles preserved, proved #

The conformality proof that is worth knowing is the one with no calculation in it.

Take a point $P$ on the sphere other than the pole $N$, and two curves crossing there. The tangent plane at $P$ meets the tangent plane at $N$ — which is the plane parallel to the image plane — in a line, and the ray $NP$ makes equal angles with the two tangent planes, because the triangle $NOP$ formed with the centre is isosceles.

That isosceles triangle is the entire proof. It makes the projection, restricted to the tangent plane at $P$, a reflection composed with a scaling — and a reflection reverses angles while preserving their size, while a scaling changes nothing. Two reflections in the composite (one at $P$, one landing in the image plane) restore the orientation. So angles come through with both their size and their sense.

The argument deserves restating in a form that shows where it could fail. Projecting from $N$ sends a small figure at $P$ to a figure in the image plane, and the map on tangent vectors factors as: project the tangent plane at $P$ to the image plane along rays through $N$. That projection between two planes is a similarity exactly when the two planes make equal angles with the ray — which is what the isosceles triangle gives, and which fails immediately for any other projection point. A one-line piece of elementary geometry is the whole reason the map is conformal, and the reason no textbook’s calculation is needed is that the calculation would be re-deriving that line.

What is worth extracting from that proof is that the map is a local similarity. At each point it scales by some factor and rotates; the factor varies from point to point, and that variation is the whole of the distortion. A map with that property is called conformal, and the next section is about the factor.

The factor, and what it destroys #

Differentiating the inverse formula gives the scaling factor at the plane point $q$:

$$\frac{2}{1 + |q|^2},$$

so lengths on the sphere are $2/(1+|q|^2)$ times lengths in the plane, and areas are multiplied by the square of that. Near the origin the factor is close to $2$; far out it decays like $2/|q|^2$, which means a patch near the pole of the sphere becomes enormous in the plane.

The distortion is unbounded, and that is not a defect that a better projection would fix. No map of any region of the sphere to the plane can preserve both angles and areas, and the reason is that a map preserving both is an isometry, while an isometry preserves Gaussian curvature — which is positive on the sphere and zero on the plane. That is Gauss’s Theorema Egregium, and it is the reason every world map is wrong in some specific way that its maker chose.

The distortion also has a clean description in terms of the sphere’s own geometry: the factor $2/(1+|q|^2)$ is exactly the conformal factor that writes the round metric of the sphere in plane coordinates. So the plane picture is the sphere, with the sphere’s own notion of distance carried along, and every statement about the sphere becomes a statement about the plane with a weight function attached. That is the standard way of doing spherical geometry in practice, and it is the same device that makes the hyperbolic plane into a disc with a weight on it.

So the choice is which property to keep. Stereographic keeps angles; the Lambert cylindrical projection keeps areas and mangles shapes; Mercator keeps angles too and is stereographic’s rival for navigation. There is no third option in which nothing is lost.

The two properties are one property #

It is worth noticing that circle-preservation and conformality are not independent, because seeing why explains what kind of map this is.

A map of the plane that takes every circle to a circle or a line, and is not a similarity, must be an inversion composed with a rigid motion — that is a classical theorem, and inversions are conformal. Conversely a conformal map defined on the whole sphere must be a Möbius transformation, and Möbius transformations take circles to circles. So on the sphere the two conditions pick out the same class of maps.

What stereographic projection actually is, seen from the plane, is an inversion. Compose the projection from the north pole with the projection from the south pole and the result is the map $q \mapsto q/|q|^2$ — inversion in the unit circle. That single sentence explains both properties at once, because inversion trades circles for circles and preserves angles and those are the two facts this rung is about.

It also explains why the exceptional point cannot be avoided. Inversion sends the centre of its circle to infinity, and the projection sends the pole to infinity, and both are the same missing point wearing the same costume. A map with these properties has to lose exactly one point, and which one is the only choice available.

Where the conformality is used #

Three uses, and each depends on the exactness rather than on the approximation being good.

Navigation and crystallography. A conformal projection lets an angle be measured on the flat picture and used directly. Crystallographers plot the directions of a crystal’s faces stereographically for exactly this reason: the angles between faces are the data — they are what identifies the crystal system, and they are what a goniometer measures — and the projection does not touch them. The whole apparatus of Wulff nets and pole figures is a stereographic plot with a rotating overlay, which is an astrolabe pointed at a mineral rather than at the sky.

Complex analysis. Identify the plane with the complex numbers and the sphere becomes the Riemann sphere, on which $1/z$ is a rotation. Conformality is then the statement that a holomorphic function is angle-preserving away from its critical points, and stereographic projection is what makes the point at infinity an ordinary point of an ordinary surface — which turns “a function blows up here” into “a function takes the value at the north pole here”, and removes an entire category of special case from the subject.

Circle packings and inversive geometry. Circles going to circles means the whole subject of circles and inversion transfers between plane and sphere without loss. That is why a circle packing realising a planar graph can be moved to the sphere, where the awkward outer face disappears and the packing becomes symmetric.

Reading the two area figures against each other settles a question the first one leaves open. A single picture of unequal patches could be an artefact of a badly chosen grid; two pictures at different resolutions with the same extreme ratio cannot be. The distortion is a function on the sphere, and a grid samples it.

What is being assumed #

Two hypotheses are doing quiet work and are worth surfacing.

The projection point must be on the sphere. Projecting from the centre — the gnomonic projection — takes great circles to lines, which is useful and is not conformal. Projecting from infinity gives the orthographic projection, which is neither. The whole of this rung is about one particular projection point, and the isosceles triangle in the conformality proof is where the choice enters.

The sphere must be a sphere. The substitution in the circle proof uses $x^2+y^2+z^2=1$ at every step; on an ellipsoid the same plane sections are ellipses and their images are not circles. The property is a property of the round sphere and of nothing near it.

And the target must be a plane, not a curved surface. Projecting the sphere onto a cylinder from its axis gives Lambert’s equal-area map, and there the isosceles argument fails and the conformality goes with it. So the two properties are not properties of “projecting”; they are properties of this projection.

Who noticed, and when #

The projection is old and its two properties were established two thousand years apart.

Hipparchus used it in the second century BC and Ptolemy describes it in the Planisphaerium, both for astronomical instruments — the astrolabe is a stereographic projection of the celestial sphere with a rotating overlay, and it works because circles stay circles, which lets the instrument be built from engraved circular arcs rather than from a computed curve at every latitude.

So the circle property was used for a millennium and a half before it was proved. The instrument makers knew it empirically; the earliest proof is usually credited to the Arab astronomer al-Farghani in the ninth century, and the conformality was not stated until Halley in 1695 — who published it precisely because he needed a projection on which angles could be measured.

That order is worth noticing. The property that is easiest to prove, and that the algebra above settles in four lines, is the one that took longest to state, because nobody had a definition of angle preserved by a map until calculus supplied one. A property is not available to be noticed until the language for it exists, which is a different obstacle from the property being hard.

What the pictures cannot show #

A residual of $10^{-9}$ is not a proof. Every figure here fits a circle or a line to hundreds of sampled points and reports how far the worst one deviates. That is a strong check on the code and no check at all on the theorem, which is the substitution above and has no picture.

Conformality is a statement about limits. The angle between two curves is the angle between their tangents, which is a limit of angles between chords. The figure computes it from a difference quotient over two neighbouring samples, which is the right approximation and is still an approximation — the tolerance it asserts against is a fifth of a degree.

The sphere is drawn in projection, which is a second projection nobody asked for. Every picture of the sphere on this page is itself a flat drawing of a round thing, made by a perspective camera — so a circle on the sphere reaches the page as an ellipse, and the claim that it is a circle is about the object rather than about the ink. That is unavoidable and worth naming, since the whole rung is about what a projection does to a circle.

And the area distortion cannot be drawn at its true size. The band touching the pole has an unbounded image, so it is omitted; the bands next to it are drawn and are already off the edge of any frame that also shows the middle. Every picture of this projection understates the distortion, because a picture has a boundary and the distortion does not.

Where the ladder goes next #

The next rung takes the plane to be the complex numbers, at which point the sphere acquires a name and the projection acquires an algebra: the Möbius transformations become the sphere’s rigid motions, and $1/z$ becomes a half-turn.

Named here as debts: the astrolabe, which is this projection engineered as an instrument and is a genuinely different kind of rung; and the general one-point compactification, which is this construction stated for any space at all and which the first rung already gestures at.

Sideways, the circle-to-circle property is what inversion in a circle has in the plane and this projection lifts to the sphere, the impossibility of keeping both angles and area is the theorem that makes every map projection a compromise of some kind, and the conformal factor $2/(1+|q|^2)$ reappears as the metric of the sphere written in the plane.

What it costs to insist on angles #

Three costs, and the first is the one that decides whether the projection is the right tool.

Nothing far from the centre can be compared with anything near it. A shape drawn near the edge of a stereographic map is the right shape and the wrong size, and no scale bar fixes that, because the scale is different at every point. Any use of the map that involves comparing extents rather than directions is a misuse of it.

The exceptional point is genuinely exceptional in use. A stereographic chart covers the sphere minus one point, so any global statement needs two charts and a rule for the overlap. That is not a technical annoyance — it is the definition of a manifold arriving because it had to — and the sphere is the standard first example precisely because two stereographic charts are the minimum and the overlap is transparent.

And the conformality is exact only for the ideal sphere. The Earth is not one, and a stereographic projection of an ellipsoid is not conformal unless the formulas are adjusted. The adjusted version exists, is standard in geodesy, and is a reminder that a property proved of a mathematical object has to be re-proved of anything it is used to model.

What is worth carrying away #

A bijection is worth very little; a bijection that preserves a structure is worth a subject.

Stereographic projection is one of infinitely many correspondences between a punctured sphere and a plane, and the only reason anybody uses it is that circles and angles come through untouched. Both properties are consequences of a single algebraic fact — the map is rational of degree one — and neither is visible in the description “shine rays from the north pole”.

The habit worth taking is to ask, of any correspondence, what it preserves rather than what it matches. Two sets in bijection have the same size and nothing else follows; two sets in a bijection preserving a structure are, for every question about that structure, one set.