A mirror ball is an equal-area fisheye

A mirror ball has no centre of projection. That is the first result of this field and it stands: the lines of sight are not concurrent, and a photograph of a mirror ball is a projection of nothing from anywhere.

It is nonetheless a picture surface in the precise sense the curved field uses the phrase — a rule taking a direction in the world to a position in a picture. Those two properties turn out to be separable, and separating them is what this rung is for, because the rule a mirror ball obeys is not a rule anybody designed and is exactly one of the four that were.

The two-line derivation #

Photograph the ball from far enough away that the incoming rays are parallel to the axis.

A point of the ball at angle $\alpha$ from that axis, measured at the ball’s centre, has its surface normal pointing at $\alpha$. A ray coming in along the axis therefore reflects at $2\alpha$ to it: the ball shows the direction $\theta = 2\alpha$ at the place whose normal is at $\alpha$.

That place appears in the picture at radius $R\sin\alpha$, because from far away the ball’s silhouette is an orthographic shadow of the sphere.

Eliminate $\alpha$:

$$\rho \;=\; R\sin(\theta/2).$$

That is the equal-area rule — usually written $2\sin(\theta/2)$, and the factor is the scale a picture is defined up to. It is not an approximation to the rule, or a family that contains it: it is the rule, and it falls out of one reflection and one silhouette.

And the measurement #

A derivation of that length is worth checking against the machinery, and the check has more in it than confirmation.

The ball’s map is computed by ray-tracing: from the camera, to the surface, reflected, and the resulting direction measured against the axis; and the picture position is where the hit point projects through the camera, which for a finite camera distance is not proportional to the radius on the ball. Each of the four named rules is then fitted to that curve with one free scale — because a rule is the shape of a curve and the size of the picture is not part of the claim — and the worst residual is reported as a percentage of the picture’s radius.

Four things in that table are doing work.

The residual falls toward zero, which makes the limit a limit rather than a coincidence at one distance. A single measurement at one camera position would have shown a good fit and proved nothing.

It reaches 0.01%, which is close enough to arithmetic that the identification is an identity rather than a resemblance.

The next-best rule is 21% out at every distance. “Nearly a fisheye” would be an empty statement if any of the four were nearly right; one of them is right and the others are a fifth of the picture away.

And close up it is measurably wrong. At three radii the ball is 4.27% from equal-area, which says the first three rows are about the geometry rather than about a curve that fits anything.

Why the camera distance matters at all #

The derivation assumed parallel rays and the measurement shows exactly how much that assumption is worth, which is the useful part.

At a finite distance two things depart from the ideal. The incoming ray at a point of the ball is not parallel to the axis — it comes from the camera, so its angle depends on where the point is — and the picture is a perspective projection of the ball rather than an orthographic silhouette, so the picture radius is not $R\sin\alpha$.

Both effects are first order in the ratio of the ball’s radius to the camera’s distance, which is exactly the pattern in the table: the residual roughly halves each time the distance doubles.

That gives a practical rule with a number in it. To use a mirror ball as an equal-area fisheye to one per cent, stand about twenty radii away — five metres from a 25 cm ball. Closer than that and the rule needs correcting; the correction is computable, since the geometry is exact, but it is no longer a named rule.

What it shows: very nearly everything #

The other property worth measuring is the field of view, and it is the one that makes a mirror ball worth using at all.

A convex mirror reflects a direction back toward the camera at the vertex, and directions increasingly far around the ball as the point moves toward the silhouette. At the tangent circle — where the camera’s ray grazes the surface — the reflected direction is very nearly straight back past the camera.

So the ball shows a solid angle of nearly the whole sphere, and the measurement puts numbers on “nearly”:

That is a remarkable instrument to be able to buy in a garden centre. No designed lens reaches it: a 180° fisheye covers half the sphere, and this covers essentially all of it, at the cost of the photographer’s own reflection occupying the middle and the blind spot sitting directly behind the ball.

The three rules it is not #

It is worth being explicit about which rules the ball is not, because two of them are near misses in ways that matter.

Equidistant, $\rho \propto \theta$. This is the runner-up in every row of the table, at 21%, and it is the one most people assume a fisheye obeys. Near the axis it agrees with the ball to first order — every one of the four rules does — and it diverges steadily, so a comparison made over a narrow field would identify the ball as equidistant with a small residual and be wrong.

Stereographic, $\rho \propto 2\tan(\theta/2)$. This one is unbounded at 180° and the ball reaches 180°, so it cannot be the rule at all — the fit is forced, and the residual reflects that rather than a near miss.

Orthographic, $\rho \propto \sin\theta$. This is the interesting near miss, because $\sin\theta$ and $\sin(\theta/2)$ agree in shape over the first quadrant and then part company completely: $\sin\theta$ turns over at 90° and comes back down, so it maps the front and rear hemispheres on top of each other. That is exactly the silhouette of the ball — the shape of its shadow — and confusing the shape of the ball with the rule of the picture it makes is the most natural mistake available here. The factor of two between the two sines is the whole of the reflection.

That last point is worth carrying. The ball’s silhouette is orthographic; the ball’s picture is equal-area; and the difference between them is precisely that reflection doubles the angle. Two closely related functions, one geometric step apart.

What it is good for, and what it is not #

Once the rule is known, the ball becomes a measuring instrument with a stated calibration, and the third column of the picture-surface table says what it is calibrated for.

Counting. Equal-area means equal picture area stands for equal solid angle, so a plain pixel count on a photograph of a mirror ball measures a fraction of the sphere directly, with no weighting. That is the one thing a fisheye is most often bought for and the one thing most fisheyes get wrong.

Environment capture. A single photograph of a mirror ball records the illumination arriving at that point from essentially every direction, at a known angular weight — which is why it is the classical way to capture a lighting environment.

Not shape. The rule bends every line that is not a radius, badly, and the area weighting that makes it good for counting is exactly what makes shapes wrong.

And not as a second view. This is where the first rung of the field bites. The picture is a projection from nowhere, so the directions it records are not all measured from the same point — they are measured from points spread over the ball’s surface, and the spread is the miss the first rung measures. For directions, at a distance where the objects are far compared with the ball, that hardly matters. For anything close, it does.

Reading it against a designed lens #

The comparison worth making is not between the ball and the ideal rule but between the ball and the instrument somebody would buy instead, and on three counts the ball does surprisingly well.

Field. A fisheye lens sold as 180° covers half the sphere. The ball covers essentially all of it from any reasonable distance. On coverage alone nothing designed competes.

Weighting. A lens marketed as a fisheye may obey any of the four rules and its documentation frequently does not say which. The ball obeys one rule exactly, derivable from its shape, needing no calibration data and no trust in a specification.

Cost and robustness. A chrome ball is cheap, has no glass to scratch in a way that matters geometrically, and can be replaced without recalibration because its rule depends only on its being a sphere.

Against that, three real costs. The photographer is in the middle of every picture. The angular resolution at the rim is dreadful, because a whole hemisphere is compressed into a thin annulus. And the picture has no viewpoint, so anything close to the ball is recorded from a smear of positions rather than from one.

Which sorts the uses cleanly. For counting a distant environment — sky, canopy, the illumination arriving at a point — the ball is an excellent instrument with an exactly known calibration. For anything requiring resolution near the rear, or any measurement of nearby geometry, it is the wrong tool for reasons that are structural rather than a matter of quality.

The identification, and why it is not a coincidence #

Two rows of this site’s work were chosen separately — one about the surfaces a picture can be made on, one about mirrors that are not cameras — and they meet at this rule. It is worth saying why that is structural rather than lucky.

The equal-area rule exists because $\sin(\theta/2)$ is the function whose Jacobian is constant on the sphere. The mirror ball produces $\sin(\theta/2)$ because reflection doubles the angle and an orthographic silhouette takes the sine. Those are two different facts and they meet in one function.

The deeper reason is that both are statements about the sphere’s own geometry rather than about anything designed. A ball is a sphere; the set of directions is a sphere; and the map between them that reflection produces is the one that respects the sphere’s area, because reflection at a sphere is an isometry of the sphere composed with a doubling of the angle at the centre — and doubling an angle at the centre is exactly what halves a solid angle uniformly.

So the identification is what should have been expected, and the measurement is what turns “should have been expected” into a number with a limit attached.

Where the near-field correction goes #

At a finite camera distance the ball is not equal-area, and since the geometry is exact the departure is computable rather than merely present. Where it goes is worth knowing, because it decides which part of the picture a close-up shot spoils.

The two finite-distance effects work in the same direction and both are largest near the rim. The incoming ray’s angle at a point of the ball departs from the axis by more the further that point is from the vertex; and the perspective foreshortening of the ball’s own surface is also concentrated at the rim, where the surface is nearly edge-on. So the middle of a close-up mirror-ball photograph is very nearly equal-area and the outer annulus is not.

That means the practical remedy is not always distance. Counting something that lies within, say, 90° of the camera’s own axis uses only the inner half of the picture’s radius, and the near-field error there is a small fraction of the 4.27% quoted for the whole frame. Counting a whole sphere uses the rim and needs the distance.

Stated as a rule: the correction is a rim effect, so how far back to stand depends on how much of the sphere is being counted.

What to check on a real ball #

Three things, in order of how much they cost.

Stand far enough back. Twenty radii for one per cent. A long lens rather than a short one, for the same framing.

Mask the photographer. The middle of the picture is the camera’s own reflection, and it is exactly the direction a count usually cares least about — the one behind the camera is the blind spot, and it is at the rim.

And know that the rim is the worst part. Near the silhouette the map is compressed enormously: a large range of directions lands in a thin annulus, so the angular resolution there is poor even though the area weighting is right. The area is correct and the detail is not, and those are different complaints.

That last is the honest limit of the instrument. Equal-area says a count is unbiased; it says nothing about how finely the count can resolve, and near the rim of a mirror ball it cannot resolve at all.