Angles, and what they cost
Lay the photograph on the complex plane and warp it with a function w = f(z).
If that function is holomorphic — built from +, ×, ÷, exp, log and powers,
with a derivative that isn’t zero — then near any point it does exactly one thing:
rotate and scale. It cannot shear. Angles survive, small circles stay circles,
and faces come out bigger or smaller but never squashed. Such a map is called
conformal , and a surprising number of the effects people want are of this
kind: the little planet, the Droste spiral, the glass bead, the whirlpool.
The number
Conformality is measurable rather than a matter of opinion. Take the map’s
Jacobian at a point and its two singular values σ₁ ≥ σ₂; the ratio
K = σ₁/σ₂ is the quasiconformal dilatation . It says what shape a tiny
circle comes back as: K = 1 is a circle, K = 2 an ellipse twice as long as it is
wide. Everything in this tool is measured that way, so the label on each map is a
claim you can check, not a decoration. The dilatation view paints K over the
picture; cold is 1, hot is shear.
Why a fisheye can’t win
Every real lens projection — equidistant, equisolid, orthographic, and even the
stereographic one — is a radial map r ↦ g(r). Such a map stretches by g′(r)
along the radius and by g(r)/r around it, and those two agree only when
g(r) = cr, which is a plain zoom. So no fisheye and no radial bulge can keep
shape. That isn’t an implementation failure, it is a small theorem, and the
measurement will show it to you: switch the lens projection with the dilatation
view on and watch which one costs least. (Stereographic, at moderate fields —
it is the projection photographers reach for when faces must survive the corners.)
The conformal family has a magnifier too, and it shows the price from the other
side. The Möbius bulge enlarges without a trace of shear, but it must
move what it enlarges: a conformal map cannot swell one region and leave
the rest where it was. Compare it with pinch & bulge , which holds
everything in place and shears instead. Something always gives.
The little planet is the exception worth knowing
Wrap the picture onto a sphere by stereographic projection, turn the sphere, and
project back down. Rotations of the sphere correspond exactly to Möbius
transformations of the plane, so the whole operation is conformal — which is why
the little planet, of all the extreme wide-angle looks, is the one where every
doorway and every face is still the right shape.
Two honest caveats
The measurement is taken at a fraction of a pixel around each sample point, so
where the map moves further than about sixty source pixels between neighbouring
output pixels — at a planet’s horizon, at a branch seam, at the pole of an
inversion — it stops meaning anything. Those points are counted as beyond
measurement and left out of the statistics rather than quietly averaged in.
And worst K can be owned by a single seam pixel, which is why the 99th
percentile is reported beside it.
Resampling uses a mip pyramid, and that is another dividend of conformality:
when σ₁ = σ₂ the correct filter footprint is a circle , so an isotropic mip
lookup is exactly right — no anisotropic filtering needed, and no boiling in the
shrunken parts of a Droste ring.
Everything runs in your browser; the photograph never leaves the tab.
Part of photo.mino.mobi , next to
glass and
glitch .
close