Is orange peel a sinusoidal map?
Nearly, and the way it falls short is the point. Peel an orange in segments from pole to pole,
n of them, and lay each flat. If each parallel keeps its length on the table, a segment's half
width at latitude φ is (π / n) cos φ: that is the sinusoidal projection
(+proj=sinu) on the segment's own central meridian. Lay the segments side by side,
touching at the equator, and they are the interrupted sinusoidal, the map of gores on
Globals and interrupteds.
But peel can hardly stretch, and what it would like to keep is length in every direction at
once. No flat piece of a sphere can do that, however small (Gauss's Theorema Egregium,
1827: the sphere's curvature is a property of its lengths alone, and the plane's is zero), so
every gore is a compromise. The table compares three, each true along its central meridian.
The sinusoidal keeps area exactly but shears: towards the poles its edges lean in, and with 12
gores angles there are out by up to 15°. Cassini (cass, true across the gore)
and transverse Mercator (tmerc, true in angle) never stretch more than 3.5% for 12
gores. Halve the width and the sinusoidal's shear halves, while the others' stretch falls to a
quarter (test/check-peel.mjs checks both rates). So a real segment of peel, which
can stretch only a little, sits nearer a Cassini or transverse Mercator gore than a sinusoidal
one; all of them meet as the gores narrow, and the sinusoidal is the one that keeps area and
lines up into a whole map.
Globe makers have printed gores to paste onto a ball since Waldseemüller's set of 1507,
thought to be the first printed (Library of Congress). Their gores often stop short of the poles,
which get round caps instead, where a narrow gore is all shear.
The spiral peel
Peel it the other way, in one strip of even width w wound from pole to pole, latitude rising
by w each turn. The strip is about 4π / w radii long, the sphere's area over its width. Laid
flat, a narrow strip keeps the curvature its centre line had on the sphere (its geodesic
curvature), and at latitude φ that is tan φ. Measured by arc length t from the equator,
sin φ = t / a with a = 2π / w, so the flat line turns at t / √(a² − t²).
Near the equator that is t / a, curvature growing in step with arc length: an Euler
spiral (a clothoid), whose coordinates are the Fresnel integrals, scaled by √(2a). This
is Bartholdi and Henriques (2012), Orange peels and Fresnel integrals. The blue line is
the Euler spiral; the readout says how much of the strip, from the middle out, stays within
half a strip's width of it. The ends, near the poles, curl tighter than the Euler spiral does,
and the narrower the strip, the smaller the share it matches but the closer the match there
once scaled.
A spiral peel is not a map projection in the usual sense: it is one long piece, every point of
the sphere is on it once, and nothing on it is straight. The gores are the opposite trade: many
pieces, each nearly a map.
How it works
Gores: each one is a grid in longitude and latitude, flattened by its projection
(sinu, cass or tmerc on the sphere, following PROJ and
checked against PROJ 9.8 in test/check-peel.mjs). The peel opens the flower: each
gore is wrapped on a sphere of radius 1 / (1 − peel) that touches the orange at its south
pole, with each point's distance from the pole and its direction there blended from the orange
to the flat gore. At 0 it is the orange, at 1 the flat flower.
Spiral: the flat centre line is integrated from its heading a − √(a² − t²),
and the strip is the centre line with its width either side (held inside the radius of
curvature near the poles, where the flat strip would fold over itself). The peel comes off from
the south pole; the peeled part lies flat in the plane that touches the orange where it is
coming away, matched to the strip's direction there.
The distortion figures are worked from the gore's own formulas by finite differences:
area is the worst error in areal scale, angles the worst angular distortion,
stretch the largest error in scale in any direction, and edge how much longer the
gore's flat edge is than the meridian it came from (π radii).
Hash keys: cut (gores, spiral), n,
method (sinu, cass, tmerc), lay
(flower, row), turns, p (the peel, 0 to 1),
lon0, skin, euler, fruit, spin,
cam.
- Bartholdi, L. and Henriques, A. (2012) Orange peels and Fresnel integrals, The Mathematical
Intelligencer 34(3), 1-3, arXiv:1202.3033.
- Gauss, C. F. (1827) Disquisitiones generales circa superficies curvas (the Theorema
Egregium).
- Snyder, J. P. (1987) Map Projections: A Working Manual, USGS Professional Paper 1395
(sinusoidal, Cassini and transverse Mercator on the sphere), as
PROJ has them.
- Library of Congress, Globe gores
(Waldseemüller's 1507 gores; Coronelli's 1688 globe of 24 gores and two polar caps).
- Coastlines and land from Natural Earth.