How large a circle each indicatrix describes, at 30°, 40°
The radius at which the image of a circle departs from the ellipse Tissot's construction predicts by 1 per cent, measured by bisection at 30°, 40° on six projections. Albers equal-area conic keeps its ellipse honest to 829 kilometres and Gnomonic to 46 — a factor of 17.8, decided by how fast each projection's own scale factor changes rather than by how large its distortion is.
It is drawn by indicatrix-figure with
show: "departure-rate" — one member of a family of
8 figures
that share a generator, so the drawing above is what that generator returns when it is asked
for this one and given nothing else.
2 essays call it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.
Where it is called
Changing this changes every one of these figures.
Measuring distortion
The indicatrix is a limit
Tissot's ellipse describes an infinitesimal circle, and every published one is drawn finite. The error of the description falls as the radius rather than as its square — halving the circle halves the lie — and on the Robinson projection at a table entry it does not fall at all: sixty metres and six kilometres are both 1.17 per cent wrong.
Measuring distortion
The indicatrix at a point that has none
Ten essays on this ladder take the derivative for granted. Rung three proved the indicatrix is a limit and every figure since has assumed the limit exists. There are points on the maps this collection draws where it does not, and they are not exotic: the horizon of a gnomonic map, and every corner of every polyhedral net.