What a standard parallel buys
Gall–Peters is exact at 45°. Behrmann is exact at 30°. Lambert’s cylindrical equal-area is exact at the equator. All three are equal-area, all three are cylindrical, and they look completely different.
The difference between them is one number, and the number is a design decision rather than a mathematical consequence.
What “exact” means here
At a standard parallel, both principal scale factors equal one: . The map is locally an isometry there — angles right, areas right, distances right in every direction.
The theorem forbidding a faithful map is a statement about regions, not points. A projection can be perfect along a curve, and every projection with a standard parallel is. What cannot happen is perfection across a patch of non-zero area.
So a standard parallel is a line of contact between the map and the truth, and the distortion grows in both directions away from it.
Tangent and secant
The traditional picture makes the choice geometric. A cylinder resting on the equator touches along one line, and that line is the standard parallel — the tangent case.
Push the cylinder inward so it cuts the sphere, and it intersects along two parallels. Both are standard, the map is exact along both, and between them it is slightly compressed rather than stretched — the error changes sign. That is the secant case, and it is the usual choice in practice.
Secant does not reduce the total distortion. It splits it: instead of one zero line with error growing monotonically away, there are two zero lines with a small opposite-sign error between them and growing error outside. For a region spanning a known band of latitude, placing the two standard parallels inside that band roughly halves the worst-case error across it.
That is the standard rule of thumb for conics, and it is why almost every national mapping conic has two standard parallels chosen from the country’s own latitude range.
What the choice does not change
The projection’s property.
All three projections in the hero figure are equal-area, exactly, everywhere. Their areal error measures between and — noise. Moving the standard parallel from the equator to 45° does not make any of them more or less equal-area; that property is built into the construction and the standard parallel is orthogonal to it.
What changes is the shape distortion and where it falls. At the equator, Lambert’s version squashes vertically and Gall–Peters stretches vertically; at 45° the reverse. The aspect ratio of the whole map changes with it, from very wide at the equator to nearly square at 45°.
Which is a reminder worth having: two projections can share a property exactly and look nothing alike.
The Peters choice, reconsidered
This bears on the Mercator–Peters argument in a way that is rarely raised.
Peters advocated an equal-area projection on the grounds that Mercator’s areal distortion misrepresented the tropics. The projection he presented is exact at 45° — which is a temperate latitude, and which makes the temperate zones look right in shape while stretching the tropics vertically.
Behrmann’s version, exact at 30°, is equally equal-area and treats the tropics better in shape. So is a version exact at 23.5°, and there is a continuum.
The choice of 45° is free, it is not a consequence of the equal-area requirement, and it is an odd one for a projection argued for on the grounds of fairness to the tropics. This is not a devastating objection — the areal argument stands regardless — but it does illustrate that “equal-area” underdetermines a map by a great deal.
Scale factor and the surveyor’s version
There is a variant of the same idea used at large scales, and it is worth knowing because it looks like a fudge and is not.
A transverse Mercator zone — a UTM zone, or a national grid — is conformal, and exact along its central meridian in the tangent case. That leaves the error growing to the zone’s edges and zero in the middle.
The standard practice is to multiply the whole projection by a scale factor slightly less than one: 0.9996 for UTM, 0.9996012717 for the British national grid. That shrinks the map everywhere, making it slightly too small in the middle and slightly too large at the edges — which converts a tangent construction into a secant one and roughly halves the worst-case scale error across the zone.
The oddly precise constants in national grid definitions are exactly this: a deliberate uniform error introduced so that the maximum error is smaller.
Choosing one
The rule that follows from all of the above is short.
Find the latitude range the map must cover. Place the standard parallels inside it, conventionally at about a sixth and five sixths of the way through the range. The error is then zero twice, small and negative between, and small and positive at the edges.
For a world map there is no such range, which is why the choice becomes aesthetic and why equal-area world projections differ so much in appearance while agreeing exactly on their property.
Where to put them, quantitatively
The rule of thumb has a derivation worth sketching, because it explains why the answer is not the region’s edges.
Put the standard parallels at the edges of the region and the distortion is zero there and negative in the middle, with the worst error in the centre. Put them in the middle and the error is zero there and positive at both edges. Somewhere between is a placement where the worst error — at the edges and in the centre — is equalised, and that is the minimum of the maximum.
For a region spanning latitudes to , the classical answer places the standard parallels at about one sixth and five sixths of the way through the range. Kavrayskiy’s variant and several others differ in the details and all land near the same place.
That is a minimax criterion, and it is the same reasoning behind the scale factor applied to a transverse Mercator zone: introduce a uniform error deliberately so that the worst case is smaller.
The property that is not free
One clarification, because the essay’s argument could be overread.
Changing the standard parallel does not change whether a projection is equal-area, and it does change the projection’s overall proportions substantially. A cylindrical equal-area map with a standard parallel at the equator is roughly 3.1 times as wide as it is tall; at 45° it is about 1.57 times.
So the choice is not invisible. It changes what the map looks like more than almost any other single parameter, which is why three projections with identical properties can be mistaken for different families.
The same idea in three places
The pattern — introduce a deliberate uniform error so the worst case shrinks — turns up repeatedly and is worth recognising.
Secant constructions, as above: two zero lines instead of one, with a small opposite-sign error between them.
Scale factors on grid systems: UTM’s 0.9996 multiplies the whole projection down so the central meridian is slightly too small and the zone edges are slightly too large.
Minimax approximation generally: the same principle behind Chebyshev polynomials, where a deliberate equal-ripple error beats a Taylor series that is exact at one point and diverges away from it.
All three are the same trade. Perfection at one place is worth giving up if the price is a smaller error everywhere else, and the optimum has the error equalised across the region rather than concentrated at its edges.
What it looks like on the map
The visible consequence of the choice is the map’s proportions, and the range is larger than most people expect.
A cylindrical equal-area projection with its standard parallel at the equator is about 3.14 times as wide as it is tall. At 30° it is about 2.36. At 45° it is about 1.57. At 60° it is about 0.79 — taller than it is wide.
All of them are equal-area, exactly. The property is identical and the maps are barely recognisable as relatives.
Which is why “equal-area” underdetermines a map so heavily, and why the Peters argument was about a particular choice within a family rather than about the family itself.
Why the numbers are so specific
A closing note on the strange precision of published grid parameters, since it is a common source of puzzlement.
A scale factor of 0.9996012717 looks like a measurement. It is not — it is a value chosen once, for a particular zone width and a particular error tolerance, and then frozen because everything downstream depends on it.
Once a national grid is published, every map, every land title, every survey mark and every dataset is expressed in its coordinates. Changing any parameter would invalidate all of it. So the numbers are exact not because anyone measured them to ten digits but because they are definitions, and a definition is as precise as it is written.
That is a different kind of number from the ones elsewhere on this site, and it is worth recognising when it appears: a parameter that looks empirical and is administrative.
The choice is also the easiest thing to get wrong by inheritance. A dataset arrives in some projection with some standard parallels, gets reused for a different region, and nobody revisits the parameters — so the map is now optimised for a place it no longer shows. Checking that the standard parallels match the region is a ten-second operation and it is skipped constantly.
The parameter is also the cheapest thing to get right. Choosing the family and the property requires knowing the map’s purpose; choosing the standard parallels requires only knowing the region’s latitude range, which is always available. It is the one decision in this subject where the correct answer can be read off the data.
The parameter also propagates further than it looks. A dataset’s standard parallels are recorded in its coordinate reference system definition and travel with it, so a reprojection that loses them silently substitutes defaults — and a map that was optimised for one latitude band ends up optimised for the equator with nothing indicating that anything changed.
There is a final asymmetry worth noticing. A badly chosen standard parallel degrades a map gracefully — the distortion grows smoothly and the map remains usable. A badly chosen property does not: an equal-area map used for navigation or a conformal map used for a density choropleth is wrong in a way no tuning repairs. The parameter is the forgiving decision and the property is not.
A last observation about how the choice is usually made in practice: it is not. Most maps inherit their standard parallels from a template, a national grid definition or the first dataset loaded, and the parameter that could have been fitted to the region in ten seconds is instead whatever arrived. That is not always wrong — a national grid’s parallels were fitted to the country by somebody — and it is worth knowing which case applies.
The choice is also the clearest small demonstration of the site’s general theme. Three projections share a property exactly, differ in one number, and look nothing alike — so the property is not the map, the parameters matter, and reading either from the projection’s name alone gets both wrong.
That is worth keeping alongside the essay’s main point. The property is a constraint the construction satisfies; the standard parallel is a choice left over; and both have to be read to know what a map does.
What was computed here
The standard-parallel figure carries an assertion that ties the picture to the claim: each variant’s angular deformation must reach its minimum at that variant’s own standard parallel, to within two degrees. If the minimum ever moved, the figure’s caption would be wrong and the build stops.
All three variants are also required to pass the equal-area test, so the essay’s central point — that changing the standard parallel does not change the property — is measured rather than asserted.
The distortion profiles are sampled at one-degree intervals from the equator to 80°, using the projections’ own derivatives.
What the pictures cannot show
The secant case, drawn as a geometry. This site’s figures show projections rather than constructions, and the two standard parallels of a secant conic appear as the two latitudes where the distortion curve touches zero rather than as a cylinder cutting a sphere.
The essay also cannot show the aspect-ratio consequence at useful size. Three equal-area cylindrical world maps at their true proportions range from extremely wide to nearly square, and drawing them side by side at readable scale means scaling them differently, which hides the very difference being discussed.
Who found it, and when
The secant construction is old and was understood geometrically long before it was analysed. Ptolemy’s conic projections in the Geography, around AD 150, already place the cone to fit the region being mapped rather than tangentially at an arbitrary parallel.
Lambert’s 1772 treatise gives the analytic treatment of standard parallels for both the conformal conic and the cylindrical equal-area, and the modern practice of choosing them from the mapped region’s latitude range descends directly from it.
The scale-factor trick on transverse Mercator zones is twentieth-century and administrative in origin — it comes from national survey organisations minimising the worst case across a legally defined zone, which is why the constants are so specific and so arbitrary-looking.
Where this goes next
The families these choices sit inside are cylinders, cones and planes. The other free choice usually left unexercised is the aspect. And the argument this essay complicates is Mercator against Peters.