Ben Joffe

A magic new Y-value formula:

y = \tanh^{- 1} (\sin (l a t) \cdot a) \cdot \frac{1}{a}

y_{a = 1} \Rightarrow Mercator Conformal

lim_{a \to 0} y \Rightarrow Lambert Equal Area

Values in-between \Rightarrow These New Maps

See the full formula for more.

A New Equal-Area Map for Interactive Computer Use

Seamlessly transitioning to Mercator upon zoom (and some new compromise cylindrical maps)

15 September 2026

Recently the UN passed the non-binding "Correct the Map" resolution which encourages the use of the Equal Earth projection in place of Mercator (which represents country-sizes disproportionately).

The Mercator projection is well suited to zoomable/navigation maps due to being "conformal", meaning it represents directions accurately.
At a global scale Equal Earth may look pretty good, but it's not what web and mobile based apps typically require.

I present an alternative. The map below is equal-area at a global scale, and upon zoom, it transitions seamlessly to the Mercator projection.
It is defined by an unusually simple math formula. The hybrid nature of the map is baked into the very design.

Formulas for the Ellipsoid Deep link

The formulas presented are modelled for a spherical Earth, just as is the case for "Web Mercator". Maps that model the ellipsoidal Earth can be easily supported by tweening from authalic latitude to conformal latitude prior to projection.

I plan to expand on the mathematical properties of this projection in a future blog post.

How and why I made this Deep link

Map projection design is certainly a niche hobby. Too mathematical to be of interest to many map nerds. Too constrained and ugly to be of interest to many mathematicians.

Well, the thing that led to this rabbit hole was actually me just wanting a nice 2:1 aspect ratio map for the world, that showed country shapes well. I was surprised that I could not find any such map projection to meet my needs.

The simulation to the right is the content that I wanted that 2:1 map for. It is a representation of how a boat travelling west, circling the world roughly once every 37 days, will keep an identical clock to a person on Mars. This visualisation was intended for use in my article about time on Mars, however I never ended up including it as I ended up making that page static so it could be shared as a PDF.

When searching for the math that would create this look I wanted, it was clear to me that it needed to be similar to Mercator near the equator, and similar to an equal-area map near the poles.

Loading simulation…

After looking at the definition of the Mercator projection, I could see it was usually defined in terms of log and tan, but one definition usually further down the list stood out to me: the atanh sin x version. This definition is perfect because within it lies the function for equal area: sin x. Scaling this function down, and then back up again (after atanh), is a fairly simple way to create this tweening function.

Novelty Deep link

This is certainly not the first aspect-adaptive map projection, in fact one of the examples above referred to "Aspect-Adaptive Miller", which can also adapt to various aspect ratios. What I think is novel is the combination of algorithmic simplicity and aesthetic result.

People have tried to create a more "tame" version of the Mercator projection by doing some kind of interpolation, whether that's interpolation of Mercator to Equirectangular (see Miller Cylindrical Projection ), or linear interpolation of Lambert cylindrical equal-area to Mercator (see Arden-Close Cylindrical (1947) ). Such interpolations usually leave the polar regions much larger than desired, leading to later "workarounds" like the "Compact Miller", which attempt to flatten the poles through specific mathematical tweaks layered on top.

A more flexible way of blending projections was outlined in Blending world map projections with Flex Projector (Jenny and Patterson, 2013). The tool discussed in the paper allows tweaking and blending map projections in realtime, a tool which was used by the same authors along with Bojan Šavrič to create the Equal Earth projection in 2018. Jenny also published a related but distinct paper on Adaptive Composite Map Projections .

Mapbox has had Adaptive Projections since 2021, allowing on-the-fly transitions between map projections.

Others have come up with ways to blend Lambert cylindrical equal-area and Mercator in mathematical non-linear ways. See Revolvable Indoor Panoramas Using a Rectified Azimuthal Projection (Fong, 2012). See the section: "A2.3 A Blended Cylindrical Map Projection".
The formula presented in the paper is significantly more complex than the one presented here, and the paper states "the inverse equation for this projection is particularly gnarly".

The formula atanh(sin(lat)a)/a is so simple it's hard to imagine I'm the first to use it. I have dug through Snyder's seminal work "Map projections: A working manual", as well as his book "Flattening The Earth" and found no reference to this formula (as well as online search, and LLM queries). Nonetheless, I put this out in the world now, and if somebody knows of this formula being used in a prior map projection I would be happy to hear of it and update this page accordingly.

Annexure A – Equal Area Proof Deep link

Write $φ$ for latitude and $λ$ for longitude. At $z = 0$ (before the Mercator tween) the projection is:

y = \frac{1}{a} \tanh^{- 1} (a \sin φ), x = b λ (1 - a^{2} \sin^{2} φ)

A projection is equal area if every small patch of the sphere lands on the map with its area multiplied by the same constant. So take a tiny patch spanning $d φ$ of latitude and $d λ$ of longitude, and compare its area on the sphere with its area on the map.

On the sphere

On a unit sphere the patch is $d φ$ tall, and $\cos φ d λ$ wide, because circles of latitude shrink by $\cos φ$ towards the poles:

sphere area = \cos φ d φ d λ

On the map

The limit of the shape on the map will be a rectangle or a parallelogram because the projection draws horizontal lines of latitude. Either shape's area can be obtained from width × height (vertical rise).

Width: Along a line of latitude, $φ$ is fixed and $x$ is simply $λ$ times a constant, so:

width = b (1 - a^{2} \sin^{2} φ) d λ

Height: $y$ depends only on $φ$ , so the height is $\frac{d y}{d φ} d φ$ . Using $\frac{d}{d t} \tanh^{- 1} (t) = \frac{1}{1 - t^{2}}$ and the chain rule:

\frac{d y}{d φ} = \frac{1}{a} \cdot \frac{a \cos φ}{1 - a^{2} \sin^{2} φ} = \frac{\cos φ}{1 - a^{2} \sin^{2} φ}

height = \frac{\cos φ}{1 - a^{2} \sin^{2} φ} d φ

Area: Width × Height; the $(1 - a^{2} \sin^{2} φ)$ factors cancel:

map area = b (1 - a^{2} \sin^{2} φ) d λ \cdot \frac{\cos φ}{1 - a^{2} \sin^{2} φ} d φ = b \cos φ d φ d λ

This is exactly $b$ times the area on the sphere, wherever the patch sits. So the projection is equal area, with a uniform scale factor of $b$ .

Notice that $a$ cancelled out entirely: any value of $a (0. .1)$ gives an equal area map. It controls the shape of the map, not its area properties.

Next Time

This is just the first of several map related articles I will be publishing over the coming months. A future post will explore the geometric meaning of projecting a map this way, including how it can be applied to conic map projections, leading to maps that look a little like the one to the right. This is a custom map projection I have had on my law firm website (my day job) - National Probate and Estates Group. I suspect this is the only law firm in Australia with a custom map projection!

Notice how the lines of latitude are curved, but the states still appear nearly rectangular?

If you would like to be notified when these articles come out, follow me on X

ACTNT NSWSA QLDTASVICWA

Revision history:

15 September 2026 - First release

Map textures and vector data on this page are derived from Natural Earth. Free vector and raster map data @ naturalearthdata.com.

A New Equal-Area Map for Interactive Computer Use

Seamlessly transitioning to Mercator upon zoom (and some new compromise cylindrical maps)

The Formula Deep link

Side-by-Side Screenshots Deep link

United States and Central America

Australia and New Zealand

Japan and Korea

Africa

South America

Europe

Why These Constants (a & b) Deep link

Compromise Cylindrical Projections Deep link

Domino (2:1)

Golden Rectangle (Φ:1)