[oofabz]

When you project a map, there are three properties you would like to maintain: shape (aka conformality), size (aka equal-area), and direction. But you can have at most two of these properties.

Mercator preserves shape and direction at the expense of size. Peters preserves size and direction at the expense of shape. Peirce Quincuncial preserves shape and size at the expense of direction. Here's a transverse Peirce Quincuncial map I generated: http://frammish.org/tpq.jpg

Many other projections try to combine these, like the Miller projection maintains direction but strikes a balance between shape and size, getting neither one right, but neither is horribly wrong either. The Winkel Tripel projection tries to balance all three attributes.

[JacobiX]

This is a consequence of theorem discovered by Gauss: Theorema Egregium http://en.wikipedia.org/wiki/Theorema_Egregium

[anonymous]

Something else I'm interested in is if somebody has ever tried using Pierce's projection for texturing spheres or doing environment mapping. It seems like it would be perfect for the job - it's square so it would fit perfectly in a texture, it can be tiled and the formula for mapping a point on the sphere from spherical coordinates to uv coordinates on the texture seems simple enough.

[pmjordan]

The main problem I see is that it's a nonlinear projection and thus generating one procedurally on the GPU (render to texture) is rather challenging. Hardware support for cube maps places them in a sweet spot between efficiency, accuracy, practicality for generating and practicality for applying.

[wtallis]

That example isn't an equal-area projection, it's just better on average at preserving size than Mercator. It does still have significant size distortions, such as making Sri Lanka look massive, when it's really slightly smaller in total area than Ireland.

[cLeEOGPw]

Your example does not preserve size, because it makes Arabian peninsula (area ~2.3 million km2) look almost as big as Europe (area ~10 million km2). Also compare India with Australia. I think this map preserves shape and direction, but not size.

[vanderZwan]

I like how there no real sense of up or down in the Peirce Quincuncial map.

[SimHacker]

I like your Peirce Quincuncial projection. It makes Florida look much more excited.