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Eckert IV projection of the world.

The Eckert IV projection is a pseudocylindrical map projection. The length of polar line is half that of the equator, and lines of longitude are semiellipses, or portions of ellipses. It was first described by Max Eckert in 1906.

Formulas

Forward formulas

Given a radius of sphere $R$, central meridian ${\lambda }_0$ and a point with polar coordinates $(\phi ,\lambda )$, $x$ and $y$ can be computed using the following formulas:

$x=\frac{2}{\sqrt{4\pi +{\pi }^2}}R(\lambda -{\lambda }_0)(1+\cos \theta )\approx 0.4222382R(\lambda -{\lambda }_0)(1+\cos \theta )$,

$y=2\sqrt{\frac{\pi }{4+\pi }}R\sin \theta \approx 1.3265004R\sin \theta$,

where $\theta +\sin \theta \cos \theta +2\sin \theta =(2+\frac{\pi }{2})\sin \phi$. This equation can be solved numerically using Newton's method.

Inverse formulas

$\theta =\arcsin \left[y\frac{\sqrt{4+\pi }}{2\sqrt{\pi }R}\right]\approx \arcsin \left[\frac{y}{1.3265004R}\right]$

$\phi =\arcsin \left[\frac{\theta +\sin \theta \cos \theta +2\sin \theta }{2+\frac{\pi }{2}}\right]$

$\lambda ={\lambda }_0+x\frac{\sqrt{4\pi +{\pi }^2}}{2R(1+\cos \theta )}\approx {\lambda }_0+\frac{x}{0.4222382R(1+\cos \theta )}$

See also

• List of map projections
• Eckert II projection
• Eckert VI projection
• Max Eckert-Greifendorff

References

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External links

• Eckert IV projection at Mathworld