Bonne projection

The Bonne projection maintains accurate shapes of areas along the central meridian and the standard parallel, but progressively distorts away from those regions. Thus, it best maps "t"-shaped regions. It has been used extensively for maps of Europe and Asia.

The projection is defined as: $${\displaystyle \begin{array}{cc}x& =\rho \sin E\\ y& =\cot {\phi }_1-\rho \cos E\end{array}}$$ where $${\displaystyle \begin{array}{cc}\rho & =\cot {\phi }_1+{\phi }_1-\phi \\ E& =\frac{(\lambda -{\lambda }_0)\cos \phi }{\rho }\end{array}}$$ and φ is the latitude, λ is the longitude, λ₀ is the longitude of the central meridian, and φ₁ is the standard parallel of the projection.

Parallels of latitude are concentric circular arcs, and the scale is true along these arcs. On the central meridian and the standard latitude shapes are not distorted.

The inverse projection is given by: $${\displaystyle \begin{array}{cc}\phi & =\cot {\phi }_1+{\phi }_1-\rho \\ \lambda & ={\lambda }_0+\frac{\rho }{\cos \phi }\arctan \left(\frac{x}{\cot {\phi }_1-y}\right)\end{array}}$$ where $${\displaystyle \rho =\pm \sqrt{x^2+{\left(\cot {\phi }_1-y\right)}^2}}$$ taking the sign of φ₁.

Special cases of the Bonne projection include the sinusoidal projection, when φ₁ is zero (i.e. the Equator), and the Werner projection, when φ₁ is 90° (i.e. the North or South Pole). The Bonne projection can be seen as an intermediate projection in the unwinding of a Werner projection into a Sinusoidal projection; an alternative intermediate would be a Bottomley projection.