Nicolosi globular projection

Abū Rayḥān Muḥammad ibn Aḥmad Al-Bīrūnī, who was the foremost Muslim scholar of the Islamic Golden Age, invented the first recorded globular projection for use in celestial maps about the year 1000.[1] Centuries later, as Europe entered its Age of Discovery, the demand for world maps increased rapidly, sparking a vast experimentation with diverse map projections. Globular projections were one category that received early attention, with inventions by Roger Bacon in the 13th century, Petrus Apianus in the 16th century, and also in the 16th century by French Jesuit priest Georges Fournier. In 1660, Giovanni Battista Nicolosi, a Sicilian chaplain in Rome, reinvented Al-Biruni's projection as a modification of Fournier's first projection. It is unlikely Nicolosi knew of al-Biruni's work, and Nicolosi's name is the one usually associated with the projection.[2]

Nicolosi published a set of maps on the projection, one of the world in two hemispheres, and one each for the five known continents. Maps using the same projection appeared occasionally over the ensuring centuries, becoming relatively common in the 19th century as the stereographic projection fell out of common use for this purpose. Use of the Nicolosi projection continued into the early 20th century. It is rarely seen today.

Nicolosi developed the projection as a drafting technique. Translating that into mathematical formulae yields:[3]

${\displaystyle {\begin{aligned}b&={\frac {\pi }{2\left(\lambda -\lambda _{0}\right)}}-{\frac {2\left(\lambda -\lambda _{0}\right)}{\pi }}\\c&={\frac {2\varphi }{\pi }}\\d&={\frac {1-c^{2}}{\sin \varphi -c}}\\M&={\frac {{\frac {b\sin \varphi }{d}}-{\frac {b}{2}}}{1+{\frac {b^{2}}{d^{2}}}}}\\N&={\frac {{\frac {d^{2}\sin \varphi }{b^{2}}}+{\frac {d}{2}}}{1+{\frac {d^{2}}{b^{2}}}}}\\x&={\frac {\pi }{2}}R\left(M\pm {\sqrt {M^{2}+{\frac {\cos ^{2}\varphi }{1+{\frac {b^{2}}{d^{2}}}}}}}\right)\\y&={\frac {\pi }{2}}R\left(N\pm {\sqrt {N^{2}-{\frac {{\frac {d^{2}}{b^{2}}}\sin ^{2}\varphi +d\sin \varphi -1}{1+{\frac {d^{2}}{b^{2}}}}}}}\right)\end{aligned}}}$

Here, ${\displaystyle \varphi }$ is the latitude, ${\displaystyle \lambda }$ is the longitude, ${\displaystyle \lambda _{0}}$ is the central longitude for the hemisphere, and ${\displaystyle R}$ is the radius of the globe to be projected.

In the formula for ${\displaystyle x}$, the ${\displaystyle \pm }$ sign takes the sign of ${\displaystyle \lambda -\lambda _{0}}$, i.e. take the positive root if ${\displaystyle \lambda -\lambda _{0}}$ is positive, or the negative root if ${\displaystyle \lambda -\lambda _{0}}$ is negative.

In the formula for ${\displaystyle y}$, the ${\displaystyle \pm }$ sign takes the opposite sign of ${\displaystyle \varphi }$, i.e. take the positive root if ${\displaystyle \varphi }$ is negative, or the negative root if ${\displaystyle \varphi }$ is positive.

Under certain circumstances, the full formulae fail. Use the following formulae instead:

When ${\displaystyle \lambda -\lambda _{0}=0}$,

${\displaystyle {\begin{aligned}x&=0\\y&=R\varphi \end{aligned}}}$

When ${\displaystyle \varphi =0}$,

${\displaystyle {\begin{aligned}x&=R\left(\lambda -\lambda _{0}\right)\\y&=0\end{aligned}}}$

When ${\displaystyle |\lambda -\lambda _{0}|={\frac {\pi }{2}}}$,

${\displaystyle {\begin{aligned}x&=R\left(\lambda -\lambda _{0}\right)\cos \varphi \\y&={\frac {\pi }{2}}R\sin \varphi \end{aligned}}}$

When ${\displaystyle |\varphi |={\frac {\pi }{2}}}$,

${\displaystyle {\begin{aligned}x&=0\\y&=R\varphi \end{aligned}}}$

• List of map projections

• Image of Nicolosi's world map at the David Rumsey collection.