Return to Answer

The earliest application is the Mercator projection which was introduced long before the complex exponential was defined in the way we define it nowadays. $z\mapsto e^z$ is considered as a map from the plane $C$ to the Riemann sphere $S$, where the plane is equipped with the usual metric, and the sphere with the spherical metric. Then $e^z$ is the inverse of the Mercator projection.

The map can be characterized by two properties: a) it is conformal, and b) meridians and parallels correspond to straight lines in the plane.

Discovered by Mercator in 1569, this was the second non-trivial example of a conformal map that was considered historically. The first one was the stereographic projection discovered in antiquity (but not known then to be conformal).

Referemce: Robert Osserman, Conformal mapping from Mercator to the Millennium.

The earliest application is the Mercator projection which was introduced long before the complex exponential was defined in the way we define it nowadays. $z\mapsto e^z$ is considered as a map from the plane $C$ to the Riemann sphere $S$, where the plane is equipped with the usual metric, and the sphere with the spherical metric. Then $e^z$ is the inverse of the Mercator projection.

The map can be characterized by two properties: a) it is conformal, and b) meridians and parallels correspond to straight lines in the plane.

Discovered by Gerard Mercator* in 1569, this was the second non-trivial example of a conformal map that was considered historically. The first one was the stereographic projection discovered in antiquity (but not known then to be conformal).

*Not to be confused with his children Arnold and Rumold, also cartographers, or with the mathematician Nicholas Mercator, a contemporary of Newton.

Reference: Robert Osserman, Conformal mapping from Mercator to the Millennium.

The earliest application is the Mercator projection which was introduced long before the complex exponential was defined in the way we define it nowadays. $z\mapsto e^z$ is considered as a map from the plane $C$ to the Riemann sphere $S$, where the plane is equipped with the usual metric, and the sphere with the spherical metric. Then $e^z$ is the inverse of the Mercator projection. So this is the second non-trivial example of a conformal map that was considered historically. The first one was the stereographic projection.

Referemce: Robert Osserman, Conformal mapping from Mercator to the Millennium.

The earliest application is the Mercator projection which was introduced long before the complex exponential was defined in the way we define it nowadays. $z\mapsto e^z$ is considered as a map from the plane $C$ to the Riemann sphere $S$, where the plane is equipped with the usual metric, and the sphere with the spherical metric. Then $e^z$ is the inverse of the Mercator projection.

The map can be characterized by two properties: a) it is conformal, and b) meridians and parallels correspond to straight lines in the plane.

Discovered by Mercator in 1569, this was the second non-trivial example of a conformal map that was considered historically. The first one was the stereographic projection discovered in antiquity (but not known then to be conformal).

Referemce: Robert Osserman, Conformal mapping from Mercator to the Millennium.

The earliest application is the Mercator projection which was introduced long before the complex exponential was defined in the way we define it nowadays. $z\mapsto e^z$ is considered as a map from the plane $C$ to the Riemann sphere $S$, where the plane is equipped with the usual metric, and the sphere with the spherical metric. Then $e^z$ is the inverse of the Mercator projection. So this is the second non-trivial example of a conformal map that was considered historically. The first one was the stereographic projection.

Referemce: Robert Osserman, Conformal mapping from Mercator to Millenum.

The earliest application is the Mercator projection which was introduced long before the complex exponential was defined in the way we define it nowadays. $z\mapsto e^z$ is considered as a map from the plane $C$ to the Riemann sphere $S$, where the plane is equipped with the usual metric, and the sphere with the spherical metric. Then $e^z$ is the inverse of the Mercator projection. So this is the second non-trivial example of a conformal map that was considered historically. The first one was the stereographic projection.

Referemce: Robert Osserman, Conformal mapping from Mercator to the Millennium.