Mercator Projection

The Mercator projection is a cylindrical map projection originating from the 16th century. It is widely recognized as the first regularly used map projection. It is a conformal projection where the equator projects to a straight line at constant scale. A rhumb line, or course of constant heading, projects to a straight line, making it suitable for navigational purposes.

Classification: Conformal cylindrical

Available forms: Forward and Inverse, spherical and ellipsoidal

Defined area: Global, but best used near the equator

Alias: merc

Domain: 2D

Input type: Geodetic coordinates

Output type: Projected coordinates

Projection String

+proj=merc
Copy

Usage

The Mercator projection is often used for equatorial regions and navigational charts. It is not suitable for world maps due to significant area distortions. For example, Greenland appears larger than South America in the projection, despite Greenland's actual area being approximately one-eighth of South America's.

Examples:

• Using latitude of true scale:

  $ echo 56.35 12.32 | proj +proj=merc +lat_ts=56.5
  3470306.37    759599.90
  Copy

• Using scaling factor:

  $ echo 56.35 12.32 | proj +proj=merc +k_0=2
  12545706.61    2746073.80
  Copy

Note: +lat_ts and +k_0 are mutually exclusive. If both are used, +lat_ts takes precedence over +k_0.

Parameters

• lat_ts: Latitude of true scale
• k_0: Scaling factor
• lon_0: Longitude of origin
• x_0: False easting
• y_0: False northing
• ellps: Ellipsoid
• R: Radius of the sphere

Mathematical Definition

Spherical Form

• Forward Projection: $$x = k_0 \cdot R \cdot \lambda$$ $$y = k_0 \cdot R \cdot \psi$$ where $$\psi = \ln\left(\tan\left(\frac{\pi}{4} + \frac{\phi}{2}\right)\right)$$
• Inverse Projection: $$\lambda = x / (k_0 \cdot R)$$ $$\psi = y / (k_0 \cdot R)$$ $$\phi = \frac{\pi}{2} - 2 \cdot \arctan\left(\exp(-\psi)\right)$$

Ellipsoidal Form

• Forward Projection: $$x = k_0 \cdot a \cdot \lambda$$ $$y = k_0 \cdot a \cdot \psi$$ where $$\psi = \ln\left(\tan\left(\frac{\pi}{4} + \frac{\phi}{2}\right)\right) - 0.5 \cdot e \cdot \ln\left(\frac{1 + e \cdot \sin(\phi)}{1 - e \cdot \sin(\phi)}\right)$$
• Inverse Projection: $$\lambda = x / (k_0 \cdot a)$$ $$\psi = y / (k_0 \cdot a)$$ $$\phi = \arctan(\tau)$$ where $$\tau = \tan(\phi)$$

Further Reading

• Wikipedia: Mercator Projection
• Wolfram Mathworld: Mercator Projection