Equal-area projection

In order for a map projection of the sphere to be equal-area, its generating formulae must meet this Cauchy-Riemann-like condition:[1]

${\displaystyle {\frac {\partial y}{\partial \varphi }}\cdot {\frac {\partial x}{\partial \lambda }}-{\frac {\partial y}{\partial \lambda }}\cdot {\frac {\partial x}{\partial \varphi }}=s\cdot \cos \varphi }$

where ${\displaystyle s}$ is constant throughout the map. Here, ${\displaystyle \varphi }$ represents latitude; ${\displaystyle \lambda }$ represents longitude; and ${\displaystyle x}$ and ${\displaystyle y}$ are the projected (planar) coordinates for a given ${\displaystyle (\varphi ,\lambda )}$ coordinate pair.

For example, the sinusoidal projection is a very simple equal-area projections. Its generating formulæ are:

${\displaystyle {\begin{aligned}x&=R\cdot \lambda \cos \varphi \\y&=R\cdot \varphi \end{aligned}}}$

where ${\displaystyle R}$ is the radius of the globe. Computing the partial derivatives,

${\displaystyle {\frac {\partial x}{\partial \varphi }}=-R\cdot \lambda \cdot \sin \varphi ,\quad R\cdot {\frac {\partial x}{\partial \lambda }}=R\cdot \cos \varphi ,\quad {\frac {\partial y}{\partial \varphi }}=R,\quad {\frac {\partial y}{\partial \lambda }}=0}$

and so

${\displaystyle {\frac {\partial y}{\partial \varphi }}\cdot {\frac {\partial x}{\partial \lambda }}-{\frac {\partial y}{\partial \lambda }}\cdot {\frac {\partial x}{\partial \varphi }}=R\cdot R\cdot \cos \varphi -0\cdot (-R\cdot \lambda \cdot \sin \varphi )=R^{2}\cdot \cos \varphi =s\cdot \cos \varphi }$

with ${\displaystyle s}$ taking the value of the constant ${\displaystyle R^{2}}$.

For an equal-area map of the ellipsoid, the corresponding differential condition that must be met is:[1]

${\displaystyle {\frac {\partial y}{\partial \varphi }}\cdot {\frac {\partial x}{\partial \lambda }}-{\frac {\partial y}{\partial \lambda }}\cdot {\frac {\partial x}{\partial \varphi }}=s\cdot \cos \varphi \cdot {\frac {(1-e^{2})}{(1-e^{2}\sin ^{2}\varphi )^{2}}}}$

where ${\displaystyle e}$ is the eccentricity of the ellipsoid of revolution.

These are some projections that preserve area:

• Azimuthal
  • Lambert azimuthal equal-area
  • Wiechel (pseudoazimuthal)

Albers projection of the world with standard parallels 20° N and 50° N.

• Conic
  • Albers
  • Lambert equal-area conic projection

Bottomley projection of the world with standard parallel at 30° N.

• Pseudoconical
  • Bonne
  • Bottomley
  • Werner

Lambert cylindrical equal-area projection of the world

• Cylindrical (with latitude of no distortion)
  • Lambert cylindrical equal-area (0°)
  • Behrmann (30°)
  • Hobo–Dyer (37°30′)
  • Gall–Peters (45°)

Equal Earth projection, an equal-area pseudocylindrical projection

• Pseudocylindrical
  • Boggs eumorphic
  • Collignon
  • Eckert II, IV and VI
  • Equal Earth
  • Goode's homolosine
  • Mollweide
  • Sinusoidal
  • Tobler hyperelliptical
• Other
  • Eckert-Greifendorff
  • McBryde-Thomas Flat-Polar Quartic Projection[7]
  • Hammer
  • Strebe 1995
  • Snyder equal-area projection, used for geodesic grids.

• Authalic latitude
• Authalic radius
• Equiareal map (mathematics)
• Measure-preserving dynamical system
• Geodesic polygon area