Equal-area projection

By Gauss's Theorema Egregium, an equal-area projection cannot be conformal. This implies that an equal-area projection inevitably distorts shapes. Even though a point or points or a path or paths on a map might have no distortion, the greater the area of the region being mapped, the greater and more obvious the distortion of shapes inevitably becomes.

Description

In order for a map projection of the sphere to be equal-area, its generating formulae must meet this Cauchy–Riemann-like condition: $${\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }\phi }\cdot \frac{\mathrm{\partial }x}{\mathrm{\partial }\lambda }-\frac{\mathrm{\partial }y}{\mathrm{\partial }\lambda }\cdot \frac{\mathrm{\partial }x}{\mathrm{\partial }\phi }=s\cdot \cos \phi }$$ where ${\displaystyle s}$ is constant throughout the map. Here, ${\displaystyle \phi }$ represents latitude; ${\displaystyle \lambda }$ represents longitude; and ${\displaystyle x}$ and ${\displaystyle y}$ are the projected (planar) coordinates for a given ${\displaystyle (\phi ,\lambda )}$ coordinate pair.

For example, the sinusoidal projection is a very simple equal-area projection. Its generating formulae are: $${\displaystyle \begin{array}{cc}x& =R\cdot \lambda \cos \phi \\ y& =R\cdot \phi \end{array}}$$ where ${\displaystyle R}$ is the radius of the globe. Computing the partial derivatives, $${\displaystyle \frac{\mathrm{\partial }x}{\mathrm{\partial }\phi }=-R\cdot \lambda \cdot \sin \phi ,\frac{\mathrm{\partial }x}{\mathrm{\partial }\lambda }=R\cdot \cos \phi ,\frac{\mathrm{\partial }y}{\mathrm{\partial }\phi }=R,\frac{\mathrm{\partial }y}{\mathrm{\partial }\lambda }=0}$$ and so $${\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }\phi }\cdot \frac{\mathrm{\partial }x}{\mathrm{\partial }\lambda }-\frac{\mathrm{\partial }y}{\mathrm{\partial }\lambda }\cdot \frac{\mathrm{\partial }x}{\mathrm{\partial }\phi }=R\cdot R\cdot \cos \phi -0\cdot (-R\cdot \lambda \cdot \sin \phi )=R^2\cdot \cos \phi =s\cdot \cos \phi }$$ 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: $${\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }\phi }\cdot \frac{\mathrm{\partial }x}{\mathrm{\partial }\lambda }-\frac{\mathrm{\partial }y}{\mathrm{\partial }\lambda }\cdot \frac{\mathrm{\partial }x}{\mathrm{\partial }\phi }=s\cdot \cos \phi \cdot \frac{(1-e^2)}{(1-e^2{\sin }^2\phi {)}^2}}$$ where ${\displaystyle e}$ is the eccentricity of the ellipsoid of revolution.

Statistical grid

The term "statistical grid" refers to a discrete grid (global or local) of an equal-area surface representation, used for data visualization, geocode and statistical spatial analysis.