Pseudocylindrical Family·Inventor: Karl Brandan Mollweide·Year: 1805

Mollweide Projection

MATHEMATICAL SPECIMEN · MOLLWEIDE

"Equal-area pseudocylindrical projection with elliptical meridians and straight parallels"

Published in Leipzig as an improvement on Mercator for astronomical and global geographic distributions, Mollweide represents the entire spherical world within an exact 2:1 ellipse.

Forward Mathematical Formulation

X-COORDINATE:

x = \frac{2 \sqrt{2}}{\pi} (\lambda - \lambda_0) \cos \theta

Y-COORDINATE:

y = \sqrt{2} \sin \theta

DIFFERENTIAL METRIC:

2 \theta + \sin(2\theta) = \pi \sin \phi \quad \text{(transcendental equation solved via Newton-Raphson)}

Cartographic Application & Critique

Primary Applications: Cosmic Microwave Background (CMB) sky maps (WMAP / Planck), Global planetary science datasets.

Critical Analysis: A staple of astrophysics and climatology because its smooth elliptical boundary naturally represents complete celestial or planetary spheres without rectangular corner artifacts.

TISSOT INDICATRIX (Lat: 60°N)Area: 1.00×

MERIDIAN (h)

0.75

PARALLEL (k)

1.33

ANGULAR (2θ')

31.9°

Tissot Profile Across Latitudes

LATITUDE	SCALE h	SCALE k	AREA hk	2θ' DEFORM
0°N	1.23	0.81	1.00×	26.5°
40.7°N	1.00	1.00	1.00×	0.0°
70°N	0.72	1.39	1.00×	39.8°
90°N	0.45	2.22	1.00×	68.2°

Interdisciplinary Geometric Links

Astronomical EphemerisSTANDARDIZES

Celestial Sky Dome Mapping

Acoustics & ResonanceRESONATES_WITH

Harmonic Elliptical Boundaries

Academic Provenance: Mollweide, K. B. (1805)