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Pseudocylindrical Family·Inventor: Karl Brandan Mollweide·Year: 1805
Mollweide Projection
"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 \thetaY-COORDINATE:
y = \sqrt{2} \sin \thetaDIFFERENTIAL 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
LunologiumSTANDARDIZES
Celestial Sky Dome Mapping
SonologiumRESONATES_WITH
Harmonic Elliptical Boundaries
Academic Provenance: Mollweide, K. B. (1805)