Pseudocylindrical Family·Inventor: Oswald Winkel·Year: 1921

Winkel Tripel Projection

"The arithmetic mean of equirectangular and Aitoff projections, minimizing three distortion metrics"

Tripel ('triple') refers to Winkel's objective to simultaneously minimize three distortions: area, direction, and distance. Adopted by National Geographic in 1998 to replace Robinson.

Forward Mathematical Formulation

X-COORDINATE:

x = \frac{1}{2} \left[ \lambda \cos \phi_1 + \frac{2 \cos \phi \sin(\lambda/2)}{\text{sinc } \alpha} \right]

Y-COORDINATE:

y = \frac{1}{2} \left[ \phi + \frac{\sin \phi}{\text{sinc } \alpha} \right]

DIFFERENTIAL METRIC:

\alpha = \arccos\left[ \cos \phi \cos(\lambda/2) \right], \quad \phi_1 = \arccos(2/\pi) \approx 50.467^\circ

Cartographic Application & Critique

Primary Applications: National Geographic Society official standard world map (1998–present), Scientific global datasets.

Critical Analysis: Gold standard for uninterrupted whole-world maps, achieving the lowest composite distortion index according to Goldberg-Gott metric testing.

TISSOT INDICATRIX (Lat: 60°N)Area: 1.02×

MERIDIAN (h)

1.13

PARALLEL (k)

0.90

ANGULAR (2θ')

13.2°

Tissot Profile Across Latitudes

LATITUDE	SCALE h	SCALE k	AREA hk	2θ' DEFORM
0°N	1.00	1.00	1.00×	0.0°
30°N	1.04	0.98	1.02×	8.5°
60°N	1.12	0.91	1.02×	24.1°
80°N	1.25	0.76	0.95×	46.0°

Interdisciplinary Geometric Links

MorphologiumCOMPUTES

Geodesic Metric Optimization

VeridianumSTANDARDIZES

Global Event Coverage Basemap

Academic Provenance: Winkel, O. (1921)