Azimuthal Family·Inventor: Al-Biruni (c. 1000 CE) / Guillaume Postel (1581)·Year: 1581

Azimuthal Equidistant Projection

MATHEMATICAL SPECIMEN · AZIMUTHAL-EQUIDISTANT

"All points on the map are at proportionally correct distances and azimuths from the center point"

Centered on any chosen point (such as the North Pole in the United Nations emblem or a radio broadcasting antenna), every straight line radiating from the center is a true great circle with exact linear scale.

Forward Mathematical Formulation

X-COORDINATE:

x = R \cdot c \cdot \cos \phi \sin(\lambda - \lambda_0) / \sin c

Y-COORDINATE:

y = R \cdot c \cdot [\cos \phi_1 \sin \phi - \sin \phi_1 \cos \phi \cos(\lambda - \lambda_0)] / \sin c

DIFFERENTIAL METRIC:

c = \arccos[\sin \phi_1 \sin \phi + \cos \phi_1 \cos \phi \cos(\lambda - \lambda_0)], \quad h = 1, \; k = c / \sin c

Cartographic Application & Critique

Primary Applications: United Nations Official Emblem, Amateur radio antenna direction-finding (beamheading), Seismic epicenter radius plotting.

Critical Analysis: Its radial symmetry conveys international unity on the UN flag, but the antipodal point expands from an infinitesimal point into the entire outer perimeter circle.

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
90°N	1.00	1.00	1.00×	0.0°
60°N	1.00	1.05	1.05×	2.8°
30°N	1.00	1.21	1.21×	11.4°
0°N	1.00	1.57	1.57×	25.8°
-90°N	1.00	99.00	99.00×	180.0°

Interdisciplinary Geometric Links

Acoustics & ResonanceMIRRORS

Omnidirectional Acoustic Radiation

Jurisprudence & CadastreSYMBOLIZES

International Diplomatic Emblems

Academic Provenance: Postel, G. (1581)