Wiechel Projection¶
A polar pseudoazimuthal equal-area map projection that retains Lambert radial spacing while twisting azimuth by half the spherical colatitude, turning meridians into a pinwheel of semicircles.
Core Idea¶
The Wiechel projection is a spherical, polar, pseudoazimuthal, equal-area map projection presented by H. Wiechel in 1879. It keeps the polar Lambert azimuthal equal-area projection's radial spacing of parallels, but rotates each point around the map center by an amount that depends on its angular distance from the centered pole. The rotation bends every meridian into a semicircle. Together the meridians form the projection's characteristic pinwheel.
Let (R) be the sphere's radius, © the colatitude measured from the centered pole, and (lambda) longitude relative to a chosen origin. One canonical orientation, following Bretterbauer, is.
Scope of Application¶
The projection belongs to mathematical cartography, projection comparison, graticule design, and geospatial software. USGS classifies it as a modified azimuthal, equal-area, pseudoazimuthal projection and describes its polar use as a novelty map. That limited historical use should be stated plainly: Wiechel is not a recommended default for navigation, cadastral mapping, conventional thematic world maps, or polar scientific operations.
Its strongest use is explanatory. It demonstrates that an equal-area polar disk need not have radial meridians. By retaining Lambert's concentric equal-area parallels while changing angular placement, it isolates the distinction among area preservation, azimuth preservation, conformality, and distance behavior.
Clarity¶
A claimed Wiechel implementation should pass six checks:
- Its source coordinates are referenced to a sphere and a centered pole. 2. Its radial coordinate is the polar Lambert equal-area radius \(2R\sin(c/2)\). 3. Its map angle differs from relative longitude by exactly half the colatitude, up to declared orientation or reflection. 4. Parallels plot as concentric circles and meridians as equally spaced semicircles forming a pinwheel.
Manages Complexity¶
Wiechel compresses a complete design choice into two coupled equations. The radial equation resolves how much area each concentric band receives. The angular equation resolves how meridians are rearranged within each band. Because those roles are separated, a cartographer can see exactly which properties come from Lambert spacing and which come from the twist.
Abstract Reasoning¶
Area preservation can be derived without trusting appearance. In polar coordinates the planar area element is \(r\,dr\,d\theta\). Since
the Jacobian from \((c,\lambda)\) to the map is
Knowledge Transfer¶
Within projection design, Wiechel is a compact example of invariant-preserving shear. Begin with a known equal-area disk map, leave every point at the same radius, and change angle by a function of radius. Because the angular derivative with respect to longitude remains one, area can survive even while directions and shapes change sharply. This reasoning transfers to the analysis of other area-preserving transforms, but the name Wiechel remains restricted to its particular half-colatitude twist.
Relationships to Other Abstractions¶
Current abstraction Wiechel Projection Domain-specific
Parents (1) — more general patterns this builds on
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Wiechel Projection is a kind of Projection Prime
Wiechel is a strict domain-specific instance of Projection.
Hierarchy path (1) — routes to 1 parentless root
- Wiechel Projection → Projection → Abstraction
Neighborhood in Abstraction Space¶
Wiechel Projection sits in a sparse region of the domain-specific corpus (83rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (1565 abstractions)
Nearest neighbors
- Hammer Projection — 0.91
- Haversine Formula — 0.88
- Karlsruhe Metric — 0.81
- Box–Muller Transform — 0.80
- World Geographic Reference System — 0.78
Computed from structural-signature embeddings · 2026-09-08