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.

Version: v3 · 2026-09-07 · History
Domain-specific #: 3112
Origin domain: cartographic projection design
Subdomain: pseudoazimuthal equal-area projections

Core Idea¶

The Wiechel projection is a spherical, polar, pseudoazimuthal, equal-area map projection presented by H. Wiechel in 1879^[1]. 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^[2]

\[ r=2R\sin\frac{c}{2}, \qquad \theta=\lambda+\frac{c}{2}, \]

with planar coordinates

\[ x=r\sin\theta, \qquad y=r\cos\theta. \]

Reversing longitude sign, reflecting an axis, or choosing a different zero direction reverses or rotates the pinwheel without changing the projection's mathematical identity. What must survive is the coupling between radius and azimuth: Lambert's equal-area radius remains unchanged, while map azimuth differs from geographic longitude by half the colatitude, with sign determined by convention.

The construction preserves spherical area and scale along meridians, but it is not conformal, generally equidistant, or genuinely azimuthal. It is chiefly a recognized novelty and teaching projection rather than a standard operational basemap^[1]. Its autonomy lies in a precise reusable transformation, graticule, invariant, inverse, and distortion pattern—not in frequency of use.

Structural Signature¶

The defining roles are:

the spherical source — longitude and latitude, or longitude and colatitude, on a declared sphere;
the centered pole — the spherical point mapped to the disk center;
the Lambert radial law — \(r=2R\sin(c/2)\), so parallels retain the spacing of polar Lambert azimuthal equal area;
the radial azimuthal twist — \(\theta-\lambda=\pm c/2\), with handedness determined by longitude and axis convention;
the circular image — the centered pole is a point, parallels are concentric circles, and the opposite pole is represented by the outer circle;
the pinwheel graticule — every fixed-longitude meridian is a semicircle through the center and a point on the boundary;
the authalic invariant — equal spherical areas receive equal planar areas up to the projection's constant scale;
the meridional-scale invariant — distance measured along a mapped meridian has scale one on the reference sphere;
the residual distortion field — angle, local shape, direction from the center, and general distance are not preserved.

The role combination is the recognition test. A circular equal-area map with straight radial meridians is Lambert azimuthal equal area, not Wiechel. A pinwheel drawing whose radial law, azimuthal twist, or area invariant differs is not established as Wiechel merely by resemblance.

The inverse follows directly. For a point inside the disk, compute

\[ r=\sqrt{x^2+y^2}, \qquad c=2\arcsin\!\left(\frac{r}{2R}\right), \qquad \theta=\operatorname{atan2}(x,y), \]

then \(\lambda=\theta-c/2\) for the displayed convention and latitude \(\varphi=\pi/2-c\). At (r=2R), all azimuths represent approaches to the opposite pole; longitude there is intrinsically indeterminate.

What It Is Not¶

Wiechel is not an ordinary azimuthal projection. In a polar azimuthal map, direction from the center corresponds to longitude or initial azimuth. Wiechel deliberately changes the map angle by (c/2), so the direction from the center varies with distance. Its concentric parallels make it look azimuthal, while its curved meridians make the classification pseudoazimuthal.

It is not the Lambert azimuthal equal-area projection. Lambert supplies the same radius for every colatitude and the same equal-area guarantee, but Lambert's polar meridians are straight radii. Wiechel adds the distance-dependent angular twist that creates the pinwheel.

It is not the Hammer projection. Both are named equal-area transformations historically related to Lambert constructions. Hammer uses a half-longitude equatorial Lambert kernel and reciprocal axis scaling to make a 2:1 ellipse for a continuous whole-world view. Wiechel uses a polar disk, keeps Lambert radial spacing, and turns meridians into semicircles.

It is not conformal. Equal area does not imply preserved angles or local shapes, and the twist changes the intersection angle between meridians and parallels away from the center. It is not generally equidistant, even though meridional scale is correct. Nor is it a coordinate reference system: datum or reference body, radius, angular conventions, units, centering, and false offsets remain separate metadata.

Finally, it is not defined by a software viewer's clipping choice. MATLAB documents a 65-degree trimming limit for its particular display implementation^[3]. The mathematical equations map the full spherical colatitude range \(0\le c\le\pi\) to the full disk \(0\le r\le2R\).

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^[1]. 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. Tissot indicatrices or a plotted graticule make the introduced shear visible.

It is also a literal reusable software transformation. MATLAB's Mapping Toolbox has long exposed the wiechel identifier for spherical coordinates, and projection libraries or custom code can implement the forward and inverse formulas^[3]. Reuse across different datasets, radii, longitude origins, and north- or south-polar aspects remains the same abstraction, provided conventions are declared.

The equations may be applied to any spherical reference surface, terrestrial or otherwise. An ellipsoidal Earth requires an explicit preceding choice—such as using authalic latitude and radius—or a separately documented extension. Nothing in the canonical formula silently supplies an ellipsoidal model.

Clarity¶

A claimed Wiechel implementation should pass six checks:

Its source coordinates are referenced to a sphere and a centered pole.
Its radial coordinate is the polar Lambert equal-area radius \(2R\sin(c/2)\).
Its map angle differs from relative longitude by exactly half the colatitude, up to declared orientation or reflection.
Parallels plot as concentric circles and meridians as equally spaced semicircles forming a pinwheel.
The planar area Jacobian equals the spherical area element up to a constant scale.
Scale along each meridian is one, while conformality and general direction preservation are not claimed.

The equations supply stronger diagnostics than a thumbnail. At the centered pole, (c=0) and (r=0). At the equator, \(c=\pi/2\) and \(r=\sqrt2R\). At the opposite pole, \(c=\pi\) and (r=2R), with the whole boundary circle representing a single spherical point approached along different meridians. A map that places the antipode at one point or an ellipse requires a different method or additional transformation.

The frozen Wikipedia page's polar rewrite is not repeated. It used an unqualified expression involving (arcsin r_L), although a dimensional Lambert radius cannot be passed directly to an inverse sine and may exceed one. The safe normalized relation is \(c/2=\arcsin(r/(2R))\), followed by \(\theta=\lambda\pm c/2\).

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.

The representation also reduces projection comparison to testable invariants. Equal area can be checked from a Jacobian; meridional scale from a path derivative; pseudoazimuthality from whether map angle equals geographic azimuth; and identity from the graticule. This prevents the common mistake of classifying projections solely by outline.

For implementation, the disk boundary gives a domain test, the inverse gives a round-trip test, and the center/equator/antipode radii give anchor tests. Reflection and rotation can be treated as coordinate conventions rather than mistaken for new projection identities. Conversely, changing the coefficient of © in the angular twist changes the graticule and must be documented as a different family member or deformation.

Abstract Reasoning¶

Area preservation can be derived without trusting appearance. In polar coordinates the planar area element is \(r\,dr\,d\theta\). Since

\[ \frac{dr}{dc}=R\cos\frac{c}{2}, \qquad \frac{\partial\theta}{\partial\lambda}=1, \]

the Jacobian from \((c,\lambda)\) to the map is

\[ r\frac{dr}{dc} =2R^2\sin\frac{c}{2}\cos\frac{c}{2} =R^2\sin c, \]

which is exactly the area element of a sphere of radius (R). The added term (c/2) in \(\theta\) does not change this determinant because it shears each radial band rather than expanding or contracting its angular measure.

Meridional scale is equally direct. Along a fixed longitude,

\[ ds_{map}^2=dr^2+r^2d\theta^2 =R^2\cos^2\frac{c}{2}\,dc^2 +R^2\sin^2\frac{c}{2}\,dc^2 =R^2dc^2. \]

That equals spherical meridional arc length, so scale is correct along meridians. But mapped meridians no longer cross parallels orthogonally except at limiting positions, so this true-scale result does not confer conformality.

For fixed (lambda), eliminate © using \(r=2R\sin(\theta-\lambda)\) in the chosen orientation. This is the polar equation of a circle of radius (R) passing through the origin; the relevant half is a semicircular meridian. The pinwheel is therefore an analytic consequence, not decorative styling.

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.

The example also teaches an implementation discipline: normalize quantities before passing them to trigonometric functions, declare orientation conventions, distinguish mathematical domain from display clipping, and test preservation properties numerically rather than by visual resemblance. These practices transfer across geospatial software, yet they are consequences of broader Projection, Measurement, and Conservation abstractions, not evidence that Wiechel itself is a prime.

Literal transfer exists between Earth maps and maps of other spherical bodies because the same angular source and disk equations recur. Transfer to a swirling infographic or a general area-preserving animation is metaphorical unless it uses the full coordinate transformation and recognition invariant.

Examples¶

Polar world graticule. Set the north pole at the map center, Greenwich as \(\lambda=0\), and (R=1). Parallels become circles of radii \(2\sin(c/2)\). Every meridian becomes a semicircle through the center, and successive longitudes rotate those semicircles uniformly into the familiar pinwheel. The south pole is the boundary circle rather than a unique plotted point.

Anchor calculation. A point at latitude \(30^\circ\) north has \(c=60^\circ\), so \(r=2R\sin30^\circ=R\). In the stated orientation its map angle is its relative longitude plus \(30^\circ\). The radius records the Lambert equal-area band; the added \(30^\circ\) records the Wiechel twist.

Equal-area band. A spherical zone between colatitudes (c_1) and (c_2) has area \(2\pi R^2(\cos c_1-\cos c_2)\). Its mapped annulus has area (pi(r_2^2-r_12)), which simplifies to the same value. Twisting angular position within the annulus does not change that total.

Software implementation. A program converts latitude to colatitude, computes ® and \(\theta\), then evaluates (x,y). Round-trip tests recover the original coordinate except for conventional longitude ambiguity at the poles. A user-specified clip at \(65^\circ\) from the center is an application setting, not part of the projection identity.

Non-example—Lambert polar equal area. A disk has the same radii and preserves area, but meridians are straight rays with \(\theta=\lambda\). It fails the half-colatitude twist and is Lambert, not Wiechel.

Structural Tensions¶

Area preservation versus angular deformation. The shear leaves the area Jacobian intact while changing the angle at which coordinate curves meet. The projection is useful for teaching conservation but visually severe for shape interpretation. Diagnostic: verify the Jacobian separately from Tissot angular deformation.

Correct meridional scale versus misleading distance intuition. Each meridian is true to scale even though it is a semicircle, but arbitrary routes and parallels are not generally true to scale. Diagnostic: identify the path class before interpreting a measured map distance.

Global mathematical domain versus practical clipping. The formula maps the full sphere, with the opposite pole spread around the boundary. A software viewer may clip far from the center to avoid severe deformation. Diagnostic: treat clipping as renderer metadata and do not infer a smaller mathematical domain.

Orientation convention versus identity. East-positive and west-positive longitude, reflected axes, and zero-direction choices reverse or rotate the pinwheel. Diagnostic: reduce formulas to the invariant \(r=2R\sin(c/2)\) and \(|\theta-\lambda|=c/2\) under the declared convention.

Historical novelty versus autonomous abstraction. Limited operational use might make Wiechel look like one historical picture. Its forward and inverse transformations, exact invariants, parameterized reuse, software implementation, and stable comparison boundaries make it a method rather than a single artifact. Diagnostic: ask whether new coordinates can be generated and tested from the rule, not how often the result is published.

Structural–Framed Character¶

Wiechel is strongly structural–framed. It has typed spherical inputs, a designated center, explicit forward and inverse equations, an exact image disk, a provable conservation law, a true-scale path family, and falsifiable graticule consequences. An implementation that substitutes another radial law, removes the half-colatitude angular term, or loses equal area is a different projection, not an interpretive variant.

Its frame is nevertheless cartographic. Colatitude, longitude, graticule, map scale, equal area, azimuth, and spherical reference surface are indispensable. The name also anchors a particular historical construction. The higher-order pattern—mapping a curved source to a planar representation while preserving a chosen invariant—is already expressed by Projection and conservation-related reasoning.

The abstraction is therefore useful beyond historical description: it coordinates formula selection, implementation validation, projection teaching, distortion analysis, and comparison with Lambert and other pseudoazimuthal maps. But its vocabulary and recognition tests remain bound to mathematical cartography.

Structural Core vs. Domain Accent¶

The structural core is an area-preserving shear of a disk: keep the radius assigned by an authalic base map, then alter angle as a function of radius without changing the longitude derivative. This separates conservation of measure from preservation of directions and shapes.

The domain accent fixes the source as a sphere, the base radius as polar Lambert equal area, the angular shift as half colatitude, the image as a disk, the centered and opposite-pole behavior, semicircular meridians, concentric parallels, and cartographic distortion semantics. Remove those commitments and only generic Projection, Transformation, and Conservation remain.

This split controls variants. A common rotation, reflection, longitude-origin shift, scale change, or north/south aspect preserves Wiechel when recorded. A different radial law changes the base projection. A different distance-dependent angular function produces another disk shear. An application clip changes only the displayed extent.

Wiechel is a strict domain-specific instance of Projection. It maps a curved spherical surface into a planar representation, selecting area and one path-scale property for preservation while discarding azimuthality, conformality, and general distance fidelity. The exact transformation is not entailed by the live prime.

It also relates to Conservation, Transformation, Symmetry, Measurement, and Tradeoff. Area is conserved; a coordinate transformation performs the mapping; rotational organization structures the graticule; scale supplies measurement diagnostics; and preservation of area is traded against other geometric properties. These are explanatory relations rather than additional minimal DAG parents.

The proposed parent set contains only prime:projection. If a live Map Projection, Equal-Area Projection, or Pseudoazimuthal Projection node later exists, parent locality should be reconsidered before implementation.

Relationships to Other Abstractions¶

Current abstraction Wiechel Projection Domain-specific

Parents (1) — more general patterns this builds on

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 (83^rd 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

Not to Be Confused With¶

Lambert azimuthal equal-area projection: same radial spacing and area invariant in polar aspect, but straight meridians and preserved azimuth from the center.
Hammer projection: equal-area Lambert-derived whole-world ellipse using half longitude and axis rescaling, not a pinwheel disk.
Ginzburg III: another pseudoazimuthal family noted by USGS, with different coordinate curves and distortion structure.
True azimuthal projections: map angle from the center tracks spherical azimuth; Wiechel's distance-dependent twist intentionally breaks this property.
Conformal polar maps: preserve local angle but cannot simultaneously preserve area except trivially; Wiechel is explicitly nonconformal.
A spherical spiral or pinwheel graphic: visual similarity is insufficient without the Lambert radial law, half-colatitude twist, and authalic invariant.
Wiechel hill-shading work: H. Wiechel is also cited in the history of analytical relief shading; that is a separate cartographic contribution, not an alias for this map projection.

References¶

[1] Snyder, John P. and Voxland, Philip M. An album of map projections. U.S. Geological Survey Professional Paper 1453, 1989. Cited for the USGS Album's per-projection record of Wiechel (p. 149), which gives its classification and its 1879 origin; the entry could not be opened directly for this check, though the same descriptors and the 1879 attribution appear verbatim in the MathWorks Mapping Toolbox entry derived from it. Cited for the USGS Album's Usage note, which is what records Wiechel's standing as a novelty rather than an operational map; the 'teaching' characterisation is editorial and is not attested in the sources consulted. The USGS Album entry (p. 149) is the source of the classification and usage characterisation attributed to USGS here; the page itself could not be opened for this check, so the exact three-term classification string remains unconfirmed. registry ↩a ↩b ↩c

[2] Bretterbauer, Kurt. Die runde Erde, eben dargestellt: Abbildungslehre und sphaerische Kartennetzentwuerfe. Geowissenschaftliche Mitteilungen 59, Technische Universitaet Wien, 2002. Bretterbauer's monograph treats the Wiechel projection at sec. 3.2.3, p. 37 (section confirmed present and titled as an equal-area azimuthal projection); the orientation convention transcribed here follows that section, whose text could not be read directly in this check. registry ↩

[3] MathWorks. “wiechel — Wiechel Projection”. MATLAB Mapping Toolbox Documentation, The MathWorks, Inc., 2026. MathWorks documents the limit verbatim - 'Data greater than 65 degrees distant from the center point is trimmed' - and frames it as a property of this implementation, not of the projection. MathWorks documents the wiechel identifier as a sphere-only Mapping Toolbox projection and dates it to before R2006a, an archived 2005 copy of the same page confirming the longevity; the clause about other libraries and custom code is a general remark, not a vendor claim. registry ↩a ↩b