Bad general idea of azimuthal projections on Wikipedia

[Piotr]

In certain days of exploring map projections, I was confused about azimuthal projections. The definition seems to be that it preserves directions, but what are directions? This was especially confusing when azimuthal equidistant was only available in polar, and Lambert azimuthal equal-area was only available in equatorial — I couldn't see what properties made them both azimuthal. It was not until later that I realized that azimuthal projections are when small circles around a center point on Earth will project to circles around a center point on the map.

[daan]

Your conception is correct for earth modeled as a sphere. Modeled as an ellipsoid, it’s more complicated.

Your conception implies that great circles from the center point follow straight lines on the map projection, for earth as a sphere. Great circles on the sphere are geodesics. To generalize further, an azimuthal projection is one in which geodesics from the projection’s center radiate as straight lines subtending the same angle from north on the projection as they have on the sphere. It is this property that defines azimuthal projections, including on the ellipsoid.

— daan

[Piotr]

And your quasiazimuthal equal-area for "apple" is... modifying the meridian spacing for equal area meridians, parallels taking the map's shape and being spaced for equal area?

[daan]

Exactly.

— daan

[Piotr]

Another bad general idea of azimuthal projections is that they directly project the globe into a plane, instead of a cylinder or a cone. Flat maps by definition are on a plane, so that would make any projection except the cylindrical and conic ones azimuthal, because instead of using intermediate shapes they map the globe into a plane with equations (although cylindrical and conic projections may also be made like this).

[quadibloc]

Another bad general idea of azimuthal projections is that they directly project the globe into a plane, instead of a cylinder or a cone. Flat maps by definition are on a plane, so that would make any projection except the cylindrical and conic ones azimuthal, because instead of using intermediate shapes they map the globe into a plane with equations (although cylindrical and conic projections may also be made like this).

Ignoring the ellipsoid completely, as it is far too complicated for beginners...

The first thing to be addressed is "directly project". As was pointed out in the thread on Mercator's 500th birthday, not all azimuthal projections are perspective projections such as the Gnomonic, the Stereographic, the Orthographic, or Clarke's Minimum-Error Perspective Projection, of which the Twilight Projection is a special case.

Other than that, while oversimplified, the idea that cylindrical projections with parallel linear vertical meridians, using a cylinder as the reducible surface, conic projections with straight-line meridians, and azimuthal projections all have something in common, and the azimuthal ones are the ones based on the tangent plane, is factual enough.

So, for example, the Mercator projects the globe with an equation - in one direction. Latitude is subjected to an equation, but longitude is left at its "projected" value. Ditto with the simple conic, ditto with the polar case of Lambert's azimuthal equal-area projection. So an azimuthal projection is based on the plane in exactly the same way that a conic projection is based on the cone, and a cylindrical projection is based on the cylinder.

Thus, I am in disagreement with the portion of your objection that applies to the substance of the matter. However, that the usual wording of the explanation of this classification is confusing is something I am open to.