Planar and modified planar projections

(Azimuthal) projections
In planar (azimuthal) projections, the parallels are full concentric circles centered on a pole. (Recall that the normal aspect for planar projections is polar.) As with conics, the meridians are straight, equally-spaced radii of the circles. Some planar projections show the whole world; others show just a hemisphere.

The Stereographic (upper left), Orthographic (center), Gnomonic (lower left), Lambert Azimuthal Equal Area (upper right), and Azimuthal Equidistant (lower right) are planar projections. The first three are perspective; the last two are not.

Are there pseudoazimuthal projections?

As you might guess, a pseudoazimuthal projection is one with parallels that are concentric circles and meridians that are curved. The Weichel projection—which looks like a pinwheel—is about the only example that gets mentioned in the literature. It doesn't seem to have practical importance and is not supported in ArcGIS.

Modified planar
Many projections, with many different properties, fall under the heading of modified planar. Just a few are introduced here.

If you take an Azimuthal Equidistant projection of a single hemisphere, put it in equatorial aspect, and stretch it horizontally until it contains the whole world, you have a projection called the Aitoff. It's an aesthetically pleasing world projection without excessive distortion of shape or area.

The Aitoff projection (lower right) is a modification of the Azimuthal Equidistant (upper left).

The Hammer-Aitoff is a similar modification of the Lambert Azimuthal Equal Area projection. It looks like the Aitoff projection, but has the advantage of preserving true area.

The Winkel Tripel projection is obtained by averaging the coordinate values of the Aitoff and Equidistant Cylindrical projections. It is the world projection used by the National Geographic Society in its latest world atlas (7th edition).

All three have curved parallels and meridians. In the Aitoff and Winkel Tripel projections, the spacing of parallels and meridians is equal. In the Hammer-Aitoff, it is slightly unequal. None of the three preserves the property of true direction.

The Aitoff (upper left), Hammer-Aitoff (right), and Winkel Tripel (lower left) are modified planar projections.