Conic and pseudoconic projections

Conic projections
In conic projections, the parallels are partial concentric circles. The meridians are straight, equally spaced radii of the circles. Conic projections usually don't show the entire world; the projection outline is fan-shaped.

The Albers Equal Area Conic (top), Lambert Conformal Conic (center), and Equidistant Conic (bottom) are conic projections.

Pseudoconic projections
Pseudoconic projections are like conic projections in that their parallels are partial concentric circles. As with pseudocylindricals, the difference is that meridians are curved rather than straight. (Again, the meridians are equally spaced.) The outline is approximately heart-shaped.

The Bonne projection is pseudoconic.

There aren't many pseudoconic projections; only the Bonne is supported in ArcGIS. The Werner is another that is often illustrated, but rarely used.

Polyconic projection

Not all map projections use a single developable surface. Conceptually, the Polyconic projection is based on multiple layered cones. It is a non-perspective projection that can only be constructed mathematically.

The Polyconic projection is a non-perspective projection that can only be constructed mathematically.

You can see that the Polyconic map projection is not suitable for mapping the entire world. However, areas where the map projection is centered are represented with less distortion (e.g., North and South America, as shown above).

The Polyconic map projection is believed to have been invented around 1820 by Ferdinand Hassler, the first superintendent of the United States Coast Survey. To Hassler, polyconic wasn't a specific map projection but a methodology for creating map projections. Over time, the term polyconic has, almost exclusively, been used to describe a specific map projection that employs Hassler's methods for creating a map.

Distortion is significant in an east-west direction but less along the central meridian. For example, North America is represented better than the areas farther way from the central meridian.

The Polyconic map projection's most popular applications have been maps for North America, not for mapping the entire continent, but for smaller areas of the world such as a USGS 7.5' quadrangle. For this reason, in many parts of the world it is called the American Polyconic map projection.

The Polyconic map projection is neither conformal nor equal-area. Although distortion can be significant in an east-west direction from the central meridian, distortion is extremely small for areas near the central meridian, such as a USGS 7.5' quadrangle. Unfortunately, the Polyconic map projection has been used incorrectly to map entire states and on occasion the entire conterminous United States.

If you zoom in on North America the curving and diverging parallels can be seen more easily.