This projection is the conformal member of the family of conic projections.
The map in this illustration has its standard parallel at 40 degrees North latitude, suitable for a map of the United States.
With 50 degrees North latitude as the standard parallel, here is a projection covering the entire Northern hemisphere:
Although conic projections are most noted for their applicability to most parts of the world in their conventional case, they also offer flexibility in that their standard parallel can be varied to fit different shapes, and they can be subjected to coordinate conversions like any other projection.
Conformal maps are particularly suited to depictions of extended areas, since they avoid the shape distortions most noticeable to the eye. The following map:
shows an oblique Lambert's Conformal Conic which covers Antarctica, Africa, Eurasia, and Australia, with its standard parallel an arc that travels through them all, shown in red on the map above. As can be guessed from the fact that the map is half a circle, the standard parallel is 30 degrees in the map coordinate system.
Nearly the whole world can be projected on the Mercator projection, and the same can be done for this projection, particularly if a standard parallel such as 10 degrees in the map coordinate system is chosen:
The Lambert Conformal Conic projection is derived from the stereographic projection, which follows the rule:
90 - lat r = tan( ---------- ) 2
by knowing that the standard forms of most mathematical functions when applied to complex numbers produce conformal mappings, and the effect of raising a number to a power in complex numbers is to raise its distance from the origin to that power using conventional arithmetic, and to multiply its angle from the x axis by that power.
Since on any conic projection with a standard parallel at lat_sp, the angles are reduced by being multiplied by sin(lat_sp), then it follows we can raise the stereographic r to the power sin(lat_sp) to get the radius from the pole for any parallel of latitude in the conformal conic.
More complicated calculations allow one to determine for what standard parallel the scale at two other parallels, north and south of it, will be equal. Then, the scale of the map can be given as the scale on those parallels; although the projection is the same, this is known as the version with two standard parallels, since at those two parallels the scale is correct, even if the curvature of the parallel is correct for another parallel between them.