How it works
A cone touches the sphere along one parallel, the standard parallel. Cut along the meridian
opposite the central one and laid flat, it becomes a sector of a circle: the cone constant is
the fraction of the circle it fills, and it equals the sine of the standard parallel.
Move the standard parallel to the equator and the cone becomes a cylinder; move it to a pole and
it becomes a plane. Each property is one family across all three surfaces:
| Property | Cylinder | Cone | Plane |
|---|
| Equidistant | eqc | eqdc | aeqd |
| Conformal | merc | lcc | stere |
| Equal-area | cea | aea | laea |
Unlike the perspective projections there is no point of light: the property decides how far
along the surface each parallel lands. The links join points on the globe to where they land
on the surface. Unroll opens the cone with its apex fixed, the way a paper cone opens.
The conformal row is Lambert's own framing: he gave the conformal conic (1772) as the family
containing the polar stereographic and Mercator (Snyder 1987, section 15). On the sphere, with
t = tan(pi/4 - lat/2) and n = sin(lat_1), the polar stereographic radius is 2R t, the conic's is
F t^n, and Mercator's y is -R ln t. The cone raises the stereographic radius to the power n; as n
goes to 0, t^n is about 1 + n ln t, the apex runs off to infinity and the parallels straighten
into Mercator's. So Mercator is the logarithm of the polar stereographic. PROJ's lcc
stops short of either end, so this page switches to merc and stere
there. The guide has more.
The flat map is drawn by proj4js from the PROJ
string, so it checks the construction.