How it works
A plane touches the sphere at the tangent point. A point of light sits on the axis through the
centre of the sphere and the tangent point. Each map point is where a ray from the light, passing
through a point on the sphere, meets the plane. Where the light sits decides the projection.
| Light | Projection |
|---|
| at the antipode | stereographic |
| at the centre | gnomonic |
| above the tangent point | near-side perspective |
| infinitely far | orthographic |
Two azimuthal projections need no light. With Compass, put the point of a compass on the
tangent point, open it to reach a point on the sphere, and swing it down onto the plane: that is
the Lambert azimuthal equal-area projection. With String, lay a string along the sphere from
the tangent point to the point, then pull it flat onto the plane: that is the azimuthal
equidistant projection. The Rays slider sets how many of these are drawn, and the orange ring is
where the antipode lands.
Aspects. The tangent point sets the aspect: at a pole it is polar, on the equator
equatorial, and anywhere else oblique. PROJ has no aspect parameter for the azimuthals:
+lat_0 decides it, and gnom (like stere, ortho
and laea) keeps a branch of its formulas for each of north pole, south pole, equatorial
and oblique, the same projection with the sines and cosines of lat_0 worked out.
The aspect changes what the graticule looks like. In the gnomonic every great circle is a straight
line, so the meridians and the equator always are; each parallel is where the plane cuts the cone of
rays from the centre through it, which makes it a circle (polar), an ellipse, a parabola or a
hyperbola. Tick Cone through a parallel to see the cone and the conic it leaves on the plane.
The dashed part comes from the half of the cone behind the light: in the gnomonic, a line through
the centre meets the sphere twice, so the dashed branch of a hyperbola is also where the opposite
latitude lands.
The bright part of the coastline on the globe is what reaches the plane, and the orange ring is
the horizon beyond which rays miss it. Lights between those positions give valid perspective
projections that PROJ does not name.
The flat map is drawn by proj4js from the PROJ
string, not from the 3D geometry, so the two views check each other.
This continues the 2016 post
Three projections.
Coastlines are from Natural Earth.