The Gnomonic Projection and Operations on the Globe
Let's talk about projections---or at least about one very special projection: the gnomonic projection. Though it may not (or may?) be completely obvious, this post is related to the series of indexing posts that trailed off some months back, but it's relevance goes beyond indexing to the fundamentals of how most ellipsoidal operations work.
When we say projection, we really mean any technique by which we can take figures on the sphere and map them to figures on the plane. Of course, any such mapping will imbue the result with distortions, but many of these projections have properties that make them quite useful. The Mercator projection, for example, has nice navigational properties. As we will see, the gnomonic projection has very nice properties that makes it useful for performing geometric operations on the globe.
The gnomonic projection starts by placing a plane tangent to the sphere. For each point in the figure we're projecting, we take the ray from the center of the sphere through that point, and map it to the place where the ray intersects the tangent plane. Looking at this edge-on:
The beauty of this starts to become clear when we look at edges. Remember that we define an edge by a great circle arc (or a great ellipsoidal arc), and that these are formed by the intersection of the sphere (or ellipsoid) with the plane defined by the center of the sphere along with the two endpoints. Extending the gnomonic projection from points to edges, we find that the projection of an edge on the sphere lies along the intersection of the plane defining the edge with the chosen tangent point, and this intersection is a straight line.
Equivalently, if we use the gnomonic projection to map each of the endpoints of an edge, then the projection of the spherical edge is the same as the planar edge between the two projected endpoints. I.e., a single edge on the sphere maps exactly to a single line segment on the plane:
The inverse of this is also true: under the inverse gnomonic projection, line segments on the plane map exactly to great circle arcs on the sphere. The result of this is that unless our operation depends on distance or area, we can project it to the plane, perform our operations on the plane, and then map it back to the sphere---and these operations require no densification or approximation. This certainly makes spherical (and ellipsoidal) geometry look a little more tractable than before.
You may notice that this only works for objects in a hemisphere: e.g., if your tangent plane touches at the north pole, then points at the equator will be projected to infinity. In practice, you start to have problems before this, since objects become more and more stretched out as you move away from the point of tangency. I'll discuss how we deal with this in a future post.
Another issue is how you deal with distance and area, and on the ellipsoid, the only way is through numeric integration.