Spherical polygons on planetary surfaces

Planets are nearly spheres, so geography can be used to illustrate spherical geometry. For example, the geographic ideas of pole and equator inform (and name) geometric concepts, and small circles and great circles can be understood as parallels and meridians on the globe. Actually, planetary surfaces are marked with natural small circle and great circle arcs of substantial extent. Even the artificial arcs of the geographic grid may be visible on the Earth's surface, since differences in land use are apparent across borders defined by the geographic grid. These arcs are produced by geological processes: transform faults and subduction zone structures (including Japan!) delineate small circle arcs for kinematic reasons; crater rays, linear streaks of impact ejecta, define great circle arcs. Impact craters are approximately small circles and are best traced in a stereographic projection, which projects small circles on a sphere to circles on a plane. The trace of the terminator on a planet's surface is a great circle.

More unusual are natural spherical polygons defined by intersecting arcs. By far the most familiar are lunes, spherical biangles marked out by the half-great circles of the terminator and the visible, illuminated edge of a planet. The illuminated and visible portion of a planet (or the Moon, hence the name) is the orthographic projection of a lune. I recently found an interesting example of a spherical triangle on a planetary surface: bright rays from the young Mercurian craters Hokusai, Debussy, and Kuiper distinctly define a spherical triangle enclosing the zero point of latitude and longitude on Mercury.

Hokusai-Debussy-Kuiper crater rays define a spherical triangle
Figure 1. Hokusai-Debussy-Kuiper (bright-rayed craters, clockwise from top) crater rays define a spherical triangle on Mercury.