Geometry of the Sphere - Spatial Mathematics: Theory and Practice through Mapping - page 3

Plane A
Small circle
Plane B
O
Great circle
Figure 1.1 A great circle on a sphere is formed by a plane (such as Plane B) cutting
the sphere through the center, O, of the sphere. A small circle on a sphere is formed
by a plane (such as Plane A) cutting into the sphere at a different location from the
center, O, of the sphere.
between two locations (40N, 90W) and (40N, 0) and also one along a
great circle; the great circle route is the shorter route. What is a geo-
desic in the plane? The software used in the top image measured the
great circle route (top line) as 4562 miles and it measured the small
circle route (along the 40th parallel, bottom line) as 4786 miles.
• In the plane, the shortest distance between two points is mea-
sured along a line segment and is unique.
• On the sphere, the shortest distance between two points is mea-
sured along an arc of a great circle.
− If the two points are not at opposite ends of a diameter of the
sphere, then the shortest distance is unique.
− If the two points are at opposite ends of a diameter of the
sphere, then the shortest distance is not unique: One may tra-
verse either half of a great circle. Diametrically opposed points
are called antipodal points: Anti + pedes, opposite + feet, as in
drilling through the center of the Earth to come out on the other
side. The Earth's diameter separates pairs of antipodal points.
• To reference measurement on the Earth-sphere in a systematic man-
ner, we introduce a coordinate system ( Figure 1.3 ) . A coordinate sys-
tem requires a point of origin, a set of axis or reference lines, and a
system of addressing any point within the system.
• One set of reference lines is produced using a great circle in a
unique position (bisecting the distance between the poles): The
Equator. A set of evenly spaced planes, parallel to the equato-
rial plane, produces a set of evenly spaced small circles on the
Earth's surface, commonly called parallels. They are called paral-
lels because it is the planes that are parallel to each other (e.g.,
Planes A and B in Figure 1.1).