An area of mathematics concerned with geometric figures on a
sphere, in the same way as planimetry is concerned with geometric figures in a plane.
Every plane that intersects a sphere gives a certain circle
as section; if the intersecting plane passes through the centre
of the sphere, then a so-called
great circle
is obtained as the intersection. A unique great
circle can be drawn through any two points
and
on the sphere
(Fig. a), except when they are diametrically opposite.
Figure: s086680a
Figure: s086680b
Figure: s086680c
Figure: s086680d
Figure: s086680e
Figure: s086680f
Figure: s086680g
Figure: s086680h
The great circles of a sphere are its geodesics (cf.
Geodesic line),
and for this reason their role in spherical geometry is the same as the role
of straight lines in planimetry. However, whereas any segment of a straight line is the
shortest curve between its ends, an arc of a great circle on a sphere
is only the shortest curve when it is shorter than the complementary
arc. Spherical geometry differs from planimetry in many other senses; for example,
there are no parallel geodesic lines: two great circles
always intersect, and, moreover, they intersect in two points.
The length of a segment
on a sphere, i.e. the length of the arc
(Fig. a) of a great circle, is measured by its corresponding
central angle
.
The angle
(Fig. b) formed on the sphere by the arcs of two great circles is measured by the angle
between the tangents of the corresponding arcs at the point of intersection
or by the dihedral angle formed by the planes
and
.
When two great circles intersect on a sphere, four
spherical digons,
or
lunes,
are formed
(Fig. c). A lune is defined by specifying
its angle. The area of a lune is determined by the formula
,
where
is the radius of the sphere and
is the angle of the lune expressed in radians.
Three great circles that do not intersect in
one pair of diametrically-opposite points form eight
spherical triangles
on the sphere
(Fig. d); if the elements (angles and
sides) of one of these is known, it is easy to determine
the elements of all the others. It is therefore usual to
consider only triangles whose sides and angles are less than
(such triangles are called
Euler triangles).
The sides
of a spherical triangle are measured by the planar angles of the trihedral angle
(Fig. e); the angles
of the triangle are measured by the dihedral angles
of that same trihedral angle. The properties of spherical triangles
vary greatly from the properties of triangles on a plane (rectilinear
triangles). Thus, a fourth case of equality for triangles on a sphere
can be added to the three already known for rectilinear triangles: Two
triangles are equal if their corresponding angles are equal
(on a sphere, similar triangles do not exist).
Triangles that can be matched up by a movement around the sphere are said to be
directly congruent.
Such triangles have equal elements and the same orientation.
Triangles that have equal elements and a different orientation are called
oppositely symmetric;
the triangles
and
in
Fig. fform an example.
In every spherical (Euler) triangle, each side is less than the sum of, and more
than the difference between, the other two; the sum of all the sides is always less than
.
The sum of the angles of a spherical triangle is always less than
and more than
.
The difference
,
where
is the sum of the angles of a spherical triangle, is called the
spherical excess.
The area of a spherical triangle is defined by the formula
,
where
is the radius of the sphere. For the relationship between
the angles and sides of a spherical triangle, see
Spherical trigonometry.
The position of each point on a sphere is completely defined by
the specification of two numbers; these two numbers (coordinates) can be
defined in the following way
(Fig. g). A great circle
(the
equator)
is fixed, along with one of the two points of intersection of the diameter
of the sphere perpendicular to the plane of the
equator and the surface of the sphere, for example
(the
pole),
as well as one of the great semi-circles
that emanate from the pole (the
zero meridian).
The great semi-circles of the sphere that emanate from
are called
meridians,
while its small circles, which are parallel to the equator, are called
parallels.
One of the coordinates of the point
on the sphere is the angle
— the
polar distance
— while the other is the angle
between the zero meridian and the meridian which passes through the point
—
the
longitude,
which is counted anti-clockwise.
The length
of an arc
(Fig. h) of the curve
,
is calculated according to the formula