Incidence Relations on a Sphere.
Suppose we have two distinct points A and B on the sphere.
Together with C, the center of the sphere, we have three
points in space, and there are two possibilities. First suppose that
A and B are not antipodal points. Then
A, B, and C do not lie on the same
line in space, and consequently
determine a unique plane. This plane
passes through C, the center of the sphere, and
consequently the intersection of the plane with the sphere is a great
circle containing A and B. Thus A
and B determine a unique great circle.
If A and B are antipodal points, then
A, B, and C lie on the same
line in space. Any plane which contains this line determines a great
circle which must contain A and B. Thus there
are infinitely many great circles containing A and
B if they are antipodal.
To sum up, the first incidence relation for the sphere is:
Now suppose we have two great circles. Each of these is the
intersection of the sphere with a plane through the center. These
planes must intersect in a line in space, which of course interesects
the sphere in two antipodal points. Thus the second incidence relation is:
- If A and B are two points which are not
antipodal, then there is a unique great circle that contains both of
them. If A and B are antipodal, then there are
infinitely many great circles conatining them.
Here is an easy question that you should be able answer. Are there
parallel great circles on a sphere? To answer you will need to decide
what parallel means on the sphere.
- Two distinct great circles meet in exactly two antipodal
Spherical distance and isometries.
If A and B
are two points on the sphere, then the distance between them is the
distance along the great circle connecting them. Since this circle
lies totally in a plane, we can figure this distance using the plane
figure to our left. If the angle ACB is a, and
if a is measured in radians,
then the distance between
A and B is given by
d(A,B) = R a,
where R is the radius of the sphere.
An isometry of the sphere is a mapping of the sphere to
itself which preserves the distance between points. It is easy to see
that a rotation of the sphere around one of the sphere's diameters is
an isometry. It simply rotates the picture to the left into another
one just like it, but in a different plane.
Another example of an isometry is the antipodal map, which
maps a point onto the point on the other side of the sphere. In other
words, given a point A on the sphere, its image under the
antipodal map is the other intersection of the line AC through
the point A and the center of the sphere C, with
In the plane the simplest polygon is the triangle. There are no
interesting polygons with only two sides. This is not true on the
sphere. Any two great circles meet in two antipodal points, and
divide the sphere into four regions each of which has two sides which
are segments of great circles. We will call such a region a
lune, or a biangle.
Why is it called a lune? The name comes from the Latin word
luna, which means moon. Think about the part of the moon
that is seen at any time. That portion has to be in the hemisphere
which is illuminated by the sun, and in the hemisphere that is visible
from the earth. The intersection of two hemispheres is precisely a
Lunes are pretty simple things. However there are two things we
should notice about them.
- The vertices of a lune are antipodal points.
- The two angles of a lune are equal.
Angles on the sphere
What do we mean by an angle on the sphere?
How do we measure them? After all curves on a sphere do not lie
in a plane. However, the lines that are tangent to the two
intersecting curves are both in the plane that is tangent to the
sphere at the point of intersection. We define the angle between two
curves to be the angle between the tangent lines.
We should mention that in these notes all angles will be measured in
With a protractor
and a little practise it is possible to measure spherical angles
pretty accurately. In the case of a lune, the angle between the great
circles at either of the vertices is simply the angle between the
planes that define the great circles, and so it does not matter at
which vertex the measurement is made.
||The next section
discussion of area on the sphere.|
||The previous section discusses the basic properties of spheres.
||Table of Contents.
John C. Polking <firstname.lastname@example.org>
Last modified: Thu Apr 15 09:20:24 Central Daylight Time 1999