Chapter 2

Some Geometry

This chapter explores the geometry of shapes on a sphere. This information is of course available in any book on spherical geometry (in much greater detail), but is not properly discussed in books on meteorology. I am including some of the principles so that you won't have to go to another text for this information unless you so desire. The terms used in this chapter are helpful in understanding the configurations shown in the later chapters, since we are looking at, and measuring angles and circular arcs on a flat map, while we are actually describing what is happening on the surface of a sphere. Nevertheless, you will be able to enjoy and understand the charts in the last sections of this book even if you decide to skip this chapter. You can come back to this chapter later on if you feel the need for a deeper perspective.

Spherical geometry may be a fringe area in meteorology, but it is used extensively in sailing, astronomy, surveying, and in other areas of science. Since lows and highs also sail about on the surface of the Earth, we will take a closer look at some of the theorems and definitions of this branch of mathematics. And since we live continuously in a three-dimensional world, many of the properties and relationships of lines and planes in space are quite familiar.

**Some Differences Between Plane and Spherical Geometry**

- The spherical surface is used as the basis for spherical geometry, while the plane is used as the basis for plane geometry.
- Two points do not necessarily determine a single line. For example, the North and South poles lie on an infinite number of great circles, as can be seen on any three dimensional globe of the Earth. The same is true for the end points of any diameter on a sphere.
- A line on the surface of a sphere can not have an infinite length as in a plane, but has a maximum finite length. If the radius of the sphere is considered as being equal to 1, then the greatest possible distance on the surface between any two points is half of the circumference of 2 PI r and is equal to PI when r = 1.
- Given three points on a line around a sphere, we can not say that only one of the three points is in the middle between the other two points. Each point, in turn, can be considered as lying between the other two.
- A perpendicular to a line on a sphere from another point on the sphere, always exists (as in a plane), but is not necessarily the only perpendicular. As an example, any line joining a pole to any point on the equator is perpendicular to the equator.

**General Definitions**

A spherical surface is a curved surface on which every point is at an
equal distance from the center of the sphere, and a line joining any point
on the surface with the center is a **radius**.

When a plane intersects or cuts a sphere, the intersection at the surface of the sphere is always a circle. When the plane intersecting the sphere passes through the center of the sphere, the intersection at the surface is called a great circle; if the plane does not pass through the center, its intersection with the surface is called a small circle. For example, the equator and the longitudinal lines (or meridians) on the Earth are great circles, while the parallels of latitude are small circles, except for the equator.

An arc of any circle is measured in degrees, minutes, and seconds by the angle (alpha), subtended at its center. See Figure 2-1.

Figure 2-1.
The angle a (alpha) cuts off an arc of greater length on the larger circle (as compared with the smaller circle), but the angle is the same for both arcs. |

The angle formed in space when two planes intersect is called a **dihedral
angle**, as shown in Figure 2-2. The line of intersection (EF) between
the two planes is called the **edge**, and each of the planes is called
a **face**. To find the magnitude of a dihedral angle (see Figure 2-3),
we pass a plane which is perpendicular to the edge. This plane will intersect
the faces of the dihedral angle in straight lines and will thereby form
a plane angle (DFd) in the plane of intersection. The **magnitude of the
dihedral angle** is equal to the plane angle, (DFd).

A line (or plane) that is perpendicular to the radius of a circle (or sphere) at the outer surface, is called a tangent line (or a plane in three dimensions). See Figure 2-4.

Figure 2-4.
The line XNY is tangent to circle NESW with a radius ON. |

Angles on a Sphere

Figure 2-5 shows three great circles intersecting to form the **spherical
triangle** ABC. This surface triangle forms a spherical pyramid OABC
when the vertexes are joined to the center O by radial lines. Side AB of
the spherical triangle is measured by the plane angle BOA; side AC is measured
by the plane angle COA; and side BC is measured by the plane angle BOC.
The sides are measured as angular degrees. The *sum of the sides *of
a spherical triangle is less than 360 degrees since the angles are measured
around the vertex at O.

Figure 2-5. A spherical triangle on the surface of a sphere is part of a spherical pyramid formed by OABC.

The angle at A is measured by the dihedral angle formed by the intersecting
faces BOA and COA, the angle at B is measured by the dihedral angle formed
by the intersecting faces AOB and COB, and the angle at C is measured by
the intersecting faces of AOC and BOC. The sum of the **dihedral angles
**of a spherical triangle is greater than 180 degrees and less than 540
degrees.

How do we determine the angle between two great circles (or parts of great circles) of a sphere? There are three general approaches:

- First, we draw a tangent line to each great circle at the points where
they cross each other. The plane angle formed by these two tangent lines
is called the
**spherical angle.**In Figure 2-6, the line CD is drawn tangent

Figure 2-6.

Two tangent lines to an angle of a spherical pyramid form a tangent plane CDE.

to the arc BC while CE is the line tangent to the arc CA. The plane angle formed by points DCE is the magnitude of the spherical angle BCA.

Figure 2-7.

Two great circle arcs, PAP' and PBP', meeting at P and P'.

All great circles that go through a point, meet again at a point known
as the **antipodal point. **The crossings are also called **polar points
**or **vertexes**. These antipodal points are opposite, since there
is one at each end of the diameter of the sphere. Through two antipodal
points an infinite number of great circles can be drawn.

Since we are also concerned with the patterns of small circles in *Singer's
Lock, *it is necessary to define the angle which is formed when two
small circles or when a small and great circle cross. These crossings can't
be called spherical angles according to the definition above. Therefore,
since I have not come across an official name for this type of angle, I

will call this crossing of arcs an **arc angle, **to distinguish
it from spherical angles.

**Distances Between Points on the Surface of a Sphere**

The shortest path between two points on the surface of a sphere is an arc of a great circle. We must also consider the path of a small circle,which is not the shortest distance between two points, but is rather a path of constant curvature. There are an infinite number of small circle paths that can be taken between any two points. There can also be an infinite number of irregular or zig-zag paths, but they are not significant for our purposes. Lastly, the shortest distance between two points on a sphere in three-dimensional space (if we burrow beneath the surface of the sphere) is a chord, which is just a straight line connecting the two points.

**Definition of Polygons on a Sphere**

That part of the surface of a sphere enclosed by three or more arcs
of great circles is called a **spherical polygon. **See **Figure 2-8.**
The enclosing arcs are the **sides **of the polygon, and the vertexes
of the spherical angles are the vertexes of the polygon. A spherical polygon
with three sides is called a **spherical triangle**. The angle formed
at the center of the sphere by these sides is called a **trIhedral angle.
**A **polyhedral angle **is a general term for a similar angle formed
by three or more sides coming into the center of the sphere. The sides
of a spherical polygon are measured in degrees in the same way as a spherical
angle, because the sides of a spherical polygon are always arcs of great
circles--all great circles on the same sphere have the same size and length.

Figure 2-8.

A spherical rectangle, on the surface of a sphere, is shown extended into space to help visualize its shape.

An

Figure 2-9

Eight equal spherical triangles on a sphere.

you will see that each corner of the triangle is 90°. Since there
are three corners to a spherical triangle, the total of all the angles
is three times 90° or 270°. This is an example of the general rule
that the sum of the angles of a spherical triangle is always something
greater than 180°,but always less than 540°. Also, the amount by
which the sum of the angles of a spherical triangle exceeds 180° is
called the **spherical excess** of the triangle. The area of a spherical
triangle is given by

(2-1)> **A =pi** times **r squared**
times **E** divided by** 180**

where A is the area, E is the spherical excess measured in degrees,
and r is the radius of the sphere. The larger the area of the spherical
triangle,the greater the amount of spherical excess, as can be seen in
Figure 2-10.

Figure 2-10.

Spherical triangles of increasing size on the surface of a sphere.

With an approximate diameter of 8,000 miles, we find that the Earth'surface
is about 200 million square miles. The **icosahedral pattern, **shown
in Figure 2-11, will give twenty **equilateral triangles, **which is
the largest number that can be drawn on a spherical surface like the Earth.
Each one of these twenty equilateral spherical triangles can be subdivided
equally into six right triangles by bisectors, which are perpendicular
to the side opposite each angle. See Figure 2-11.

Figure 2-11.

An icosahedral pattern with the 20 equilateral triangles, each sub-divided into 6 right triangles.

The angles formed by the triangle in Figure 2-12 (which is one of the 120 triangles generated by an icosahedral pattern) has exactly 6 degrees of spherical excess, since the total of the three angles is 186°. The area of one of these triangles on the Earth, is approximately, 1,666,666 square miles.

Figure 2-12. The angles formed by one of the 120 triangles generated by an icosahedral pattern.