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

1. The spherical surface is used as the basis for spherical geometry, while the plane is used as the basis for plane geometry.
2. 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.

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4. 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.
5. 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.
6. 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.

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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).

 Figure 2-2. Four angles are formed when two planes intersect. The values of the dihedral angles can be represented by: DFd, dFC, CFc, and cFD. Figure 2-3. A view of a dihedral angle, formed when two planes intersect.

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.

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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.

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

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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
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Figure 2-6.

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

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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.

• Second, each great circle can be considered to be lying in a plane. Therefore, two different great circles can be considered as two different planes intersecting each other at a dihedral angle. This dihedral angle is also the spherical angle between the two great circles.
• Third, we can consider the point of crossing or vertex of two great circles as a pole. For any point P. chosen as a pole, we can draw another great circle, which is called the equator (GABH in Figure 2-7) for that point on that sphere. In Figure 2-7, we have the arcs of two great circles, PAP' and PBP' both intersecting at P and P'. The spherical angle at the intersection P (or P') is equal to the plane angle AOB which lies in the plane of the equatorial circle GABH. Every equatorial circle can be divided into 360 degrees. Therefore, the planes of the two great circles cut off the arc AB, which gives the measure of the spherical angle in degrees, on this equatorial circle. PAP' and PBP', together, form a lune which is the part of the surface of a sphere that is bounded by two semicircles of two great circles. For a spherical angle of 180°, the lune becomes half of a sphere or a hemisphere.

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Figure 2-7.

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

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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

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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.

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Figure 2-8.

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

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An arc polygon will be formed when a part of the surface of a sphere is enclosed by three or more arcs, and at least one of these three arcs is part of a small circle.
Spherical Triangles

Three mutually perpendicular planes passed through the center of a sphere, cut the surface of the sphere into eight spherical triangles, as shown in Figure 2-9. Now consider only one of the eight spherical triangles;

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Figure 2-9

Eight equal spherical triangles on a sphere.

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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.

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Figure 2-10.

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

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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.

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Figure 2-11.

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

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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.

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