Arcs and Triangles Help
Now we know how latitude and longitude are defined on the surface of the earth, and how to find the differences in latitude and longitude between points along north–south and east–west paths. Let’s look at the problem of finding the distance between any two points on the earth, as measured along the arc of a great circle between them.
There are two great-circle arcs that connect any two points on a sphere. One of the arcs goes halfway around the sphere or further, and the other goes halfway around or less. The union of these two arcs forms a complete great circle. The shorter of the two arcs represents the most direct possible route, over the surface of the earth, between the two points.
Let’s agree that when we reference the distance between two points on a sphere, we mean to say the distance as measured along the shorter of the two great-circle arcs connecting them. This makes sense. Consider a practical example. You can get from New York to Los Angeles more easily by flying west across North America than by flying east over the Atlantic, Africa, the Indian Ocean, Australia, and the Pacific Ocean.
Spherical Triangles and Polygons
A spherical triangle is defined by three vertex points that all lie on the surface of a sphere. Imagine a triangle on a sphere whose vertex points are Q, R, and S. Let P be the center of the sphere. The spherical triangle, denoted Δ sph QRS, has sides q, r, and s opposite the vertex points Q, R, and S respectively, as shown in Fig. 11-4. (Here, the uppercase Greek delta means “triangle” as in geometry, not “the difference in” as earlier in this chapter!) Each side of the spherical triangle is a great-circle arc spanning less than 360°. That means that each side must go less than once around the sphere. It is “illegal” to have a spherical triangle with any side that goes all the way around the sphere, or further.
For any spherical triangle, there are three ordinary plane triangles defined by the vertices of the spherical triangle and the center of the sphere. In Fig. 11-4, these ordinary triangles are Δ PQR , Δ PQS , and Δ PRS . All three of these triangles define planes in 3D space; call them plane PQR , plane PQS , and plane PRS . Note these three facts:
- Plane PQR contains arc s
- Plane PQS contains arc r
- Plane PRS contains arc q
These concepts and facts are important in defining the interior spherical angles of the spherical triangle .
Let’s look at the general case, for polygons on the surface of a sphere having three sides or more. A spherical polygon, also called a spherical n-gon, is defined by n vertex points that all lie on the surface of a sphere, where n is a whole number larger than or equal to 3. Each side of a spherical n-gon is a great-circle arc spanning less than 360°. That means that each side must go less than once around the sphere. It is “illegal” to have a spherical polygon with any side that goes all the way around the sphere, or further.
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