Arcs and Triangles Help (page 3)
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.
The sides of any spherical triangle are curves, not straight lines. The interior angles of a spherical triangle are called spherical angles. A spherical angle can be symbolized ∠ sph . There are two ways to define this concept.
Definition 1. Consider the three planes defined by the vertices of the spherical triangle and the center of the sphere. In Fig. 11-4, these are plane PQR, plane PQS, and plane PRS.
The angles between the arcs are defined like this:
- The angle between planes PQR and PQS, which intersect in line PQ, defines the angle between arcs r and s, whose vertex is at point Q
- The angle between planes PQR and PRS, which intersect in line PR, defines the angle between arcs q and s, whose vertex is at point R
- The angle between planes PQS and PRS, which intersect in line PS, defines the angle between arcs q and r, whose vertex is at point S
How do we express the measure of an angle between two planes, also known as a dihedral angle? It’s easy to intuit, but hard to explain. Figure 11-5 illustrates the concept. Suppose two planes X and Y intersect in a common line L. Consider line M in plane X and line N in plane Y, such that line M is perpendicular to line L, line N is perpendicular to line L, and lines M and N both meet somewhere on line L. The angle between the intersecting planes X and Y can be represented in two ways. The first angle, whose measure is denoted by u, is the smaller angle between lines M and N. The second angle, whose measure is denoted by v, is the larger angle between lines M and N. The smaller angle is acute, and the larger angle is obtuse. When we talk about the angle at the vertex of a spherical triangle, we must pay attention to whether it is acute or obtuse!
Definition 2 . This definition is less rigorous than the first, but it is easier for some people to envision. Let’s use a real-world example. On the surface of the earth, suppose two “shortwave” radio signals arrive from two different directions after having traveled partway around the planet along great-circle arcs. If the receiving station uses a directional antenna to check the azimuth bearings (compass directions) of the signals, the curvature of the earth is not considered. The observation is made locally, over a region small enough so that the earth’s surface can be considered flat within it. The angle between two great-circle arcs on any sphere that intersect at a point Q can be defined similarly. It is the angle as measured within a circle on the sphere surrounding point Q , such that the circle is so small with respect to the sphere that the circle is essentially a flat disk (Fig. 11-6).
Then the great-circle arcs seem to be straight rays running off to infinity, and the angle between them can be expressed as if they lie in the geometric plane tangent to the surface of the sphere at point Q .
Angular Sides & Equilateral Spherical Triangle Principles
The sides q , r , and s of the spherical triangle of Fig. 11-4 are often defined in terms of their arc angles (∠ SPR , ∠ QPS , and ∠ RPQ , respectively), rather than in terms of their actual lengths in linear units. When this is done, it is customary to express the arc angles in radians.
If we know the radius of a sphere (call it r sph ), then the length of an arc on the sphere, in the same linear units as we use to measure the radius of the sphere, is equal to the angular measure of the arc (in radians) multiplied by r sph . Suppose we let | q |, | r |, and | s | represent the lengths of the arcs q , r , and s of Δ sph QRS in linear units, while their extents in angular radians are denoted q , r , and s . Then the following formulas hold:
In the case of the earth, the linear lengths (in kilometers) of the sides of the spherical triangle Δ sph QRS are therefore:
Equilateral Spherical Triangle Principles
In plane geometry, if a triangle has sides that are all of the same length, then the interior angles are all of the same measure. The converse also holds true: If the interior angles of a triangle are all of equal measure, then the sides all have the same length.
There is an analogous principle for equilateral triangles on a sphere. If a spherical triangle has sides all of the same angular length, then the interior spherical angles are all of equal measure. The converse is also true: If the interior spherical angles of a spherical triangle all have the same measure, then the angular lengths of the sides are all the same. These principles are important to the solving of the two problems that follow.
The Case Of The Expanding Triangle
Imagine what happens to an equilateral spherical triangle that starts out tiny and grows larger. (An equilateral spherical triangle has sides of equal angular length and interior spherical angles of equal measure.) An equilateral spherical triangle on the earth that measures 1 arc second on a side is almost exactly the same as a plane equilateral triangle whose sides are 30.88 meters long. The sum of the interior spherical angles, if we measure them with a surveyor’s apparatus, appears to be 180°, and each interior spherical angle appears to be an ordinary angle that measures 60°. The interior area and the perimeter can be calculated using the formulas used for triangles in a plane.
As the equilateral spherical triangle grows, the measure of each interior spherical angle increases. When each side has a length that is ¼ of a great circle (the angular length of each side is π /2 rad), then each interior spherical angle measures 90°, and the sum of the measures of the interior spherical angles is three times this, or 270°. An example is shown in Fig. 11-7B. As the spherical triangle expands further, it eventually attains a perimeter equal to the circumference of the earth. Then each side has an angular length of 2 π /3 rad. The spherical triangle has become a great circle. Its interior area has grown to half the surface area of the planet. The formulas for the perimeter and interior area of a plane triangle do not work for a spherical triangle that is considerable with respect to the size of the globe.
Now think about what happens if the equilateral spherical triangle continues to “grow” beyond the size at which it girdles the earth. The lengths of the sides get shorter, not longer, even though the measures of the interior spherical angles, and the interior area of the spherical triangle, keep increasing. Ultimately, our equilateral spherical triangle becomes so “large” that the three vertices are close together again, perhaps only 1 arc second apart. We have what looks like a triangle similar to the one we started out with—but wait! There are differences. The perimeter is the same, but the interior area is almost that of the whole earth. The inside of this triangle looks like the outside, and the outside looks like the inside. The interior spherical angles are not close to 60°, as they were in the beginning, but instead are close to 300°. They must be measured “the long way around”. The sum of their measures is approximately 900°.
This is a bizarre sort of triangle, but in theory, it’s entirely “legal.” In fact, we can keep on going past a complete circle, letting the interior area and the measures of the interior angles keep growing while the perimeter cycles between zero and the circumference of the earth, over and over. Not many people can envision such a “triangle” after six or eight trips around the world. It’s definable, but it’s also incomprehensible.
The Long Way Around
The foregoing example is not the only instance of “weird spherical triangles” that can be conjured up. Imagine a spherical triangle whose vertices are close to each other, but whose sides go the long way around (Fig. 11-8).
As long as we are going to be extreme, why stop now? Suppose we free ourselves of the constraint that each side of a spherical n-gon must make less than one circumnavigation of the sphere. Any spherical polygon can then have sides that go more than once around, maybe hundreds of times, maybe millions of times. It’s not easy to envision what constitutes the interior of such a monstrosity; we might think of it as a globe wrapped up like a mummy in layer upon layer of infinitely thin tape. And what about the exterior? Perhaps we can think of the mummy-globe again, but this time, wrapped up in infinitely thin tape made of anti-matter.
Mind games like this can be fun, but they reduce to nonsense if taken too seriously. It’s a good idea to keep this sort of thing under control, if only for the sake of our sanity. Therefore, when we talk about a spherical polygon, we should insist that its size be limited as follows:
- The perimeter cannot be greater than the circumference of the sphere
- The interior area cannot be greater than half the surface area of the sphere
Any object that violates either of these two criteria should be regarded as “illegal” or “non-standard” unless we are dealing with some sort of exceptional case.
Spherical Law Of Sines
For any spherical triangle, there is a relationship among the angular lengths (in radians) of the sides and the measures of the interior spherical angles. Let ∠ sph QRS be a spherical triangle whose vertices are points Q, R, and S . Let the lengths of the sides opposite each vertex point, expressed in radians, be q, r, and s respectively, as shown in Fig. 11-9. Let the interior spherical angles ∠ sph RQS , ∠ sph SRQ, and ∠ sph QRS be denoted ψ q , ψ r , and ψ s respectively. (The symbol ψ is an italicized, lowercase Greek letter psi; we use this instead of θ to indicate spherical angles.) Then:
(sin q )/(sin ψ q ) = (sin r )/(sin ψ r ) = (sin s )/(sin ψ s )
That is to say, the sines of the angular lengths of the sides of any spherical triangle are in a constant ratio relative to the sines of the spherical angles opposite those sides. This rule is known as the spherical law of sines. It bears some resemblance to the law of sines for ordinary triangles in a flat plane.
Spherical Law Of Cosines
The spherical law of cosines is another useful rule for dealing with spherical triangles. Suppose a spherical triangle is defined as above and in Fig. 11-9. Suppose you know the angular lengths of two of the sides, say q and r , and the measure of the spherical angle ψ s between them. Then the cosine of the angular length of the third side, s , can be found using the following formula:
cos s = cos q cos r + sin q sin r cos ψ s
This formula doesn’t look much like the law of cosines for ordinary triangles in a flat plane.
Arcs and Triangles Practice Problems
A great-circle arc on the earth has a measure of 1.500 rad. What is the length of this arc in kilometers?
Multiply 6371, the radius of the earth in kilometers, by 1.500, obtaining 9556.5 kilometers. Round this off to 9557 kilometers, because the input data is accurate to four significant figures.
Describe and draw an example of a spherical triangle on the surface of the earth in which two interior spherical angles are right angles. Then describe and draw an example of a spherical triangle on the surface of the earth in which all three interior spherical angles are right angles.
To solve the first part of the problem, consider the spherical triangle Δ sph QRS such that points Q and R lie on the equator, and point S lies at the north geographic pole (Fig. 11-7A). The two interior spherical angles ∠ sph SQR and ∠ sph SRQ are right angles, because sides SQ and SR of Δ sph QRS lie along meridians, while side QR lies along the equator. (Remember that all of the meridian arcs intersect the equator at right angles.)
To solve the second part of the problem, we construct Δ sph QRS such that points Q and R lie along the equator and are separated by 90° of longitude (Fig. 11-7B). In this scenario, the measure of ∠ sph QSR, whose vertex is at the north pole, is 90°. We already know that the measures of angles ∠ sph SQR and ∠ sph SRQ are 90°. So all three of the interior spherical angles of ∠ sph QRS are right angles.
What are the measures of the interior spherical angles, in degrees, of an equilateral spherical triangle whose sides each have an angular span of 1.00000 rad? Express the answer to the nearest hundredth of a degree.
Let’s call the spherical triangle Δ sph QRS, with vertices Q, R, and S, and sides q = r = s = 1.00000 rad. Then:
cos q = 0.540302 cos r = 0.540302 cos s = 0.540302 sin q = 0.841471 sin r = 0.841471
Plug these values into the formula for the law of cosines to solve for cos ψ s , where ψ s is the measure of the angle opposite side s . It goes like this:
This means that ψ s = arccos 0.350777 = 69.4652°. Rounding to the nearest hundredth of a degree gives us 69.47°. Because the triangle is equilateral, we know that all three interior spherical angles ψ have the same measure: approximately 69.47°.
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