Geometry, Spherical

Geometry, Spherical

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Spherical geometry is the three-dimensional study of geometry on the surface of a sphere. It is the spherical equivalent of two-dimensional planar geometry, the study of geometry on the surface of a plane. A real-life approximation of a sphere is the planet Earth—not its interior, but just its surface. (Earth is more accurately called an "oblate spheroid" because it is slightly flattened at the ends of its axis of rotation, the North and South Poles.) The surface of a sphere together with its interior points is usually referred to as the spherical region; however, spherical geometry generally refers only to the surface of a sphere.

As seen in the figure on the next page, a sphere is a set of points in three-dimensional space equidistant from a point O called the center of the sphere. The line segment from point O (at the center of the sphere) to point P (on the surface of the sphere) is called the radius r of the sphere, and the radius r extended straight through the sphere's center with ends on opposite points of the surface is called the diameter d of the sphere (with a value of 2r ; that is, two times the value of the radius). As an example, the line that connects the North Pole and the South Pole on Earth is considered a diameter*.

*The average length of Earth's diameter is d = 6,886 miles.

An infinite line that intersects a sphere at one point only is called a tangent line. An infinite plane can also intersect a sphere at a single point on its surface. When this is the case the plane is also considered tangent to the sphere at that point of intersection. For example, if a basketball were lying on the floor, the floor would represent a tangent plane because it intersects the ball's surface (the sphere) at only one point.

Great and Small Circles

The shortest path between two points on a plane is a straight line. However, on the surface of a sphere there are no straight lines. Instead, the shortest distance between any two points on a sphere is a segment of a circle. To see why this is so, consider that a plane can intersect a sphere at more than one point. Whenever this is the case, the intersection results in a circle. A great circle is defined to be the intersection of a sphere with a plane that passes through the center of the sphere. For example, see the circle containing points C and D in the illustration below. Similar to a straight line on a plane, the shortest path between two points on the surface of a sphere is the arc of a great circle passing through the two points.

The size of the circle of intersection will be largest when the plane passes through the center of the sphere, as is the case for a great circle. If the plane does not contain the center of the sphere, its intersection with the sphere is known as a small circle. For example, see the circle containing points A and B in the illustration below.

As a real-world example, assume a cabbage is a sphere, and is cut exactly in half. The slice goes through the cabbage's center, forming a great circle. However, if the slice is off-centered, then the cabbage is cut into two unequal pieces, having formed a small circle at the cut.

Spherical Triangles

Consider a circle of radius r. A portion of the circle's circumference is referred to as an arc length, and is denoted by the letter s. The first illustration of this article shows a circle of radius r and arc length s. The angle θ is defined as θ = . Rearranging this equation in terms of s yields s = θr. So the arc lengths of a great circle is equal to the radius r of the sphere times the angle subtended by that arc length.

Connecting three nonlinear points on a plane by drawing straight lines using the shortest possible route between the points forms a triangle. By analogy, to connect three points on the surface of a sphere using the shortest possible route, draw three arcs of great circles to create a spherical triangle. A triangle drawn on the surface of a sphere is only a spherical triangle if it has all of the following properties:

1. the three sides are all arcs of great circles;
2. any two sides, summed together, is greater than the third side;
3. the sum of the three interior angles is greater than 180°, and
4. each spherical angle is less than 180°.

In the second illustration of the article, triangle PAB is not a spherical triangle (because side AB is an arc of a small circle), but triangle PCD is a spherical triangle (because side CD is an arc of a great circle).

The left portion of the figure directly below demonstrates how a spherical triangle can be formed by three intersecting great circles with arcs of length (a, b, c ) and vertex angles of (A, B, C ).

The right portion of the figure directly above demonstrates that the angle between two sides of a spherical triangle is defined as the angle between the tangents to the two great circle arcs for vertex angle B.

The above illustration also shows that the arc lengths (a, b, c ) and vertex angles (A, B, C ) of the spherical triangle are related by the following rules for spherical triangles.

Sine Rule:

Cosine Rule: cosa = (cosb cosc ) + (sinb sinc cosA ).

Spherical Geometry in Navigation

Spherical geometry can be used for the practical purpose of navigation by looking at the measurement of position and distance on the surface of Earth. The rotation of Earth defines a coordinate system for the surface of Earth. The two points where the rotational axis meets the surface of Earth are known as the North Pole and the South Pole, and the great circle perpendicular to the rotation axis and lying halfway between the poles is known as the equator. Small circles that lie parallel to the equator are known as parallels. Great circles that pass through the two poles are known as meridians.

Measuring Latitude and Longitude. The two coordinates of latitude and longitude can define any point on the surface of Earth, as is demonstrated within the diagram below. Great circles become very important to navigation because a segment along a great circle provides the shortest distance between two points on a sphere. Therefore, the shortest travel-time can be achieved by traveling along a great circle.

The longitude of a point is measured east or west along the equator, and its value is the angular distance between the local meridian passing through the point and the Greenwich meridian (which passes through the Royal Greenwich Observatory in London, England). Because Earth is rotating, it is possible to express longitude in time units as well as angular units. Earth rotates by 360° in 24 hours. Hence, Earth rotates 15° of longitude in 1 hour, and 1° of longitude in 4 minutes.

The latitude of a point is the angular distance north or south of the equator, measured along the meridian, or line of longitude, passing through the point.

Measuring Nautical Miles. Distance on the surface of Earth is usually measured in nautical miles, where 1 nautical mile (nmi) is defined as the distance subtending an angle of 1 minute of arc at the center of Earth. Since there are 60 minutes of arc in a degree, there are approximately 60 nautical miles in 1 degree of Earth's surface. A speed of 1 nautical mile per hour (nmph) is known as 1 knot and is the unit in which the speed of a boat or an aircraft is usually measured.

A Case Study in Measurement. As noted earlier, Earth is not a perfect sphere, so the actual measurement of position and distance on the surface of Earth is more complicated than described here. But Earth is very nearly a true sphere, and for our purposes this demonstration is still valid.

The terms and concepts that have been developed can be applied to a real-world example. Consider a voyage from Washington, D.C. ("W" in diagram below) to Quito, Ecuador ("Q" in diagram below), which is nearly on the equator at 0° latitude, 77° West longitude. The latitude and longitude of Washington, D.C. is about 37° North latitude, 77° West longitude. If the entire voyage from Washington, D.C. to Quito (on the equator) is along the great circle of longitude 77°, we can use the equation s = θr to find the distance s that the airplane travels from Washington D.C. to Quito.

For this example, θ = 37° (the angle between W and Q ). Knowing that 2 radians equals 360° (one complete revolution around a great circle), we now convert the angle from degrees to radians: (37°) = 0.628 radians. Denoting the radius of Earth as r, we use the "arc-length" equation developed earlier, that is s = θr, to compute the arc length between Washington, D.C. and Quito.

Placing the values of θ = 0.628 radians and r = 3,443 nautical miles (nmi) (the average radius-value for Earth) into the equation yields: s = θr = (0.628 rad × 3,443 nmi) = 2,163 nmi. Therefore, along the arc of the great circle of longitude 77°, from Washington D.C. to Quito, Ecuador, our trip covers a distance of 2,163 nmi.

see also Triangles; Trigonometry.

William Arthur Atkins with

Philip Edward Koth

Bibliography

Abbott, P. Geometry (Teach Yourself Series). London, U.K.: Hodder and Stoughton, 1982.

Henderson, Kenneth B., Robert E. Pingry, and George A. Robinson. Modern Geometry: Its Structure and Function. New York: Webster Division, McGraw-Hill Book Company, 1962.

Ringenberg, Lawrence A., and Richard S. Presser. Geometry. New York: Benziger, Inc. with John Wiley & Sons, 1971.

Selby, Peter H. Geometry & Trigonometry for Calculus. New York: John Wiley & Sons, 1975.

Ulrich, James F., Fred F. Czarnec, and Dorothy L. Guilbault, Geometry. New York: Harcourt Brace, 1978.

Internet Resources

The Geometry of the Sphere. Mathematics Department at Rice University, Houston, Texas. <http://math.rice.edu/~pcmi/sphere/>.

Spherical Geometry. Mathematics Department at the University of North Carolina at Charlotte. <http://www.math.uncc.edu/~droyster/math3181/notes/hyprgeom/node5.html#SECTION00500000000000000000>.