Spherical geometry studies figures drawn on the surface of a sphere, such as Earth. It matters because the shortest paths for airplanes, ships, and satellites are usually curved on a flat map but straight in the geometry of the globe. Instead of ordinary lines, spherical geometry uses great circles, which are circles made by slicing a sphere through its center.
This changes familiar ideas from plane geometry, especially the behavior of triangles.
Key Facts
- A great circle is the largest possible circle on a sphere and has the same center as the sphere.
- On a sphere, the shortest path between two points is usually an arc of a great circle.
- A spherical triangle is formed by three great-circle arcs.
- For a spherical triangle, angle sum = 180 degrees + spherical excess.
- Spherical excess E = A + B + C - 180 degrees, where A, B, and C are the triangle angles.
- For a sphere of radius R, area of a spherical triangle = E R^2 when E is measured in radians.
Vocabulary
- Spherical geometry
- Spherical geometry is the study of points, lines, angles, and shapes on the surface of a sphere.
- Great circle
- A great circle is a circle on a sphere whose plane passes through the center of the sphere.
- Spherical triangle
- A spherical triangle is a three-sided figure on a sphere whose sides are arcs of great circles.
- Spherical excess
- Spherical excess is the amount by which the angle sum of a spherical triangle is greater than 180 degrees.
- Geodesic
- A geodesic is the shortest path between nearby points on a curved surface, such as a great-circle arc on a sphere.
Common Mistakes to Avoid
- Treating latitude lines as great circles is wrong because only the equator is a great circle among lines of latitude. Other latitude lines do not pass through Earth's center and are not shortest paths.
- Assuming every spherical triangle has angles that add to 180 degrees is wrong because curvature makes the sum greater than 180 degrees. The extra amount is called spherical excess.
- Using flat map distances as shortest Earth paths is wrong because map projections distort distances and angles. Great-circle routes often look curved on a flat map but are shortest on the globe.
- Using degrees directly in the area formula area = E R^2 is wrong because E must be measured in radians. Convert degrees to radians before calculating area.
Practice Questions
- 1 A spherical triangle has angles 80 degrees, 70 degrees, and 60 degrees. Find its spherical excess in degrees.
- 2 A spherical triangle on a sphere of radius 10 cm has angles 90 degrees, 90 degrees, and 90 degrees. Convert the spherical excess to radians and find the triangle's area.
- 3 Explain why an airplane route between two distant cities may appear curved on a flat map even though it follows the shortest path on Earth.