II. SPHERICAL POLYGONS 191. Spherical Polygon. A portion of a sphere bounded by three or more arcs of great circles is called a spherical polygon. The difference between the general nature of a spherical polygon and that of a plane polygon is that the former lies on a spherical surface and has arcs of great circles as its sides, while the latter lies on a plane surface and has segments of straight lines as its sides. The terms sides, angles, vertices, diagonal, convex, concave, and congruent are used as with plane polygons. 192. Spherical Triangle. A spherical polygon of three sides is called a spherical triangle. A spherical triangle may be right, obtuse, or acute, and may also be equilateral, isosceles, or scalene. The terms spheric polygon and spheric triangle are also used. 193. Relation of Polygons to Polyhedral Angles. The planes of the sides of a spherical polygon form a polyhedral angle whose vertex is the center of the sphere, whose face angles are measured by the sides of the polygon, and whose dihedral angles have the same numerical measure as the angles of the AB polygon. Thus, the planes of the sides of the polygon ABCD here shown form the polyhedral O-ABCD. The face & BOA, COB, · ,... are measured by the sides BA, CB,... of the polygon. The dihedral angle whose edge is OA has the same measure as the spherical ▲ BAD, and so on. Hence from any property of polyhedral angles, we may infer an analogous property of spherical polygons, and conversely. Since we have considered only convex polyhedral angles in the preceding work, we shall consider only convex spherical polygons. Because of the relation between polyhedral angles and spherical polygons, we shall first consider the former. Proposition 8. Two Face Angles 194. Theorem. The sum of any two face angles of a trihedral angle is greater than the third face angle. X A Y Given the trihedral V-XYZ with face XVZ> face XVY or face YVZ. Prove that ZXVY+ZYVZ> <XVZ. Proof. In the plane XVZ let XVW = XVY, and through any point D of VW draw a line cutting VX in A and VZ in C. On VY take VB=VD. A, B, C determine a plane. $ 31, 1 Then Since AV AV, VB=VD, and ZAVB=ZAVD, then AAVB is congruent to ▲AVD (§ 7, 3), and AB=AD(§ 7, 2). Proposition 9. Sum of Face Angles 195. Theorem. The sum of the face angles of a polyhedral angle is less than four right angles. Given the polyhedral V with the face a, b, c,.... Prove that a+b+c+ <4 rt. s. Proof. Let a plane cut all the edges of V, thus forming the polygon ABC, and let P be any point within ABC... Drawing PA, PB, PC,..., there are as many ▲(PAB, PBC,) as there are faces (VAB, VBC, · · ·). Hence the sum of the s of the ▲ with vertex V equals the sum of the s of the A with vertex P. $ 8, 4; Ax. 1 Now LEAV+ZBAV><BAE, ZVBA+ZCBV><CBA, .... $194 with Hence the sum of the at the bases of the vertex V is greater than the sum of the s at the bases of the A with vertex P. or ..a+b+c+···<<APB+<BPC+<CPD+···, Ax. 8 Ax.8 a+b+c+ · < 4rt. s. ... $ 3, 2 Proposition 10. Side of a Triangle 196. Theorem. Any side of a spherical triangle is less than the sum of the other two sides. Proof. Let O-ABC be the corresponding trihedral. Then ZCOA</BOA+ZCOB. .. CA<AB+BC. Exercises. Spherical Triangles $ 194 § 193 1. Explain how you could proceed to bisect a given great-circle arc. 2. Explain how you could determine the arc that bisects a given spherical angle. 3. Draw a sphere and upon it draw freehand a spherical AABC. With A, B, C as poles draw freehand three great circles and show that these circles divide the sphere into eight triangles. Assume that the center, diameter, and radius are given. 4. Make a drawing of a sphere and on the sphere show an equilateral spherical triangle, each side of which is 90°. Then draw a triangle with the three vertices as poles. Proposition 11. Sum of Sides 197. Theorem. The sum of the sides of a spherical polygon is less than 360°. D Given the spherical polygon ABCD. Prove that AB+BC+ CD +DA <360°. Proof. Let O-ABCD be the corresponding polyhedral . Then BOA+ZCOB+ZDOC+≤DOA<360°. $ 195 § 193 198. Polar Triangle. The triangle formed by the arcs of great circles of which the vertices of a given triangle are poles is called the polar triangle of the given triangle. A Thus, if A is the pole of the great circle of which a' is an arc, B is the pole of the great circle of which b' is an arc, and C is the pole of the great B B α a' circle of which c' is an arc, then ▲A'B'C' is the polar triangle of AABC. If, with A, B, C as poles, entire great circles are drawn, these circles divide the sphere into eight spherical triangles. Of these eight triangles, that one is the polar of ▲ABC whose vertex A', corresponding to A, lies on the same side of BC as the vertex A; and similarly for the other vertices. While it is desirable to have a spherical blackboard on which the student can draw figures, any small ball will serve the purpose. |