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Extra resources for Algebraic Geometry

Example text

1) is called the polar triangle of T(x, y, z). Let a', b', c' be the lengths of the sides of T' and let a', (3', "I' be the opposite angles. By Lemma 1, we have a' = 7f - a, b' = 7f - (3, c' = 7f - "I. As T(x, y, z) is the polar triangle of T', we have a' = 7f - a, (3' = 7f - b, "I' = 7f - c. "I) - cos(7f ). sm(7f - a sm(7f - (3) o Area of Spherical Triangles A lune of 8 2 is defined to be the intersection of two distinct, nonopposite hemispheres of 8 2 • Any lune of 8 2 is congruent to a lune L(a) defined in terms of spherical coordinates (¢, 0) by the inequalities 0 ~ 0 ~ a.

Therefore la(t) - a(s)1 = It - sl b _ a la(b) - a(a)1 = It - sI- Thus a is a geodesic arc. Conversely, suppose that a is a geodesic arc. Let t be in [a, b]. Then la(b) - a(a)1 b- a b-t+t-a la(b) - a(t)1 + la(t) - a(a)l. 1. Euclidean Geometry 24 By Lemma 1, we have that a(a), a(t), a(b) are collinear with a(t) between a(a) and a(b). Therefore, there is some J(t) in [0,1] such that a(t) = a(a) Now, since J(t) = + J(t)(a(b) - a(a)). 1 and la/(t)1 = la(b) - a(a)1 b-a = 1. o Definition: A geodesic segment joining a point x to a point y in a metric space X is the image of a geodesic arc a : [a, b] -> X whose initial point is x and terminal point is y.

Now £b, R) ::; £b, R') = £(0:, P) + £((3, Q). 'YI = 10:1 + 1(31· Moreover 'Y is rectifiable if and only if 0: and (3 are rectifiable. o 1. Euclidean Geometry 30 Let X be a geodesically connected metric space and let 'Y : [a, b] ~ X be a curve from x to y. Then h'l ~ d(x, y) with equality if 'Y is a geodesic arc. Thus d(x, y) is the shortest possible length of 'Y. It is an exercise to show that bl = d(x, y) if and only if'Y maps [a, b] onto a geodesic segment joining x to y and d(x, 'Y(t)) is an increasing function of t.

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