GEOMETRY OF A SPHERE

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GEOMETRY OF A SPHERE

• RET
• Tulane-Summer 2004
• Stephanie Cox
• Laura Flora
• Cheryl Sephus
• Tom Slack

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OBJECTIVE

• Study non-Euclidean relationships of
• spherical triangles
• Law of Cosines
• Pythagorean Theorem
• AAA Congruence
• Sum of the angle measures

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DEFINITIONS

• Great Circle
• Lune
• Spherical Triangle
• Spherical Distance

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• The spherical distance between 2 points x, y
• ? S is defined to be the central angle, ,
• between the two vectors and .
• Law of Cosines z ² x ² y ² - 2xy cos
• cos
• cos

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OBJECTIVE 1 Prove Elliptic Law of Cosines
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• Consider
• spherical ?ABC
• vectors , and ? S, correspond to
• vertices A, B, and C
• the following relationships apply
• 
  
  
  cos a
• cos b
• cos c

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Computing Angle A

• Angle A is defined to be the angle between the
  planes in R3 which contain and .
• We can compute the angle by finding the angle
  between the vectors that are normal to the two
  planes by using the cross product.

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OBJECTIVE 2Pythagorean Theorem

• If A is a right angle then cos A 0.

Pythagorean Theorem states the relationship of
the length of the hypotenuse in terms of the
lengths of the legs.
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OBJECTIVE 3Angle-Angle-Angle Congruence

• To prove the side lengths of a spherical
  triangle are determined totally by the angle
  measures of the triangle.
• We will use spherical ?ABC and ?ABC with

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• We can measure ?A by measuring the angle
• between normal vectors A x B and A x C.
• cos A
• Correspondingly, we can compute the length of
  side
• a
• cos a .
• - cos A

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• Using these relationships and the Elliptic Law
• of Cosines, we have
• cos a ? cos A
• Thus, side length a is determined by angles
• A, B, and C.
• Two triangles are congruent if their
• corresponding angles are congruent (AAA).

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OBJECTIVE 4Sum of the Angles of Spherical
Triangle
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• The areas of the lunes are in the same
• proportion to the area of the sphere as their
• angles are to 2p.
• area lune A (m? A 2p) 4p
• area lune B (m? B 2p) 4p
• area lune C (m? C 2p) 4p

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• A hemisphere (H) of area 2p is formed by
• H ?ABC ?A ?B ?C.
• Union of lunes is
• lune A lune B lune C
• ?ABC ?A ?ABC ?B ?ABC ?C
• H ?ABC ?ABC
• Thus, the area of the lunes is
• area lune A area lune B area lune C
• 2p 2 area ?ABC

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• Thus, we have
• 4p m? A/2p m? B/2p m? C/2p
• 2p 2 area ?ABC
• And
• m?A m?B m?C p area ?ABC
• sum of the angles measures exceeds p by the area
  of the triangle
• area of a spherical triangle depends only on its
  angles

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CONCLUSIONS

• Elliptic Law of Cosines
• Pythagorean Theorem
• Side length depends only on angles (AAA
  congruence)
• Sum of the measures of the angle exceeds p by the
  area of the triangle