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Spherical Triangles and Girard’s Theorem Abhijit Champanerkar College of Staten Island, CUNY MTH 329, Spring 2013
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Spherical Triangle

Jun 02, 2018

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Page 1: Spherical Triangle

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Spherical Triangles and Girard’s Theorem

Abhijit Champanerkar

College of Staten Island, CUNY

MTH 329, Spring 2013

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Spherical geometry

Let  S 2 denote the unit sphere in  R3 i.e. the set of all unit vectors

i.e. the set  {(x , y , z ) ∈ R3|  x 2 + y 2 + z 2 = 1  }.

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Spherical geometry

Let  S 2 denote the unit sphere in  R3 i.e. the set of all unit vectors

i.e. the set  {(x , y , z ) ∈ R3|  x 2 + y 2 + z 2 = 1  }.

A great circle in  S 2 is a circle which divides the sphere in half. In

other words, a great circle is the interesection of  S 2 with a planepassing through the origin.

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Great circles are straight lines

Great circles play the role of straight lines in spherical geometry.

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Great circles are straight lines

Great circles play the role of straight lines in spherical geometry.

Given two distinct points on  S 2, there is a great circle passingthrough them obtained by the intersection of  S 2 with the planepassing through the origin and the two given points.

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Great circles are straight lines

Great circles play the role of straight lines in spherical geometry.

Given two distinct points on  S 2, there is a great circle passingthrough them obtained by the intersection of  S 2 with the planepassing through the origin and the two given points.

You can similarly verify the other three Euclid’s posulates forgeometry.

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Diangles

Any two distinct great circles inter-sect in two points which are nega-tives of each other.

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Diangles

Any two distinct great circles inter-sect in two points which are nega-tives of each other.

The angle between two great circles at an intersection point is theangle between their respective planes.

A region bounded by two great circles is called a diangle.

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Diangles

Any two distinct great circles inter-sect in two points which are nega-tives of each other.

The angle between two great circles at an intersection point is theangle between their respective planes.

A region bounded by two great circles is called a diangle.

The angle at both the vertices are equal. Both diangles boundedby two great circles are congruent to each other.

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Area of a diangle

Proposition

Let  θ  be the angle of a diangle. Then the area of the diangle is 2θ.

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Area of a diangle

Proposition

Let  θ  be the angle of a diangle. Then the area of the diangle is 2θ.

Proof:  The area of the diangle is proportional to its angle. Sincethe area of the sphere, which is a diangle of angle 2π, is 4π, the

area of the diangle is 2θ.

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Area of a diangle

Proposition

Let  θ  be the angle of a diangle. Then the area of the diangle is 2θ.

Proof:  The area of the diangle is proportional to its angle. Sincethe area of the sphere, which is a diangle of angle 2π, is 4π, the

area of the diangle is 2θ.

Alternatively, one can compute this area directly as the area of asurface of revolution of the curve  z  =

 1 − y 2 by an angle  θ. This

area is given by the integral  1−1

 θz  

1 + (z )2 dy .  

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Area of a diangle

Proposition

Let  θ  be the angle of a diangle. Then the area of the diangle is 2θ.

Proof:  The area of the diangle is proportional to its angle. Sincethe area of the sphere, which is a diangle of angle 2π, is 4π, the

area of the diangle is 2θ.

Alternatively, one can compute this area directly as the area of asurface of revolution of the curve  z  =

 1 − y 2 by an angle  θ. This

area is given by the integral  1−1

 θz  

1 + (z )2 dy .  

If the radius of the sphere is  r  then the area of the diangle is 2θr 2.

This is very similar to the formula for the length of an arc of theunit circle which subtends an angle  θ   is  θ.

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Spherical polygons

A spherical polygon is a polygon on  S 2 whose sides are parts of great circles.

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Gi d’ Th A f h i l i l

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Girard’s Theorem: Area of a spherical triangle

Girard’s Theorem

The area of a spherical triangle with angles α, β  and γ  is α+β +γ −π.

Gi d’ Th A f h i l i l

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Girard’s Theorem: Area of a spherical triangle

Girard’s Theorem

The area of a spherical triangle with angles α, β  and γ  is α+β +γ −π.

Proof:

Gi d’ Th A f h i l t i l

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Girard’s Theorem: Area of a spherical triangle

Girard’s Theorem

The area of a spherical triangle with angles α, β  and γ  is α+β +γ −π.

Proof:

Gi d’ Th A f h i l t i l

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Girard s Theorem: Area of a spherical triangle

Girard’s Theorem

The area of a spherical triangle with angles α, β  and γ  is α+β +γ −π.

Proof:

Girard’s Theorem: Area of a spherical triangle

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Girard s Theorem: Area of a spherical triangle

Girard’s Theorem

The area of a spherical triangle with angles α, β  and γ  is α+β +γ −π.

Proof:

Girard’s Theorem: Area of a spherical triangle

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Girard s Theorem: Area of a spherical triangle

Girard’s Theorem

The area of a spherical triangle with angles α, β  and γ  is α+β +γ −π.

Proof:

Girard’s Theorem: Area of a spherical triangle

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Girard s Theorem: Area of a spherical triangle

Girard’s Theorem

The area of a spherical triangle with angles α, β  and γ  is α+β +γ −π.

Proof:

Girard’s Theorem: Area of a spherical triangle

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Girard s Theorem: Area of a spherical triangle

Girard’s Theorem

The area of a spherical triangle with angles α, β  and γ  is α+β +γ −π.

Proof:

Area of a spherical triangle

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Area of a spherical triangle

B

A

C

F

E

D

ABC  as shown above is formed by the intersection of three greatcircles.

Vertices  A  and  D  are antipodal to each other and hence have the

same angle. Similarly for vertices  B , E   and  C , F . Hence thetriangles  ABC   and  DEF  are antipodal (opposite) triangles andhave the same area.

Assume angles at vertices  A, B  and  C  to be  α, β  and  γ  respectively.

Area of a spherical triangle

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Area of a spherical triangle

B

A

C

F

E

D

ABC R  AD    R BE    R CF 

Let  R AD ,  R BE   and  R CF  denote pairs of diangles as shown. ThenABC   and  DEF  each gets counted in every diangle.

Area of a spherical triangle

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Area of a spherical triangle

B

A

C

F

E

D

ABC R  AD    R BE    R CF 

Let  R AD ,  R BE   and  R CF  denote pairs of diangles as shown. ThenABC   and  DEF  each gets counted in every diangle.

R AD  ∪ R BE  ∪ R CF   = S 2,  Area(ABC ) =  Area(DEF ) = X .

Area of a spherical triangle

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Area of a spherical triangle

B

A

C

F

E

D

ABC R  AD    R BE    R CF 

Let  R AD ,  R BE   and  R CF  denote pairs of diangles as shown. ThenABC   and  DEF  each gets counted in every diangle.

R AD  ∪ R BE  ∪ R CF   = S 2,  Area(ABC ) =  Area(DEF ) = X .

Area(S 2) =   Area(R AD ) +  Area(R BE ) +  Area(R CF ) − 4X 

4π   = 4α + 4β  + 4γ  − 4X 

X    =   α + β  + γ  − π

Area of a spherical polygon

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Area of a spherical polygon

Corollary

Let R  be a spherical polygon with  n  vertices and n  sides with interiorangles  α1, . . . , αn. Then  Area(R ) = α1 + . . . + αn − (n − 2)π.

Area of a spherical polygon

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Area of a spherical polygon

Corollary

Let R  be a spherical polygon with  n  vertices and n  sides with interiorangles  α1, . . . , αn. Then  Area(R ) = α1 + . . . + αn − (n − 2)π.

Proof:  Any polygon with  n  sides for  n ≥ 4 can be divided inton − 2 triangles.

The result follows as the angles of these triangles add up to theinterior angles of the polygon.