Spherical Geometry Patrick Showers Michelle Zrebiec

Why is this important … One application of Spherical Geometry is that pilots and ship captains need to navigate the earth to reach their destination. For example, did you know that the shortest flying distance from Florida to the Philippine Islands is a path across Alaska? The Philippines are South of Florida – so why is flying North to Alaska a short-cut? The answer is that Florida, Alaska, and the Philippines are collinear locations in Spherical Geometry (they lie on a "Great Circle")

The Basics! Sphere A set of points in three dimensional space equidistant from the center. Radius Distance from the center to the points on the sphere Diameter Twice the radius

Points on Sphere Points are only on the surface of a sphere Points are the same concept we are used to studying in absolute and Euclidean geometry Special pairs of points: Called antipodal or polar those points that correspond to opposite ends of a diameter of the sphere. Example: N & S poles

What are lines? Shortest path on the sphere between two points on the sphere’s surface Known as great circles or arcs A great circle is a circle on a sphere which divides the sphere in 2 equal hemispheres; it contains a diameter of the sphere and hence the center as well

Planes intersecting with Sphere 3 cases involving planes and spheres The plane misses the sphere The plane hits the sphere at 1 point Known as a tangent plane The plane and the sphere meet in a circle

Planes Intersecting Sphere The circle of intersection will be largest when the plane passes through the center of the sphere Paradoxically, larger radius means less distance traveled on the surface (closest to Euclidean distance) 7

Lines - Arcs In absolute geometry, an axiom states any two points determine a unique line This is NOT true in Spherical Geometry To see this, think about the North and South poles of the Earth. Is there more than one great circle that runs trough them? Is this true for any 2 points? When are two points contained in infinitely many arcs and when is there a unique arc containing two points? Are there conditions guaranteeing uniqueness of segments connecting two points? 8

Plane of the Sphere There is only ONE plane that exists in spherical geometry The plane is the surface of the sphere 9

Distance Given two points A and B on the sphere, the distance AB* between them is length of the shortest arc-segment which connects them. Because of the way we defined spherical ‘lines’, this arc- segment must be an arc of a great circle. Joining our points A and B to the center of the sphere, to create a 'wedge' shape. We can measure the angle at the center - call it θ. The length of our arc is then rθ. 10

Distance in Picture Form B The length of our arc is then rθ. 11

Betweenness A-B-C holds if and only if A,B, & C are distinct points of a great circle AB* + BC* = AC* Activity: What does betweenness look like on a sphere? 12

Angles Imagine two great circles on the sphere. Choose one of the two points where they intersect. The angle between the two arcs is then defined to be the angle between the two planes containing the great circles. 13

Additional Angle Definition Imagine a flat, transparent plane which just touches the sphere at the point we wish to measure our angle. Looking at the plane from directly above, arcs on the sphere coincide with lines on the plane. The angle between our spherical lines is then the angle between the corresponding lines on the plane 14

Triangles A spherical triangle consists of three vertices, and three arc-segments which join them. Note that the lines do not have to be the shorter arcs. Also, if any two points are antipodal, we must specify which great circle to be the arc joining them (as there are an infinite number of great circles containing both points). 15

Question … Does the sum of the angles of a triangle equal 180 in spherical geometry?? 16

Answer …. The angle sum will never be 180° Always 180° < A + B + C < 540° B A C 17

Girard’s Theorem for Triangular Area Consider a (small) spherical triangle with vertices A, B, and C noncollinear Note that A, B, and C are the intersections of arcs a, b, and c so that we have polar points A’,B’, and C’ also A and A’ are the endpoints of two two lunes, call them Lα and Lα’. Similarly for points B, B’ and C, C’. 18

Girard’s Theorem Con’t A(Lα U Lα’) + A(Lβ U Lβ’) + A(Lγ U Lγ’) = A(sphere) + 2A(ΔABC) +2A(ΔA’B’C’) since Lβ and Lγ each overcount the area of ΔABC. (Similarly for ΔA’B’C’ with lunes Lβ’ and Lγ’) 2(2α+2β+2γ) = 4π + 4A(ΔABC) α+β+γ = π + A(ΔABC) and thus A(ΔABC) = α+β+γ - π (Girard’s Theorem) 19

Spherical Excess Given a spherical ΔABC with angles α, β, and γ, the quantity α+β+γ-π is called the spherical excess of ΔABC. Girard’s theorem tells us that area of a spherical polygon is intimately linked with the spherical excess. 20

Spherical Trigonometry 21

Theorem AAA Congruence Criterion If two triangles have three angles of one congruent, respectively, to the three angles of the other, the triangles are congruent 22

References http://noneuclidean.tripod.com/applications.html http://math.rice.edu/~pcmi/sphere/sphere.html#basic http://en.wikipedia.org/wiki/Spherical_geometry The book http://h2g2.com/dna/h2g2/A974397 http://www.math.hmc.edu/funfacts/ffiles/20001.2.shtml