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# Spherical Geometry

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by

Tweet## Britt Kirkland

on 9 March 2011#### Transcript of Spherical Geometry

Definitions Betweenness Congruence Measurement lines are great circles Everything lives on the unit sphere angles are formed by great circles intersecting lunes are regions between great circles that intersect antipodal points Antipodal points are points that are on opposite ends of a diameter of the sphere measuring distance on a sphere is the same as measuring the arc length along a great circle Spherical

Geometry tools for measuring angles and distances on a Lenart sphere disc protractor spherical ruler spherical compass the angle formed by the intersection of the two defining planes with the plane tangent to the sphere at the point of intersection angle measure distance therefore, in spherical geometry B-3 fails. Consider Axiom B-3: If A, B, and C are three distinct points lying on the same line, then one and only one of the points is between the other two Consider any great circle and let A, B, and C be any three points on that circle notice that B is between A and C but... Hence we have a contradiction Conventions A is between B and C too! gtfo ;)! wait for it... When considering the distance between two points, we will always take the minimum distance

Triangle Congruence Axioms That Fail Euclidean Geometry

SSA

AAA Spherical Geometry

AAS

SSA

Notice: AAA is a congruence axiom in spherical geometry AAS Counterexample SSA Counterexample AAA Euclid's 5th: notice that AD, AC, and AB are all congruent

angle D and angle C are both right angles

but triangles ADB and ADC are not congruent.

therefore, SSA is not a congruence axiom in spherical geometry. Notice that both the green and yellow regions are technically triangles.

We will call the yellow region a degenerate triangle Comparison with Euclidean geometry Jenn, Julia, & Britt notice that angles ADC, ACD, ACB and ABC are all right angle and that AC is congruent to itself but triangles ADB and ADC are not congruent therefore AAS is not a congruence axiom in spherical geometry again, our counterexample for SSA and AAS provides a counterexample for the parallel postulate. in fact, every pair of antipodal points define an infinite number of lines that satisfy the hypothesis of Euclid's 5th postulate. think about the lines of latitude traveling through the north and south poles Consider triangles ABC and AB'C', assume they are not congruent.

Assume angle BCA congruent to angle B'C'A

and angle ABC congruent to AB'C'.

Since there are no parallel lines in spherical geometry, BC and B'C' must coincide and hence ABC is congruent to AB'C'. On the plane

the straight line is infinite

the straight line has no center

there is one unique straight line that passes through any pair of points

the shortest distance between two points is a straight line segment

when measuring distance between two points along a straight line, there is only one distance you can measure

two distinct lines with no point of intersection are called parallel

two distinct lines have at most one point of intersection

a pair of perpendicular lines intersect one time and creates 4 right anlges

a pair of parallel straight lines have infinitely many common perpendiculars

a two-sided polygon does not exist

3 noncollinear points determine one unique triangle

the angle sum of a triangle is 180 degrees On the sphere

the great circle is a closed geodesic

the great circle has antipodal points

there is one unique great circle passing through any pair of points unless the points are antipodal

an orthodrome segment is the shortest path between two points

when measuring distance between two points along the great circle, there are two distances you can measure, we usually measure the shortest one

two distinct great circles always intersect at two points

two distinct great circles have two points of intersection

a pair of perpendicular great circles intersects 2 times and create 8 right angles

a pair of parallel great circles does not exist

a two-sided polygon does exist (a lune)

3 noncollinear points determine two unique triangles

the interior angle sum of a triangle ranges from 180 to 540

Full transcriptGeometry tools for measuring angles and distances on a Lenart sphere disc protractor spherical ruler spherical compass the angle formed by the intersection of the two defining planes with the plane tangent to the sphere at the point of intersection angle measure distance therefore, in spherical geometry B-3 fails. Consider Axiom B-3: If A, B, and C are three distinct points lying on the same line, then one and only one of the points is between the other two Consider any great circle and let A, B, and C be any three points on that circle notice that B is between A and C but... Hence we have a contradiction Conventions A is between B and C too! gtfo ;)! wait for it... When considering the distance between two points, we will always take the minimum distance

Triangle Congruence Axioms That Fail Euclidean Geometry

SSA

AAA Spherical Geometry

AAS

SSA

Notice: AAA is a congruence axiom in spherical geometry AAS Counterexample SSA Counterexample AAA Euclid's 5th: notice that AD, AC, and AB are all congruent

angle D and angle C are both right angles

but triangles ADB and ADC are not congruent.

therefore, SSA is not a congruence axiom in spherical geometry. Notice that both the green and yellow regions are technically triangles.

We will call the yellow region a degenerate triangle Comparison with Euclidean geometry Jenn, Julia, & Britt notice that angles ADC, ACD, ACB and ABC are all right angle and that AC is congruent to itself but triangles ADB and ADC are not congruent therefore AAS is not a congruence axiom in spherical geometry again, our counterexample for SSA and AAS provides a counterexample for the parallel postulate. in fact, every pair of antipodal points define an infinite number of lines that satisfy the hypothesis of Euclid's 5th postulate. think about the lines of latitude traveling through the north and south poles Consider triangles ABC and AB'C', assume they are not congruent.

Assume angle BCA congruent to angle B'C'A

and angle ABC congruent to AB'C'.

Since there are no parallel lines in spherical geometry, BC and B'C' must coincide and hence ABC is congruent to AB'C'. On the plane

the straight line is infinite

the straight line has no center

there is one unique straight line that passes through any pair of points

the shortest distance between two points is a straight line segment

when measuring distance between two points along a straight line, there is only one distance you can measure

two distinct lines with no point of intersection are called parallel

two distinct lines have at most one point of intersection

a pair of perpendicular lines intersect one time and creates 4 right anlges

a pair of parallel straight lines have infinitely many common perpendiculars

a two-sided polygon does not exist

3 noncollinear points determine one unique triangle

the angle sum of a triangle is 180 degrees On the sphere

the great circle is a closed geodesic

the great circle has antipodal points

there is one unique great circle passing through any pair of points unless the points are antipodal

an orthodrome segment is the shortest path between two points

when measuring distance between two points along the great circle, there are two distances you can measure, we usually measure the shortest one

two distinct great circles always intersect at two points

two distinct great circles have two points of intersection

a pair of perpendicular great circles intersects 2 times and create 8 right angles

a pair of parallel great circles does not exist

a two-sided polygon does exist (a lune)

3 noncollinear points determine two unique triangles

the interior angle sum of a triangle ranges from 180 to 540