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Spherical Geometry and World Navigation

Euclidean Geometry. Most people are familiar with itChildren learn shapes: triangles, circles, squares, etc.High school geometry: theorems concerning parallelism, congruence, similarity, etc.Common, easy to understand, and abundant with applications; but only a small portion of geometry. Euclid'

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Spherical Geometry and World Navigation

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    1. Spherical Geometry and World Navigation By Houston Schuerger

    2. Euclidean Geometry Most people are familiar with it Children learn shapes: triangles, circles, squares, etc. High school geometry: theorems concerning parallelism, congruence, similarity, etc. Common, easy to understand, and abundant with applications; but only a small portion of geometry

    3. Euclid’s Five Axioms 1. A straight line segment can be drawn joining any two points.

    4. Euclid’s Five Axioms 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line.

    5. Euclid’s Five Axioms 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

    6. Euclid’s Five Axioms 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent.

    7. Euclid’s Five Axioms 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

    8. Euclid’s 5th Axiom more common statement equivalent to Euclid’s 5th axiom given any straight line and a point not on it, there exists one and only one straight line which passes through that point parallel to the original line

    9. Euclid’s 5th Axiom more common statement equivalent to Euclid’s 5th axiom given any straight line and a point not on it, there exists one and only one straight line which passes through that point parallel to the original line 5th axiom has always been very controversial Altering this final axiom yields non-Euclidean geometries, one of which is spherical geometry.

    10. Euclid’s 5th Axiom more common statement equivalent to Euclid’s 5th axiom given any straight line and a point not on it, there exists one and only one straight line which passes through that point parallel to the original line 5th axiom has always been very controversial Altering this final axiom yields non-Euclidean geometries, one of which is spherical geometry. This non-Euclidean geometry was first described by Menelaus of Alexandria (70-130 AD) in his work “Sphaerica.”

    11. Euclid’s 5th Axiom more common statement equivalent to Euclid’s 5th axiom given any straight line and a point not on it, there exists one and only one straight line which passes through that point parallel to the original line 5th axiom has always been very controversial Altering this final axiom yields non-Euclidean geometries, one of which is spherical geometry. This non-Euclidean geometry was first described by Menelaus of Alexandria (70-130 AD) in his work “Sphaerica.” Spherical Geometry’s 5th Axiom: Given any straight line through any point in the plane, there exist no lines parallel to the original line.

    12. Great Circles Straight lines of spherical geometry circle drawn through the sphere that has the same radii as the sphere Occurs when a plane intersects a sphere through its center Shortest distance between two points is along their shared great circle

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