Difference between revisions of "The Axioms of Spherical Geometry Exploration"

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* Objective: Understand the axioms of spherical geometry.
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{{Exploration}}
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{{time|30}}
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{{objective|Understand the axioms of spherical geometry.}}
  
  
 
The axioms of Euclidean geometry give us the basic statements from which the rest of Euclidean geometry can be deduced. They basically describe the behavior of lines, segments, circles, angles and parallel lines.
 
The axioms of Euclidean geometry give us the basic statements from which the rest of Euclidean geometry can be deduced. They basically describe the behavior of lines, segments, circles, angles and parallel lines.
  
Below write down the respective axioms for spherical geometry.
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'''Below write down the respective axioms for spherical geometry. Illustrate with example(s0 when appropriate.'''
  
 
Remember that the role of lines is played by the great circles - also known as geodesics - in spherical geometry.
 
Remember that the role of lines is played by the great circles - also known as geodesics - in spherical geometry.
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'''Spherical Geometry:''' <br>
 
'''Spherical Geometry:''' <br>
Case 1: Points P and Q are not antipodal. <br>
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Case 1: Points A and B are not antipodal. <br>
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[[Image:AB-not-anti.jpg|250px]]
  
  
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Case 2: Points P and Q are antipodal. <br>
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Case 2: Points A and B are antipodal. <br>
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[[Image:AB-antipode.jpg|250px]]
  
 
===Segments===
 
===Segments===
 
Euclidean Geometry: Any straight line segment can be extended indefinitely in a straight line.
 
Euclidean Geometry: Any straight line segment can be extended indefinitely in a straight line.
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'''Spherical Geometry:''' <br>
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[[Image:Spheresegment.jpg|250px]]
  
  
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Euclidean Geometry: Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
 
Euclidean Geometry: Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
  
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'''Spherical Geometry:''' <br>
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<br>
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<br>
  
 
===Angles===
 
===Angles===
 
Euclidean Geometry: All right angles are congruent.
 
Euclidean Geometry: All right angles are congruent.
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'''Spherical Geometry:''' <br>
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<br>
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<br>
  
 
===Parallel Lines===
 
===Parallel Lines===
 
Euclidean Geometry: Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.
 
Euclidean Geometry: Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.
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<br>
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<br>
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Are the following curves examples of spherical parallel lines? Explain why or why not.
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[[Image:Fake-lines.jpg|250px]]
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{{handin|A copy of this sheet with answers to the questions.}}
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[[category:Non-Euclidean Geometry Explorations]]

Latest revision as of 16:40, 20 October 2009


Time-30.svg

Objective: Understand the axioms of spherical geometry.


The axioms of Euclidean geometry give us the basic statements from which the rest of Euclidean geometry can be deduced. They basically describe the behavior of lines, segments, circles, angles and parallel lines.

Below write down the respective axioms for spherical geometry. Illustrate with example(s0 when appropriate.

Remember that the role of lines is played by the great circles - also known as geodesics - in spherical geometry.

A geodesic on a sphere

Lines

Euclidean Geometry: Any two points can be joined by a straight line. (This line is unique given that the points are distinct).

Spherical Geometry:
Case 1: Points A and B are not antipodal.
AB-not-anti.jpg




Case 2: Points A and B are antipodal.
AB-antipode.jpg

Segments

Euclidean Geometry: Any straight line segment can be extended indefinitely in a straight line.


Spherical Geometry:
Spheresegment.jpg


Circles

Euclidean Geometry: Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.


Spherical Geometry:


Angles

Euclidean Geometry: All right angles are congruent.

Spherical Geometry:


Parallel Lines

Euclidean Geometry: Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.

Are the following curves examples of spherical parallel lines? Explain why or why not.

Fake-lines.jpg


Handin: A copy of this sheet with answers to the questions.