   |
| Intro |
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| intro 2 |
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| planes 1 |
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| planes 2 |
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| planes3 |
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| 2-D |
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| 2-D (cont) |
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| 2-D last |
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| App A |
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| End |
This is an introduction to relationships between planes
in Hyperbolic Geometry. This is the result of over a quarter of a
century's worth of asking questions and then starting to answer them
myself. Actually, what is examined is the orthogonal (perpendicular)
projection of one plane upon another.
Mathematical Musings
John C. Longnecker
Professor Emeritus, Mathematics
Beginnings
In Hyperbolic Geometry, all of
the basic truths of Euclidean Geometry still hold until one hits a parallel
postulate. [Note: Euclid was wise enough to consider it a postulate
rather than a truth.] Instead of having only one line through a
point parallel to a given line, we have at least two - an immediate consequence
is that we have an unlimited number of parallels to a given line
through a given point (and these are all coplanar). The study
of plane Hyperbolic (as it is known) Geometry is common - extensions into the
third dimension are few. This is one of those few.
At this time, all that will be examined are the
relationships between a pair of planes, along with the orthogonal projections of
one on the other. 3-D figures may be introduced at some later date.
Of most interest will be the boundaries of those projections - it is claimed
that each is a curve of constant curvature.
