| Class Schedule |
- New model for a geometry: Points are points on the hemisphere bounded
by a great circle h. Lines are arcs of circles orthogonal to h -- these
are semicircles with center on h.
- Is there exactly one line through any two points A and B?
- How do lines intersect? How many points of intersection possible?
- Two kinds of parallels: asymptotic parallels when semicircles
are tangent on h and ultraparallel when semicircles do not touch
in the hemisphere or on h.
Review of connections among previous models by relationship to
stereographic projection (from a sphere G, center of projection N and
point S opposite N).
- S model (spherical geometry) is the stereo image of G, this
is usual E plane with point Inf added at infinity. The circles on S
through N are mapped to lines, the others to circles. This model also
has a speci al circle e that is equator. Knowing e one can identify
images of great circles.
- I model (inversive plane) is the stereo image of the geometry
of circles on the sphere G. This is the same as the S-model, except
that the great circles are not singled out.
- E plane (a model of Euclidean geometry) is the stereo image
of G with N removed. E-lines are images of circles on G through N.
- D model (a model of Euclidean geometry) is the stereo image
of G with S removed; the image of S is the special point O that is removed
from the plane. The point Inf at infinity is the image of N and is a
D-point.. D-lines are images of circles on G through S, i.e., E-circles
or E-lines through O.
- Hemisphere model (a model of hyperbolic non-Euclidean geometry)
is explained above. This is a model on the sphere G.
- Poincaré disk model (a model of hyperbolic non-Euclidean
geometry) is the stereo image of of the hemisphere model, where
the hemisphere is the "southern hemisphere" centered at S.
So the points are points interior to a circle h and the lines are arcs
of circles (or segments) orthogonal to h. More details are found in
Lab 8.
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