Download spherical experiments_sol

Experiment with spherical geometry (solutions):
Plotting: applet at
http://www.math.psu.edu/dlittle/java/geometry/spherical/toolbox.html
Q1:
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Plot two points.
Plot a line through them.
Attempt to draw a parallel (plot two more points that look like
they should give you a line parallel to the first).
Plot a line through them.
Any way to do this and NOT have the lines intersect?
Nope. They're always going to cross somewhere as they travel
around the sphere.
Q2: Try constructing a line, a perpendicular, and a perpendicular to the perpendicular.
Any luck at getting parallel lines this way? Do the perpendiculars look "right"?
Blue is perpendicular to red, and green is perpendicular to blue. And
red and green are distinctly not parallel, and the perpendiculars don't
even look like right angles, especially the green/blue one.
Q3: Since the lines look like circles, you might want to go see what a circle looks like.
Plot one. Oddly enough, the circle looks like a ... ? Since a circle is defined as the set of
all points an equal distance from the center, this actually tells you something about how
distances are measured in spherical geometry – would you guess that the distances are
still measured in essentially the Euclidean metric, or not?
The circle looks like a circle. How about that? You measures
distances in a perfectly normal way; it's the Euclidean distance along
the curved surface of the sphere (which involves an arc length
calculation). Spherical geometry has a finite metric – you can't have
a distance any large than 2π (that assumes a sphere of radius 1, and
uses circumference = 2πr.
Measuring triangles: applet at
http://www.walter-fendt.de/m11e/sphertriangle.htm
Q4: The sum of the measures of the angles of a spherical triangle is always between what
value and what value? What theorem of absolute geometry does this run counter to?
Triangle angle sum always strictly greater than 180°, and strictly less than 540°. Runs
counter to Saccheri-Legendre, which says in absolute geometry, angle sum ≤ 180°.
Spherical geometry not absolute.
Q5: Drag the triangle vertices around and attempt to get the angle sum as close to 180 as
possible. There are two ways to do this: make the triangle _very small_ (but still
triangular), or _flatten into a straight line_.