Parallel Transported Lines
Symmetries of Parallel Transported LinesConsider two lines, r and r’, that are parallel transports of each other along a third line, l. Consider now the geometric figure that is formed by the three lines and look for the symmetries of that geometric figure.What can you say about the lines r and rl Do they intersect? If so, where? Look at the plane, spheres, and hyperbolic planes.If a transversal cuts two lines at congruent angles, are the lines, in fact parallel in the sense of not intersecting?Suggestions can be found on the book.Problem 9.1, 9.2 and 10.1 Due date is March 6thProblem 9.1 Side-Side-Side (SSS)Are two triangles congruent if the two triangles have congruent corresponding sides?SuggestionsStart investigating SSS by making two triangles coincide as much as possible, and see what happens. For example, in Figure 9.2, if we line up one pair of corresponding sides of the triangles, we have two different orientations for the other pairs of sides as depicted in Figure 9.2. Of course, it is up to you to determine if each of these orientations is actually possible, and to prove or disprove SSS. Again, symmetry can be very useful here. 6n a sphere, SSS doesn’t work for all triangles. The counterexample in Figure 9.3 shows that no matter how small the sides of the triangle are, SSS does not hold because the three sides always determine two different triangles on a sphere. Thus, it is necessary to restrict the size of more than just the sides in order for SSS to hold on a sphere. Whatever argument you used for the plane should work for suitably defined small triangles on the sphere and all triangles on a hyperbolic plane. Make sure you see what it is in your argument that doesn’t work for large triangles on a sphere. for more information on Parallel Transported Lines check on this:https://en.wikipedia.org/wiki/Parallel_transport
Top-quality papers guaranteed
100% original papers
We sell only unique pieces of writing completed according to your demands.
We use security encryption to keep your personal data protected.
We can give your money back if something goes wrong with your order.
Get free features with our reliable essay writing service
We offer you a free title page tailored according to the specifics of your particular style.
Include your preferred formatting style when you order from us to accompany your paper.
Get a list of references to go with your ordered paper.
24/7 support assistance
Reach out to our support agents anytime for free assistance.
Calculate how much your essay costs
What we are popular for
- English 101
- Business Studies