Math 621: Elementary Geometry |
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Exercises 1.1.1 and 1.1.2 Sketch pictures illustrating the definitions in section 1.2 for which a picture is appropriate. Sketch a proof of theorem 1.3.12. Finish the proof of the Star Trek Lemma, theorem 1.3.14. Exercises 1.1.3, 1.3.4, 1.3.5, 1.3.8 Exercises 1.4.1, 1.4.2, 1.4.3, 1.4.5, 1.4.8 Exercises 1.5.1, 1.5.2, 1.5.3, 1.5.7, 1.5.8, 1.5.17, 1.5.18 Exercises 1.6.1--1.6.4 Read all of the proofs in Section 1.7. Which is your favorite? |
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Exercises 2.2.1--2.2.4, 2.2.9 Fill in any missing details of proofs we sketched in class. Exercises 2.3.1--2.3.6, 2.3.10, 2.3.14, 2.3.16 |
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Exercises 2.5.1, 2.5.3, 2.5.4, 2.5.5, 2.5.6? (at least draw a picture
and note the result), and 2.5.7 Exercises 2.6.1, 2.6.2, 2.6.3, 2.6.5 |
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Exercises 2.7.1 -- 2.7.6 Read the proof of "uniqueness" of Theorem 2.7.3. (We did existence in class.) Exercises 2.8.1 -- 2.8.8 |
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Write down the matrix representation of a relfection through a line \(\ell\)
passing through the origin; i.e., \(\ell\) is given by \(y = mx\). Show that if \(f\) and \(g\) are isometries of the plane, then \(f^{-1}\) and \(f\circ g\) are also isometries of the plane. Construct the centers of similitude of two nonconcentric circles. Construct the image of a triangle \(\Delta ABC\) under each of the types types of transformations: \(T(AB), R(P,\theta), R(P), R(\ell), H(P,k), H(P,k,\theta), G(\ell,AB),\) and \(S(P,k,\ell)\). Exercises 3.2.1--3.3.3 (3.3.3 refers to the three circles from 3.3.2) |
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Read sections 4.1 and 4.2. Consider the extended plane model of the Riemann sphere. Sketch some triangles that pass through the point at infinity. Sketch some triangles in the hyperbolic plane and Poincare disk. Try to think of all possibilities of vertex alignment. Exercises 4.3.1 -- 4.3.4. |
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Exercises 4.5.1--4.5.14 (i.e., all of them) Sketch the unit circles for the norms: \(\Vert\cdot\Vert_{\tfrac{1}{2}}\), \(\Vert\cdot\Vert_{1}\), \(\Vert\cdot\Vert_{2}\), and \(\Vert\cdot\Vert_{\infty}\). |
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