Math 621: Elementary Geometry


Good Problems


On this page you will find the Good Problems (written assignments), including due dates. Don't forget to check the Recommended Exercises page frequently as well. Remember, exam questions will be taken from both the Good Problems and the Recommended Exercises.

Here is a sample write-up: Proposition 1.1.2


The date listed to the right of each assignment number is the initial due date. For each good problem, something must be turned in by the initial due date. You may only rewrite a good problem if you turned in an initial "attempt." If you turned in an initial attempt, and you wish to rewrite a good problem, all rewrites must be submitted by the beginning of class on the date specified. You may only submit one rewrite per assignment.


Chapter 4--

23. W 14 Nov Construct a triangle through 3 points in the Poincaré disk, one of which is the origin.
24. W 14 Nov Construct a triangle in the hyperbolic plane that passes through 3 points, none of which lie on the same vertical line.
25. M 19 Nov Exercise 4.3.3.
26. M 19 Nov Exercise 4.3.4.
27. M 19 Nov Compute the Poincaré (upper half plane) distance between the points \(P\left( \tfrac{\sqrt{3}}{2},\tfrac{1}{2}\right)\) and \(Q\left(-\tfrac{\sqrt{2}}{2}, \tfrac{\sqrt{2}}{2}\right)\).
28. M 26 Nov Exercise 4.5.5.
29. M 26 Nov Exercise 4.5.13.


Chapter 4 re-writes due by W 5 Dec



Chapter 3--

17. M 29 Oct Write down the matrix representation of a relfection through a line \(\ell\) passing through the origin; i.e., \(\ell\) is given by \(y = mx\).

18. M 29 Oct Let \(O(\mathbb{R}^2)\) denote the set of all \(2 \times 2\) orthogonal matrices: \(Q^T = Q^{-1}\). Show that the linear transformation corresponding to each \(Q \in O(\mathbb{R}^2)\) is an isometry. Recall, this means you need to check that \(\Vert Q \mathbf{x} \Vert = \Vert\mathbf{x}\Vert\) for all \(\mathbf{x}\) in \(\mathbb{R}^2\).

Hint: use the linear algebra definition of dot product: \(\mathbf{x}\cdot\mathbf{y} = \mathbf{x}^T\mathbf{y}\).

19. W 31 Oct Let \(y = f(x)\) be a function whose graph is a plane curve. Determine the transformation of \(\mathbb{R}^2\) that carries the unit circle to the osculating circle of the curve at the point \((x_0,f(x_0))\).

Hint: first map the unit circle to a circle centered at the origin of correct radius, then map to the circle with the correct center.

20. W 31 Oct Let \(\mathrm{Iso}_0(\mathbb{R}^2)\) denote the set of all isometries of \(\mathbb{R}^2\) that fix the origin; \(f(\mathbf{0}) = \mathbf{0}\). This is called the isotropy subgroup of isometries. Prove that \(\mathrm{Iso}_0(\mathbb{R}^2)\) indeed forms a group under composition.

That is, show that if \(f\) and \(g\) are two isometries in \(\mathrm{Iso}_0(\mathbb{R}^2)\), then \(f \circ g\) and \(f^{-1}\) are also both isometries that fix the origin.

21. W 07 Nov Construct the nine point circle of a non-equilateral triangle. You should list the steps, but you do not need to describe the construction in detail.

22. M 12 Nov Start with a triangle \(\Delta ABC\), a line \(\ell\) that does not intersect the triangle, and a point \(P\) that is not on either the triangle or the line. Construct the image of the triangle under the transformation \(S(P,2,\ell)\). List the steps.


Chapter 3 re-writes due by M 26 Nov


Chapter 2--

10. W 26 Sep Exercise 2.2.4
11. W 26 Sep Exercise 2.2.9
12. M 01 Oct Exercise 2.3.3
13. M 01 Oct Exercise 2.3.10 (Hint: Identify the Gergonne point of a larger triangle and apply Desargues's Theorem.)
14. W 03 Oct Exercise 2.5.4
15. M 08 Oct Exercise 2.6.3
16. W 10 Oct Exercise 2.7.5

Chapter 2 re-writes due by W 31 Oct


Chapter 1--

1. M 27 Aug Complete exercise 1.1.1, clearly describing all steps.
2. W 29 Aug Exercise 1.3.7
3. W 29 Aug Use the result of exercise 1.3.5 to complete exercise 1.3.9.
4. W 05 Sep Exercise 1.4.6 -- find at least 2 different circles that solve the problem
5. W 05 Sep Exercise 1.4.7
6. M 10 Sep Exercise 1.5.4
7. M 10 Sep Exercise 1.5.13
8. W 12 Sep Exercise 1.6.1
9. W 12 Sep Construct the number \(\sqrt{2}\) using the method of Exercise 1.6.4.

Chapter 1 re-writes due by M 24 Sep



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