Math 621: Elementary Geometry


Written Assignments


On this page you will find the 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 Written Assignments and the Recommended Exercises.

All students must complete all un-starred exercises below. Students taking this course for graduate credit must also complete the starred exercises.

Here is a sample write-up: Proposition 1.1.2


"Unit 5" Assignments--

30. W 29 Nov Consider the circle in the complex plane centered at the point \(c = a + bi\) with radius \(r\). Write the formula for the inversion \(I(c,r^2)(z)\) in complex coordinates.
31. W 29 Nov Exercise 3.4.2.
32. W 29 Nov Exercise 3.4.5.



Midterm Exam 4--

22. Construct a triangle through 3 points in the Poincaré disk, one of which is the origin.
23. Construct a triangle in the hyperbolic plane that passes through 3 points, none of which lie on the same vertical line.
24. Construct a triangle in the extended plane (the model plane for the Riemann sphere) that has two right angles.
25. Exercise 4.3.3.
26. Exercise 4.3.4.
27. 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. Exercise 4.5.5.
29. Exercise 4.5.13.
29.* Sketch ellipses with eccentricity \(\tfrac{1}{2}\) as seen by the norms \(\Vert\cdot\Vert_\infty\)   and   \(\Vert\cdot\Vert_{\tfrac{1}{2}}\).
30.* Show that the only \(p\)-norm that corresponds to an inner product on \(\mathbb{R}^2\) is the 2-norm.


Chapter 4 re-writes due by M


Midterm Exam 3--

18. 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 \(\mathbb{R}^2\) and that \(\theta(Q\mathbf{x},Q\mathbf{y}) = \theta(\mathbf{x},\mathbf{y})\) for all \( \mathbf{x}, \mathbf{y}\) in \(\mathbb{R}^2\).
Hint: use the linear algebra definition of dot product: \(\mathbf{x}\cdot\mathbf{y} = \mathbf{x}^T\mathbf{y}\).
19. Let \(y = f(x)\) be a function whose graph is a plane curve. Determine the transformation of \(\mathbb{R}^2\) that carries the unit cirlce 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 translate to the correct center.
19.* 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.
20. 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.
21. 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 6 Nov


Midterm Exam 2--

10. Exercise 2.2.4
11. Exercise 2.2.9
11.* Fill in the details in the book's proof of Menelaus's Theorem; i.e., justify all unproved claims.
12. Exercise 2.3.3
13. Exercise 2.3.10 (Hint: Identify the Gergonne point of a larger triangle and apply Desargues's Theorem.)
13.* Exercise 2.3.14
14. Exercise 2.5.4
14.* Exercise 2.5.7
15. Exercise 2.6.3
16. Exercise 2.7.5
17. Exercise 2.8.5 including a construction of the radical axis.
17*. Exercise 2.8.6

Chapter 2 re-writes due by W 18 Oct


Midterm Exam 1--

1. Complete exercise 1.1.1, clearly describing all steps.
1.* Grad students only: Complete exercise 1.1.2, clearly describing all steps.
2. Exercise 1.3.7
3. Use the result of exercise 1.3.5 to complete exercise 1.3.9.
4. Exercise 1.4.6 -- find at least 2 different circles that solve the problem
5. Exercise 1.4.7
5*. Exercise 1.4.11
6. Exercise 1.5.4
7. Exercise 1.5.13
7*. Exercise 1.5.18. Cleary describe why your construction works with a collapsible compass.
8. Exercise 1.6.1
9. Construct the number \(\sqrt{2}\) using the method of Exercise 1.6.4.
9*. Exercise 1.6.2

Chapter 1 re-writes due by M 18 Sep



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