The explanations in the text are fairly detailed. So you are expected to study the text, before and after the lectures, on your own. I will only have time to discuss the most essential aspects of the theory; you should read the explanations and examples to fill in details not carefully discussed in class. If you find places that need clarification, please raise these points. This way, our discussion will be more focused.

The problems to be assigned are grouped in to recommended and required. The recommended problems are scattered in the body of the text in each section, and have complete solutions in the back of the text, so will not be collected; the required problems are listed in the last section at the end of each chapter, and are collected for grading.

• Chapter I. Note that the word gradient in the text is the British English word for slope in American English. Please review the concepts of linear transformations, orthogonal matrices, eigenvalues and eigenvectors of matrices, and diagonalization of symmetric matrices. We will be needing these concepts mostly in the two or three dimensional setting.
  • Sections 1.1, 1.2, due: Monday, Sept. 12.
    • Recommended: 1.1 : 3,6,7,8,9; 1.2: 2,4,7,8;
    • Required: 1.1: 4,5; 1.2: 3,5.
    • Here you will find the solutions to HW 1.
  • Sections 1.2, 1.3, due: Monday, Sept. 19.
    • Recommended: 1.3 : 1,2,3.
    • Required: 1.3: 1(a); 1.2: Let AB be a diameter of a circle and P be a point on the same circle, other than A or B. Prove that AP is perpendicular to PB, and the angle PAB is equal to the angle between BP and the tangent to the circle at P.
    • Here you will find the solutions to HW 2.
• Chapter II. The most important geometric concepts of this chapter are Parallel Projections and Affine transformations. All geometric properties of affine transformations can be described by those of parallel projections, because Each parallel projection is an affine transformation (Theorem 5 in Section 2.2 ) and Any affine transformation can be expressed as the composite of two parallel projections (Theorem 6 of Section 2.2). The most important properties of affine transformations are the three listed on p. 57. With the Fundamental Theorem of Affine Geometry (on p.69), we can use an affine transformation to transform a given geometric problem into a more special one, where perhaps one can understand the situation more easily. All of Section 2.4 are about the application of this idea to geometric problems involving rectilinear figures. Section 2.5 contains applications of this idea to conic sections. For Section 2.5, we will only discuss Theorems 1-4 and the Corollary on p. 86 (and perhaps Theorem 7).
  • Sections 2.1, 2.2, due: Monday, Sept. 26.
    • Recommended: 2.1 3, 4, 5, 6, 7; 2.2 1, 2, 3.
    • Required: 2.1 2,5; 2.2 3, 4(a).
    • Here you will find the solutions to HW 3.
  • Sections 2.3, 2.4, due: Oct. 3
    • Recommended: 2.3: 2, 4, 5; 2.4: 1, 2, 4, 6.
    • Required: 2.3: 2(a), 4; 2.4: 4, 5.
    • Here you will find the solutions to HW 4.
  • Sections 2.4, 2.5, due: Oct. 10 (Some problem of 3.2 also due at the same date, see below).
    • Recommended: 2.4: 1, 2, 4, 6; 2.5: 1.
    • Required: 2.4: 6; 2.5: 1.
• Chapter III. The algebraic manipulations in the text becomes a bit more difficult, although the exercises do not involve much complicated algebra. The more difficult issue is perhaps the constant translation between a geometric concept (there are many new concepts in this chapter) and its algebraic representation. It is important not to get buried in the computations and forget about the geometry behind. Section 3.1, especially pp. 98--101, are worthy of repeated reading. In order not to be overcome by the algebra and abstract conception, it may be worth while to first read and understand the geometric meaning of the following theorems: Theorems 1 and 2 of 3.2 on p.106, and Theorem 3 of 3.2 on p.110. The really important geometric properties for us are refleted in Theorem 2 of 3.3 on p.119 and Theorem 4 of 3.3 on p.127. The only quantitative invariant of projective geometry is introduced through Theorems 3 and 4 of 3.5 on pp.140-141. I have writen up some notes on projective geometry. Hopefully they complement the discussion in the text.
  • Section 3.2, due Oct. 10.
    • Recommended: 1, 2, 3, 4, 5, 6;
    • Required: 2(a), 3(a).
    • Here is the solution to this set of homework.
    • The first midterm is scheduled to be on Monday, Oct. 17. It will cover the material up to and including 3.2. Here are some practice problems.
  • Sections 3.2 and 3.3, due Oct. 24.
    • Recommended: Section 3.2: 8,9,12; Section 3.3: 1, 2, 4, 5, 7.
    • Required: Section 3.2: 4(a), 5; Section 3.3: 3, 4.
  • Here is the solution to the first midterm.
  • Sections 3.4 and 3.5, due Oct. 31 ---For 3.5, we will only discuss the applications of Theoresm 3 and 4 on pp.140-141 through their applications in 3.5.2, 3.5.3. Read 3.5.2, 3.5.3 on pp.143--146 carefully.
    • Recommended: Section 3.4:1, 2; Section 3.5: 4, 5, 6, 7, 8;
    • Required: Section 3.4: 2; Section 3.5: 4, 5; From class: Suppose that A, B, C, and A', B', C' are collinear, respectively; and that BC' is parallel to B'C, that AC' is parallel to A'C. Prove that AB' is parallel to A'B. (Hint: You may use either the synthetic method or the coordinate method. If you use the synthetic method, be aware that the lines ABC and A'B'C' may intersect or be parallel, so you need to cover both cases; if you use the coordinate method, you may apply an affine transformation to assume that C=(0,1), C'=(0,0), B'=(1,0), and A'=(a,0) for some a. You then need to set up an equation for the line ABC.)
• Chapter VII. We will cover 7.1, 7.2, 7.3.1, 7.3.3.
  • Sections 7.1, 7.2, Due Nov. 7
    • Recommended: Section 7.1: 1, 2, 3, 4; Section 7.2: 1, 4, 5, 6.
    • Required: Section 7.1: 2, 3; Section 7.2: 1, 3.
    • Here is the solution to Homework 7, and Here, together with additional solution, are the solution to Homework 8.
• Chapter VII. Section 7.3 and Euclidean Geometry
  • Download the homework due for Nov. 14 here.
  • Here is the solution to the homework assignment above. Reminder: our second midterm is scheduled on Monday, Nov. 21. Final Exam Schedule: According to university schedule, our final exam will be from 8AM---11AM, Wednesday, Dec. 21.
  • Here is the solution to the second midterm, and here is the homework assignment due Monday, Dec. 5. And here is the solution to the Dec. 5 homework.
  • Here are some practice problems for the final exam.
  • Here are some notes on non-Euclidean geometry.

  We are now moving into the examination of some issues in Euclidean geometry, in particular, in relation to spherical and other geometries. I want to caution you that this is not a systematic study of plane Euclidean geometry---we only have time to examine selectively some issues in Euclidean geometry. Our emphasis in this part of the course is the synthetic, or axiomatic method. In our earlier discussions, we have freely used some properties from Euclidean geometry. In this part of the course, we need to understand the similarities and differences between the axiomatic approach and the analytic approach that we have been mostly using so far. The most visible difference is the way they define the basic objects of study: points, lines, circles, etc. You need to look at the different approaches from different perspectives to understand, appreciate, and apply both approaches well. You should re-read the handout distributed at the beginning of the semester.

  In the axiomatic approach, you will need to understand the structure and logic of the axiomatic method, and understand the logical dependence relations between different axioms and propositions, and learn to use the axiomatic method to construct proofs in simple settings. For this purpose, I am asking students to sign up for specific propositions to report in class. Your report should consists of three parts: (i) explanation of the assumptions and conclusions of the propositions; (ii) proof of the propositions, either by following Euclid, or by your own method; (iii) an analysis of how the propositions depend on other propositions (i.e., what properties you used in your proof). You should plan to complete the reporting in 5 minutes for each proposition.

  I have divided the propositions to be reported into groups (mostly with a common theme), and encourage you to sign up in groups of two to three students to work on the preparation of your group of propositions. Here they are

  • I. 11--12 (construction of perpendicular lines)
  • I. 13--15 (angle relations between two intersecting lines)
  • I. 16--17 (angle relations in a triangle)
  • I. 18--20 (angle/side inequalities in a triangle): already signed out.
  • I. 33--34 ( properties of parallelograms)
  • I. 35--36 (area equalities for parallelograms)
  • I. 37--38 ( area equalities for triangles )
  • I. 39--40 ( area equalities for triangles )
  • more to come