Some Reading Guides to "Geometry (2nd edition)" by Brannan et al, and Homework
Assignments
Some Reading Guides to "Geometry (2nd edition)" by Brannan et al, and Homework
Assignments
The explanations in the text are fairly detailed. So you are expected to
study the text on your own, before and after the lectures. I will only have time to discuss the
more difficult and subtle 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 to me either in or before class. This way, our discussion in the lecture time will be more focused.
The problems to be assigned are grouped into 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---but
you should try to work out these problems on your own without consulting the solutions; you may be fooled into
thinking that you understand the solutions easily
by just reading through the solutions, yet may retain very little
of the ideas and techniques behind the solutions unless you work through the problems on your own and
learn from the mistakes and setbacks.
The required ones
are listed in the last section at the end of each chapter, labeled as Exercises,
and are collected for grading. The exams will have problems similar to both the
recommended and
required ones.
- Chapter I.
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.
- Some aspects of Linear Algebra are used regularly in this course, starting from the very beginning.
I have provided some review problems on our sakai site.
- Sections 1.1, 1.2, due: Friday, Sept. 11.
- Recommended: 1.1: 3,6,7,8,9; 1.2: 2,4,7,8;
- Required: 1.1: 2, 4, 5, 6(b), (c)
- Sections 1.2, due: Friday, Sept. 18.
- Recommended: 1.2: 2,4,7,8.
- Required: 1.2: 3, 4, 5, 6
- Section 1.3. due: Friday, Sept. 25.
- Recommended: 1.3: 1,2,3.
- Required: 1.3: 1(a),(c), 2(a),(c)
- 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. 73. With the Fundamental Theorem of Affine Geometry
(on p.88), 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.111 (and perhaps Theorem 7).
- Sections 2.1, due: Friday, Oct. 2.
- Recommended: 2.1: 3, 4, 5, 6, 7;
- Required: 2.1: 2,3,4(b), 5;
- The first midterm is scheduled to be on Friday Oct. 9. It will cover the material up
to and including 2.3. Here
are some practice problems. The following
formula sheet
will be made available on the exam.
- Section 2.2: (This set from 2.2 and next set 2.3 will not be collected
for grading, but may be tested on the first midterm).
- Recommended: 1, 2, 3.
- Required: 3, 4(a).
- Section 2.3
- Recommended: 2.3: 2, 4, 5 .
- Required: 2.3: 2(a), 4.
- Section 2.3, 2.4.1-2, due: Friday, Oct. 16.
- Recommended: 2.3: 2, 4, 5 ; 2.4: 1, 2, 4, 6.
- Required: 2.3: 5, 6; 2.4: 2, 5.
- Section 2.4.1-2; 2.5.1 (up to p.111), due: Friday, Oct. 23.
- Recommended: 2.4: 1, 2, 4, 6; 2.5: 1.
- Required: 2.4: 4, 6, 7; 2.5: 2.
- 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. 131--134,
and Section 3.2.3
are worthy of repeated reading. In order not to be overcome by the algebra and abstract conception,
it is worth while to first read and understand the geometric meaning of the following theorems:
Theorems 1 and 2 of 3.2 on p.141--142, and Theorem 3 of 3.2 on p.146.
The text provides plenty of details for the algebraic approach in 3.3, which will be the
basis for computations in computer graphics, but in our course we should draw attention to two
important geometric theorems of 3.3 : Theorem 2 of on p.157 and
Theorem 3 of on p.162.
The only quantitative invariant of projective geometry is
introduced through Theorems 3 and 4 of 3.5 on pp.184. We will explain their applications
in 3.5.2-3.
I have writen up some notes
on projective geometry. Hopefully they complement the discussion in the text. You are strongly
adviced to read at least the Introductory Remarks in the notes before our first lecture on this chapter.
- Sections 3.2 and 3.3 (geometric discussion of Theorems 2, 3 only), due Oct. 30.
- Recommended: Section 3.2: 1, 2, 3, 4, 5, 6, 8, 9, 12; Section 3.3: 1, 2, 5.
- Required: Section 3.2: 2(a), 3(a), 5; Section 3.3: 3, 4.
- Section 3.5.2-3, due Nov. 6 .
We will only discuss the
applications of Theorems 3 and 4 on pp.184 through their applications in 3.5.2, 3.5.3.
Read 3.5.2, 3.5.3 on pp.187--191 carefully.
- Chapter VII. We will cover 7.1, 7.2, 7.3.1, 7.3.3, and 7.3.4.
- Sections 7.1, 7.2, Due Nov. 13.
- Recommended: Section 7.1: 1, 2, 3, 4; Section 7.2: 1, 3, 4, 5, 8, 9.
- Required: Section 7.1: 2, 3; Section 7.2: 1, 3.
- Second midterm is scheduled on Tuesday, Nov. 17
- Section 7.3.1 (Theorems 2 and 3), 7.3.3 (Theorems 5, 6, 7), and 7.3.4 (Theorem 8), Due Nov.
25.
- Recommended: 2, 5, 6, 7, 8;
- Required: 2(a)-(b) (Hint: construct a median
from one vertex and use Theorem 3 and one of the rules in Theorem 7 of 7.3.3), 3, 5, 6.
- Euclidean and non-Euclidean Geometry. In the remainder of the semester,
our focus will be shifted to examining a few central issues in classical Euclidean Geometry
and how the study of Euclid's Parallel
Postulate led to the birth of a non-Euclidean geometry, called hyperbolic geometry today.
I have written up some notes
to help you put things into perspective. You should download these notes and study them before
our discussions. For the lecture materials, we will be using Euclid's "Elements",
Book I,
and the following
notes on non-Euclidean geometry
that I prepared; chapter 6 of the text presents more technical aspects of hyperbolic geometry.
- Euclidean Geometry
- Required: Exercises 2, 3, 4, 5 in
the notes. Due December 4.
- non-Euclidean geometry