Isometries
composed by K. E. Feldman
1
Introduction
One of the beauties of mathematics is that though mathematics consists of very abstract notions and follows very strict rules, it is an experimental science and it does describe various phenomena of natu
MAT 360: SOLUTIONS REVIEW FOR MIDTERM; SPRING
2017
In this review sheet (and on the midterm) a model shall always consist
of a set of points P, a set of lines L such that each line l L is just a subset
of P, and the incidence relation whereby a point p P
HW 5 FOR MAT 360: SPRING 2017
You may hand in HW 5 during the lecture on Thursday 3/2.
(1) In this problem we assume that we have a model for a geometry (P=points,
L=lines, where the lines are just subsets of points) and this model satisfies
the 3 inciden
HW 6 FOR MAT 360: SPRING 2017
You may hand in HW 6 during the lecture on Thursday 3/9.
(1) Let A,B,C denote the vertices of a triangle ABC. For any two points
P,Q let lP,Q denote the line containing P,Q. Let SC denote the side of lA,B
which contains C; le
MAT 360: REVIEW FOR MIDTERM; SPRING 2017
Our midterm is scheduled for Tuesday March 21, in our usual classroom
at our usual class time. This exam will cover materials contained in the first
3 chapters of the text. The exam will contain 4 or 5 problems sim
HW 3 FOR MAT 360: SPRING 2017
You may hand in HW 3 during lecture any of the lectures of the week
2/13-2/17.
(1) Let P denote a set of points, let L denote a set of lines, and let
I P L denote an incidence relation. Assume that P, L and I satisfy
the Inci
HW 6 FOR MAT 360: SPRING 2017
Please hand in HW 8 during the lecture on Thursday 3/30.
(1) Do exercise #3 at end of chapter 4 (i.e. proof of Proposition 4.1).
(2) Do exercise #4 at end of chapter 4 (i.e. proof of Proposition 4.2).
(3) An angle is obtuse i
HW 6 FOR MAT 360: SPRING 2017
You may hand in HW 7 during the lecture on Thursday 3/23.
(1) It is not necessary to hand in this HW problem. Do exercise
#31 in Chapter 3.
(2) Do problem #32 in Chapter 3. (Hint: a hint is given in the textbook;
also, you mi
Isometries
1. Isometries versus moves
Isometries
In this course, the notion of move is initial and undefinable. The
notion of congruent figures was introduced in terms of moves: two figures
are called congruent if there exists a move mapping one of them t
Stony Brook University
Mathematics Department
Oleg Viro
Geometric Structures
MAT 360
February 8, 2014
Homework 1
Problems 63, 66, 72, 76, 103 from the textbook.
Stony Brook University
Mathematics Department
Oleg Viro
Geometric Structures
MAT 360
February 16, 2014
Homework 2
Problems 90, 91, 92, 95, 110, 111 from the textbook.
Stony Brook University
Mathematics Department
Oleg Viro
due on February 27
Geometric Structures
MAT 360
February 21, 2014
Homework 3
Problems 130, 136, 138, 140, 152 from the textbook.
SIMILARITY
BASED ON NOTES BY OLEG VIRO, REVISED BY OLGA PLAMENEVSKAYA
Euclidean Geometry can be described as a study of the properties of geometric
gures, but not all kinds of conceivable properties. Only the properties which do
not change under isometrie
Isometries.
Congruence mappings as isometries. The notion of isometry is a general notion
commonly accepted in mathematics. It means a mapping which preserves distances.
The word metric is a synonym to the word distance. We will study isometries of the
pl
Circle Inversions and Applications to Euclidean Geometry
Kenji Kozai Shlomo Libeskind
January 9, 2009
Contents
1
Chapter 0 Introduction
We have seen that reections and half turns are their own inverses, that is 1 2 Ml1 = Ml and HO = HO , or equivalently,
1. Inversions.
1.1. Denitions of inversion. Inversion is a kind of symmetry about
a circle. It is dened as follows. The inversion of degree R2 centered at
a point O maps a point A = O to the point B on the ray OA such that
R is the geometric mean of OA an
SOLUITONS FOR MIDTERM FOR MAT 360: SPRING 2017
(1) Let P, L denote an affine plane; that is, this model satisfies the 3 incident
axioms and also satisfies Euclids parallel axiom. (Euclids parallel axiom
states that for any line l and any point p not on l,