Mathematics 130
Isometries of Euclidean Plane Geometry
Last modified: January 20, 2009
(These notes omit many proofs that can be found in Ryan, Chapter 1)
Let d(X, Y) denote the distance between two points X and Y of the Euclidean plane E2. An
isometry of
Studyguide, Exam
Study homework assignments, lecture notes. Be prepared to give examples, and answer true/false questions. There will also be some proofs on
the exam. I tried to indicate what you should know how to prove in this
studyguide.
Picking up wi
Mathematics 130
A Brief History of Geometry
Last modified: January 20, 2009
Thales (640-546 B.C.)
abstraction of points and lines
Euclid (about 300 B.C.)
writes the "Halliday and Resnick" of geometry texts
Postulates (as worded by Ryan's teacher H. S. M.
Mathematics 130
Galilean Geometry
Last revised: January 20, 2009
Reading: Yaglom, pages 15-25 (especially 23-25) and 33-53.
As a model for Galilean geometry, consider uniform motions along the y axis, described by
y = kt + y0 (k is the velocity, y 0 the i
Mathematics 130
Faculty Senate Affine Geometry
Last revised: January 20, 2009
Reference: Bennett: Affine and Projective Geometry (on reserve in Cabot)
To save time in proving results about inverses and identities, we start with an independent set of
axiom
Mathematics 130
Linear Algebra and Affine Geometry
Last Modified: January 20, 2009
A collineation is a bijection (1-to-1 onto mapping) T: E2 -> E2 that carries lines into lines: points P,
Q, R are collinear if and only if their images TP, TQ, TR are colli
Mathematics 130
Measuring Angles and Distances in Three Different Ways
Last modified: January 20, 2009
We want to measure the angle between the x axis and the line y = mx. This is defined as twice
the area of the region between the two lines bounded by
2
Mathematics 130
Projective Geometry
Last revised: January 20, 2009
In projective geometry, there are no parallel lines.
From any statement of projective geometry, we can form a dual statement by interchanging the
terms "instructor" and "committee" ("point
Mathematics 130
Hyperbolic Geometry
Last revised: January 20, 2009
Cross products and polarities
Ryan introduces a new cross product x b and then comments that it will be denoted simply by x "if
b is clear from the context." This makes it difficult to con
Mathematics 130
Polarities and Symmetric Bilinear Functions
Last revised: January 20, 2009
Preliminary - inverting a matrix A using cross products
Suppose A has columns u, v, and w, which we can think of as instructors.
Form the cross products = v x w, =
Studyguide for hyperbolic geometry section of the nal
Know about: standard bilinear form on R3 , funny bilinear form b : R3 R3
R that gives rise to H2 , funny cross product that comes from the funny
bilinear form, what geometry do we get if we use b(x, y