Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
UNIVERSITY OF TORONTO SCARBOROUGH
Computer & Mathematical Sciences
MAT D02S
Winter 2014
Problem Set V
Due: Thurs., April 4th at beginning of class
1. (page 164; #7.79) (originally Question 6 from Prob
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
Axioms for Plane Euclidean and Hyperbolic Geometry
POSTULATED CONCEPTS
Points (a set)
Lines (a set)
Between (a relation)
Congruence (a relation written )
On (or Through; a relation)
AXIOMS
Incidence A
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
TERM TEST
MATD02F Classical Plane Geometries & their Transformations
Examiner: P. Selick
Date: October 19, 2012
Time: 1
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
Geometry
1. Axioms for Plane Euclidean and Hyperbolic Geometry
POSTULATED CONCEPTS
Points (a set)
Lines (a set)
Between (a relation)
Congruence (a relation written )
On (or Through; a relation)
AXIOMS
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
MATD34 Midterm Practice Questions
Winter 2015
(1) Show that if f (z) is analytic near z = 0 and has a zero of order
n > 0 at z = 0 then there is an analytic function g so that
f (z) = g(z)n near z = 0
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
Midterm Review MATD34
(1) Material from Assignment 1:
(a) Topology: open, closed, connected, compact, Hausdorff (as in MATB43)
(b) Definition of Riemann surface:
A Riemann surface is a topological spa
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
University of Toronto at Scarborough
Division of Physical Sciences, Mathematics
MAT C34F
2002/03
Complex Variables
Instructor:
Prof. L. Jeffrey
Office: S504C
Telephone: (416)2877265
Email: [email protected]
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
MATD34 Midterm Practice Questions  Solutions
Winter 2015
(1) Show that if f (z) is analytic near z = 0 and has a zero of order
n > 0 at z = 0 then there is an analytic function g so that
f (z) = g(z)
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
MAT C34F
2013/14
Complex Variables
Instructor:
Prof. L. Jeffrey
Office: IC474
Telephone: (416)2877265
Email: jef
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
Dept. of Computer & Mathematical Sciences
UNIVERSITY OF TORONTO SCARBOROUGH
MAT D02
Solution Sketch: Midterm Exam
1.
a) A theorem says that given a line L and a point A not on L, there are two rays
r
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
FINAL EXAMINATION
MATD02F Classical Plane Geometries and their Transformations
Examiner: P. Selick
Date: December 7, 20
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
Dept. of Computer and Mathematical Sciences
University of Toronto at Scarborough
MAT D02S
201314
General Information
Lectures:
Tuesday 23 pm
Thursday 24 pm
Room IC320
Room IC320
Lecturer:
P. Selick
O
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
UNIVERSITY OF TORONTO SCARBOROUGH
Computer & Mathematical Sciences
MAT D02S
Winter 2014
Problem Set IV
Due: Thurs., March 21st at beginning of class
1. (page 144; #7.30)
2. Find a fractional linear tr
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
UNIVERSITY OF TORONTO SCARBOROUGH
Computer & Mathematical Sciences
MAT D02S
Winter 2014
Problem Set III
Due: Thurs., March 4th at beginning of class
1. In Euclidean Geometry, consider the triangle ABC
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
Department of Computer & Mathematical Sciences
University of Toronto Scarborough
MAT D02S
Winter 2014
Solution #5
1. Let F3 = # of triangular faces and let F4 = # of squarefaces
Then
2E = 4V
3F3 = V
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
DEPT. OF COMPUTER & MATHEMATICAL SCIENCES
UNIVERSITY OF TORONTO SCARBOROUGH
MAT D02S
Problem Set I
Due: Thurs., Jan. 23 at beginning of class
1. (page 133; #7.2)
2. We know that in the Poincare upper
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
Department of Computer & Mathematical Sciences
University of Toronto Scarborough
MAT D02S
Solution #2
1. AB = AC so ABC is isosceles and ABC = ACB.
BD, CE are angle bisectors, so DBC = 21 ACB.
The
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
Department of Computer & Mathematical Sciences
University of Toronto Scarborough
MAT D02S
Winter 2014
Solution #4
1. Let A , B , C , D be the images of A, B, C, D under inversion in a circle of
radius
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
Department of Computer & Mathematical Sciences
University of Toronto Scarborough
MAT D02S
Winter 2014
Solution #3
1. (a) G is the centroid so A G = 21 AG = 12 2 = 1
(b) Set x = GD = B D. As in
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
Department of Computer & Mathematical Sciences
University of Toronto at Scarborough
MAT D02S
Solution #1
1. Parameterize by (t) = (2, t)
s=
2. .
Z
2
1/7
2 t 1/7.
2
1 1
0 + 12 dt = ln t1/7 = ln 2 ln(1/
Classical Plane Geometries and their Transformations
MATH MATD02

Winter 2014
Riemann Surfaces
Dr C. Teleman1
Lent Term 2003
1
Originally LATEXed by James Lingard please send all comments and corrections to
[email protected]
Lecture 1
What are Riemann surfaces?
1.1 Proble