PMath 360
10
Set 10
Train, Necklace and Porism
10.1
Train of circles
10.1.1
Foundation line
Let L be any line. A horizontal line will do. Hide the point(s) used to dene
L. Let P1 and P2 be any two, new, distinct points on L. Let and L1 and L2
be lines per

PMATH 360 Geometry, Assignment 3
1: (a) Let u =
(b) Let u =
(c) Let u =
1 (1, 1, 2). Express the isometry Fu in matrix form.
6
1
3 (1, 2, 2) and let = 2 . Express the isometry Ru, in
1 (1, 1, 1), = , v = 1 (2, 1, 1) and x = 1 (1, 0, 1).
3
3
6
2
1 (1, 1, 0

PMath 360
1 of 3
Set 91
More reection and inversion
9.1
Poincar reection
e
Let be the circle that bounds a Poincar model of the hyperbolic plane.
e
Let O be the centre of . Let A and B be two P-points in this hyperbolic
plane. Suppose A, B, and O are not

PMath 360
Set 20
Set 201
Some Inversions
20.1
Two tangent circles
Let C1 and C2 be two distinct circles, one inside the other, and tangent to
each other at a common point t.
20.1.1
A circle for inversion
Find a circle centred at t that both circles C1 and

PMath 360
P. 1 of 7
Set 81
Pencils of circles
Poincar model of Hyperbolic Plane
e
This set of questions is intended to give you experience with the three kinds
of pencils2 of circles. In 8.2 you will be studying two pencils of circles in
which each circle

PMath 360
Set 7
P. 1 of 6
Set 7
Constructing inverses1
7.1
7.1.1
A construction using an orthogonal circle
The steps of the construction
Let O and R be distinct points and let = circle (O, R), the circle with
centre O and radius point R. Construct the inv

PMath 360
6
Set 6
The Pascal Conguration
An Exploration
Use the GeoGebra tool for a conic through 5 Points to construct a conic determined by any ve distinct points, with no 3 collinear. It is suggested that to
avoid congestion and confusion later, you ei

PMath 360
Set 5
Set 5 : Conics, Poles and Polars,
a guided exploration1
5.1
A particular conic
Let be the conic whose Cartesian equation is 2x2 + 3y 2 + 2x 4y = 0.
1. Find the 3 by 3 symmetric matrix M that represents .
Solution: We must nd the matrix M s

PMath 360
Set 4
Set 4: Functions and Harmonic sets
4.1
Functions of the form (ax + b)/(cx + d)
The purpose of this exercise is to complete a proof that the cross ratio
(x1 , x2 ; x3 , x4 ) is invariant under the map f dened by
f : x (ax + b)/(cx + d)
(1)

PMATH 360 Geometry, Assignment 5
Due Mon July 25
1: (a) Let C be the circle in R2 with diameter from a = (1, 4) to b = (3, 2), and let D be
the circle in R2 with diameter from c = (4, 2) to d = (3, 3). Find the (Euclidean) area of
the image of D under the

PMATH 360 Geometry, Assignment 4
Due Tues July 12
1: (a) Let x = (2, 1, 3) and y = (5, 1, 4). Find dP [x], [y] .
(b) What fraction of the area of P2 is covered by a triangle with angles 3 , 3 and 2 ?
(c) Find the area of the circle on P2 which is circumsc

PMATH 360 Geometry, Assignment 1
Due Thurs May 19
1: (a) Let u = (1, 1, 2), v = (2, 1, 3) and x = (4, 1, 1). Find ProjU (x) where U = Spancfw_u, v.
(b) Let a = (2, 1, 3), b = (1, 2, 1), u = (1, 3, 2) and v = (2, 0, 1). Find the distance between
the line x

PMath 360
9
Set 9
p.1/3
More reection and inversion
9.1
Poincar reection
e
Let be the circle that bounds a Poincar model of the hyperbolic plane.
e
Let O be the centre of . Let A and B be two P-points in this hyperbolic
plane. Suppose A, B, and O are not

PMath 360
p.1/8
Set 81
Pencils of circles
Poincar model of Hyperbolic Plane
e
This set of questions is intended to give you experience with the three kinds
of pencils2 of circles. In 8.2 you will be studying two pencils of circles in
which each circle in

PMath 360, Set 7
p.1/7
Set 7
Constructing inverses1
7.1
Using an orthogonal circle
7.1.1
The steps of the construction
Let O and R be distinct points and let = circle (O, R), the circle with
centre O and radius point R. Construct the inverse of P with res

PMath 360
4
Set 4
Conics, poles and polars and Pascals Theorem
4.1
(*) A particular conic, some poles and polars.
Let be the conic whose homogeneous equation is
16x1 2 4x2 2 10x1 x3 6x2 x3 = 0.
1. Write the 3 by 3 symmetric matrix A that represents .
2. F

PMath 360
2
Set 2
Set 2
2.1
Desargues Theorem and Duality
Usually, in constructing a Desargues conguration, people think of following
an outline that goes something like this:
1. Start with a point V.
2. Pick three lines x, y, and z on the line V
3. Selec

PMATH 360 Geometry, Assignment 2
Due Thurs June 2
1: (a) Let u = 15 (2, 0, 1), v = (2, 1, 4) and w = (1, 3, 2). Find the oriented angle (v, w)
from v to w in the tangent space Tu .
(b) Let u = 16 (1, 1, 2), v = 114 (2, 1, 3) and w = 111 (1, 3, 1). Find uv

PMath 360
Set 3
Set 3
3.1
The Far O Point (Desargues)
Imagine that you are given a page and on that page is a point and two
non-intersecting lines. Imagine further that you have a ruler without markings and a writing surface to work on, but it is not much

PMath 360
Set 2
Set 2
2.1
Desargues Conguration and Duality
To construct a Desargues conguration one may follow this sequence of steps:
1. Start with a point V.
2. Pick any three distinct lines x, y, and z on the point V.
3. Select two distinct points on

PMath 360
11
Set 11
Page 1 of 2
Plane Sphere
In this section, let S be the sphere in R3 with centre (0,0,0) and radius 1. Let
N : (0, 0, 1) S be the point of projection on S so that a point z = x + iy
in the complex plane corresponds to a point (u, v, w)

PMath 360
10
Set 10
Page 1 of 6
Train, Necklace and Porism
10.1
Train of circles
10.1.1
Foundation line
Let L be any line. A horizontal line will do. Hide the point(s) used to dene
L. Let P1 and P2 be any two, new, distinct points on L. Let and L1 and L2

PMath 360
7
7.3
7.3.1
Set 7 b
3 of 6
Constructing Inverses Part b
A self inverse construction
The best construction (?).
On a new gure, again let be a circle with centre O, and let P be any point
distinct from O. Construct Q as follows.
L1 = line ( O,
L2

PMath 360
5
Set 5
Conics, poles and polars
5.1
(*) A particular conic
Let be the conic whose homogeneous equation is
16x1 2 4x2 2 10x1 x3 6x2 x3 = 0.
1. Write the 3 by 3 symmetric matrix A that represents .
2. For S : s = (1, 2, 0) and Q : q = (1, 2, 1),

PMath 360
8
Set 8
1 of 6
Pencils of circles
Poincar model of Hyperbolic Plane
e
This set of questions is intended to give you experience with the three types of
pencils (families) of circles. There are also examples of mutually orthogonal
pencils of circl

PMath 360
7
Set 7 a
1 of 2
Constructing Inverses Part a
7.1
The inverse of a point
7.1.1
Construction using an orthogonal circle
Let O and R be distinct points and be a circle with centre O and radius
point R. Let P be any point distinct from O.
Consider

PMath 360
4
4.1
Set 4
Functions and Harmonic sets
Functions of the form (ax + b)/(cx + d)
The purpose of this exercise is to complete the proof that the cross ratio
(x1 , x2 ; x3 , x4 ) is invariant under the map f dened by
f : x (ax + b)/(cx + d)
(1)
whe

PMath 360
6
Set 6
1 of 3
The Pascal Conguration:
An Exploration
Use the GeoGebra tool called Conic through 5 points to construct a conic
named determined by ve distinct points, no 3 collinear. It is suggested
that to avoid a gure that goes o the page, you