Table of Contents
Module 8
The underlying structure will be
Felix Klein wrote that
We’ll be starting with a whole lot of new
definitions and we’ll be in Euclidean geometry initially. This way of presenting
geometry has ties to modern algebra and to computer science.
Superposition
The issue is:
It is utterly necessary that we use congruence as a
concept…many of our proofs rely on finding or creating congruent angles, lengths, or
triangles. Congruence by moving one object over another or near requires that the
“moving” be precisely defined.
Invariant
Examples of invariants in mathematics:
Examples of properties that are NOT invariant:
For the background vocabulary and structure,
we’ll need definitions for:
isometries in the plane
Defining a geometry
Getting to “Transformation”
The set A in the preceding definition is called the
domain of f and the set B is called the codomain of f.
A mapping f: A ?? B is called
Familiar graphs of functions:
In keeping with the notion of saying why do we
care…there are functions that are not 1:1 or not onto or not both.
A function that is not onto and not 1:1:
thus 1/2 has no preimage (not onto)
1:1 function that is not onto:
An onto function that is not 1:1 and not from a set
to itself:
Transformation
example
Well, take (p, s) and (p, r). If s ? r, it’s
not a function.
checking onto:
Unless you know a list of mathematical instructions
is a transformation you have to check that it is a function that is 1:1 and onto on your
own.
3 ? 14
Fixed Points
To find the fixed point, use the definition:
When you have a transformation in two dimensions
you are using the set
Some caution is required when checking for fixed
points of transformations in
Looking for a fixed point, t(x ) = x and t(y) = y,
use:
This function leaves (1,1) alone, possibly all the
other points move around.
Example 2, page 133, our text
What is happening?
Let’s do exercise 2, p134: check for a fixed
point:
Checking this:
Stable Sets
Definition
Composition
Definition
example
Theorem
Identity Transformation
hint
The identity transformation is the transformation,
I, on a set S such that
The identity function:
Inverse Transformation
The characteristic behavior of inverse functions is
that composition with them results in the identity.
A Group
3. There are inverses
Closure means that using the operation on set
elements results in a set element.
Identity element means that there has to be a set
element that functions just like the definition of identity. The identity element is
unique.
Inverses means that for each set element there is a
unique companion set element such that composition in either order results in the identity
element.
Associativity means that the order of combining
three set elements is irrelevant.
A Group
T = the set of all transformations from a given set
to itself
2. Is there an identity?
3. Are there inverses? I’ve asserted it, and
we’ll see them soon.
Notebook Problems
Hint on stable lines
In Class work
Start first with (0,0)
Up to now, we’ve just talked abstractly about
transformations. The ones we’ll focus on are
There is one more adjective, a very restrictive
adjective, that we’ll apply to these important geometric transformations:
En Anglais, naturellement,
Isometries
Definition
As often happens, the definition also includes a
way to check or test if a transformation is an isometry.
Looking at the rotation from the preceding section:
(0,0) ? (2,2)
((2-x2) - (2-x1))2 =
Therefore, this transformation is an isometry.
Plane Motions
Each of these is an isometry.
2. Angle measure is invariant under an
Definitions
Usually symbolized “R(P,a)” a rotation of
a (with the ccw motion understood to be positive) about the point P.
The collection of all rotations about a given point
P is a group:
The product of two rotations is a rotation.
An isometry, t, is called a translation iff for all
points P and Q, the points P, Q, T(P) and T(Q) would form a parallelogram if joined
appropriately.
On an intuitive level, a translation is a
correspondence between domain points and image points that moves each point in the domain
the same distance and in the same direction.
A translation has no fixed points. The stable lines
of a translation are parallel to the direction of motion.
Theorem:
The collection of a translations is a group:
The inverse of a reflection is the same reflection
(a condition known as “involutory” or self-inverse).
The identity glide reflection is the composition of
the identities of the reflection and translation involved so that each point is it’s
own image.
At last
PPT Slide
Show that it’s inverse is an isometry which he
does by selecting image points for two arbitrary points and using the definition of
inverse:
Notebook problems
Theorem
Proof
If F(D) = D, then F is the identity isometry. Why?
Because we can, let’s construct three circles,
each centered at one of our original noncollinear points A, B, and C.
Since A ? B, the circles intersect at two points at
most, one of which is D.
Suppose there’s exactly two intersections D
and E and F(D) = E. Then AB is the perpendicular bisector of DE.
This means that both D and E are the same distance
from any point on the extended line AB.
Since C isn’t on the line AB (from the
hypothesis), d(C,E) isn’t the radius length d(C,D). This says that E can’t be
F(D).
Therefore, there’s only one point of
intersection of the circles centered at A and B: F(D) = D.
Theorem
Theorem
This is a modern realization. There is only one
motion…the others come about by composing it cleverly.
The Results
Product of three reflections:
If two lines of reflection intersect at a point on
the third, then the motion is a rotation.
Uses of these motions
Theorem
Proof:
illustration
Since T is an isometry, side AC is parallel to BC,
thus m?ACB = m? CBC’.
Substituting
Recasting statements
Using transformational geometry:
Congruence
Theorem
Isosceles Triangles
Proof:
Consider the reflection, r, of DABP about the line
AP.
Since r is an isometry, AB ? AX
Theorem
Proof
There are two types of isometries that map A onto D
and B onto E: one is direct and the other is opposite. The direct isometry finishes the
proof since, then, one triangle is the isometric image of the other.
Let P be the isometry that maps C to C’ with
C’ in the half plane opposite F. Since P is an isometry, angle measures are invariant
and all corresponding angles are congruent. However, P is an orientation reversing motion.
Let rDE be a reflection about DE, which will then
be a fixed line for the reflection.
Test 3 suggestions
Know the definition of isometry and be able to come
up with examples of planar motions that are and are not examples.
Be able to identify some invariant properties of
geometric objects:
Know the proof of “the composition of two
transformations is a transformation”.
Know the proof of “A Euclidean plane isometry
is determined by what it does to three noncollinear points.”
Be able to rewrite an isometry as a composition of
reflections.
Some sample problems:
Illustration
Construct the bisector of angle BAE and call it ray
AP with the point P at the intersection of the triangle side BE.
Since the reflection is an isometry, AX ? AD and AD
? AC, by hypothesis.
Under the reflection, D goes to C and we do know
that angle ADE is congruent to angle ACB. So the image of E is on CB as well as AB. Thus
the image of E is B since B is the only point that satisfies both criteria.
This means that the image of triangle ADE under the
isometry is triangle ACB, which means that the triangles are congruent.
Things to notice:
Problem Two
Problem Three
Problem Four |