§15.1 Euclid’s Superposition Proof and
Plane Transformations.
The student will learn:
the basic concepts of
transformations,
isometries and
reflections.
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Relations
Definition. A relation f from the plane Π
into itself is a pairing of points of Π with
certain other points of Π. If (P, P’) is an
ordered pair then P’ is called the image
of P, and P is called the pre-image of P’,
under f.
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Function
Definition. A function, or mapping, is a relation
f for which each point P has a unique image.
1. f is one-to-one: If P  Q then f (P)  f (Q).
2. f is onto: Every point R in the plane has a
preimage under f, that is, there exist a
point S such that f (S) = R.
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Transformation
Definition. If a mapping f: Π → Π (from a
plane Π to itself) is both one-to-one and onto,
then f is called a plane transformation.
Definition. If a transformation maps lines
onto lines, it is called a linear transformation.
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Transformation Example - Play
Let the point (x, y) be mapped to the point
(x’, y’) by
x’ = 2x and y’ = y + 3
Find the images of (1, - 1) and (- 3,4).
(1, - 1) → (2, 2) and (- 3, 4) → (- 6, 7)
Find the preimage of (2, 9).
(x, y) → (2, 9) so 2x = 2 and y + 3 = 9
so x = 1, y = 6.
Is this mapping a transformation?
Yes, it is both one-to-one and onto.
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Inverse Transformation
The inverse mapping of a transformation f,
denoted f -1, is the mapping which
associates Q with P for each pair of points
(P, Q) specified by f. That is, f -1 (Q) = P iff
f (P) = Q.
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Theorem 1. The inverse of a linear
transformation is a linear transformation.
Since f is a linear transformation there exist
three collinear points P, Q, and R so that f (P),
f (Q) and f (R) are collinear. However by
definition f is one-to-one and onto and hence
f -1 is also one-to-one and onto. So
f -1 f (P) = P , f -1 f (Q) = Q, and f -1 f (R) = R
and hence f -1 is also a linear transformation.
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Fixed Points
Definition. A transformation f of the plane is
said to have P as a fixed point iff f (P) = P.
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The Identity
Definition. A transformation of the plane is
called the identity mapping iff every point of
the plane is a fixed point. This transformation
is denoted e.
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Introduction to Line Reflections
Definition. Let l be a fixed line in the
plane. The reflection R (l) in a line l is the
transformation which carries each point
P of the plane into the point P’ of the
plane such that l is the perpendicular
bisector of PP’. The line l is called the
axis (or mirror or axis of symmetry) of
the reflection.
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l
P
P’
Reflection in a line.
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Introduction to Point Reflections
Definition. Let C be a fixed point in the
plane. The reflection R (C) in a point C is
the transformation which carries each
point P of the plane into the point P’ of
the plane such that C is the midpoint of
PP’. The point C is called the center of
the reflection.
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Reflection in a point.
P
C
P’
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Isometry
Definition - A transformation of the plane
that preserves distance is called an isometry.
If P and Q are points in the plane and a
transformation maps them to P’ and Q’
respectively so that PQ = P’Q’ then that
transformation is an isometry.
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Isometry Facts
Lemma 1. An isometry preserves collinearity.
Lemma 2. An isometry preserves betweenness.
If A, B, and C are points then A, B, and C,
are collinear iff AB + BC = AC. This also
means that B is between A and C which is
written A-B-C.
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Lemma. An isometry preserves collinearity.
If A, B, and C are pints then A, B, and C, are collinear
iff m(AB) + m(BC) = m(AC). This also means that B
is between A and C which is written A-B-C.
What do we know?
AB + BC = AC
We have an isometry so
AB = A’B’
BC = B’C’
AC = A’C’
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What do we need to prove?
A’B’ + B’C’ = A’C’
How do we prove this?
YES! By substitution
AB + BC = AC
A’B’ + BC = AC
A’B’ + B’C’ = AC
A’B’ + B’C’ = A’C’
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More on Isometries.
Lemma. An isometry maps a triangle ABC
into a congruent triangle A’B’C’.
Lemma. An isometry preserves angle
measure.
Theorem. The identity map is an isometry.
Proof for homework.
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Theorem 1
Reflections are
A. Angle-measure preserving.
B. Betweeness preserving.
C. Collinearity preserving.
D. Distance preserving.
We need only show distance preservation to
get the other three.
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Proof: Theorem 1 - Reflections are
Distance preserving.
Case where A and B are on same side of line or reflection.
Opposite side case similar.
Prove: AB = A’B’
SASAS
CPCFE
Given: A and B reflected in line l.
(1) ABYX ≅ A’B’YX
(2) AB = A’B’
B
A
X
Y
A’
B’
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Point Orientation
Def. Given a triangle ABC in the plane, the
counterclockwise direction is called the
positive orientation of its vertices, while the
clockwise direction is the negative orientation.
Definition. A transformation is called direct
iff it preserves the orientation of any
triangle, and opposite iff it reverses the
orientation of any triangle.
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Theorem 2: Orientation Theorems
Theorem 2: The product (composition) of an
even number of opposite transformations is
direct, and the product of an odd number of
opposite transformations is an opposite
transformation.
Proof through the application of the
definition of direct and opposite
transformations.
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AND
Theorem 2b. A reflection is an opposite
transformation.
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Theorem 3
The product of two line reflections R (l) and
R (m), where l and m are parallel is distance
and slope preserving and maps a given line
n into one that is parallel to it.
A line reflection is an isometry by lemma
and hence distance preserving. We will
prove slope preserving but first let’s look at
a figure of what this product of two line
reflections looks like.
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That is
R (l) • R (m)
B’
B
A
B”
A”
A’
l
m
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Theorem 3. The product of two line reflections
R (l) and R (m), where l and m are parallel is
distance and slope preserving and maps a given
line n into one that is parallel to it.
Proof: If AB is parallel to l and m the
theorem is proven.
B”
B’
B
A
l
A’
m
A”
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Theorem 3. The product of two line reflections R
(l) and R (m), where l and m are parallel is
distance and slope preserving and maps a given
line n into one that is parallel to it.
Proof: AB intersects l and m with B on l. 1 ≅ 2 ≅
 3 ≅ 4 ≅ 5 by either angle preservation of
reflections or corresponding angles of parallel lines.
Since 1 ≅ 5, AB and
A”B” are parallel and slope
is preserved.
B’
B 1
l
B”
2
m
3 4
5
A A”
A’
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Summary.
• We learned about relations, functions, and
transformations.
• We learned about linear transformations.
• We learned about inverse transfromations.
• We learned about the identity transformation.
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Summary.
• We learned about line and point reflections.
• We learned about isometries.
• We learned about point orientation.
• We learned about direct and indirect
transformation.
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