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§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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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.
Relations
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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).
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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TransformationDefinition. 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.
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.
Inverse Transformation
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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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Pl
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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P
C
P’
Reflection in a point.
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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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Lemma 1. An isometry preserves collinearity.
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.
Isometry Facts
Lemma 2. An isometry preserves betweenness.
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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 1Reflections 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.
Given: A and B reflected in line l. Prove: AB = A’B’
B
A
A’
B’
(1) ABYX ≅ A’B’YX SASAS
(2) AB = A’B’ CPCFE
X Y
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Point 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.
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.
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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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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.
Theorem 3
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’
l m
B”
A A’
B
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: If AB is parallel to l and m the theorem is proven.
B’
l m
B”
A A’
B
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’
l
m
B”
A A’
B
A”
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3 45
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Summary.
• We learned about relations, functions, and transformations.
• We learned about inverse transfromations.
• We learned about linear transformations.
• We learned about the identity transformation.
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Summary.
• We learned about line and point reflections.
• We learned about point orientation.
• We learned about isometries.
• We learned about direct and indirect transformation.
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