11.1 Rigid Motions. Symmetry
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- Abel McKenzie
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1 11.1 Rigid Motions
2 Rigid Motions We will now take a closer look at the ideas behind the different types of symmetries that we have discussed by studying four different rigid motions. The act of taking an object and moving it from some starting position to some ending position without altering its shape or size is called a rigid motion. A rigid motion preserves distances.
3 Rigid Motions The figure below shows a rigid motion. The position of the face has changed, but the shape has not.
4 Rigid Motions The motion in the figure below is not a rigid motion. In this motion, both the position and the shape have been altered.
5 11.2 Reflections
6 What is a Reflection? A reflection is a motion that moves an object to a mirror image of itself. The mirror is called the axis of reflection, and is given by a line m in the plane.
7 What is a Reflection? To find the image of a point P under a reflection, draw the line through P that is perpendicular to the axis of reflection m. TheimageP will be the point on this line whose distance from m is the same as that between P and m. m P
8 What is a Reflection? To find the image of a point P under a reflection, draw the line through P that is perpendicular to the axis of reflection m. TheimageP will be the point on this line whose distance from m is the same as that between P and m. m P 2 2 P
9 Examples Find the image of P under the reflection given by the line m. m P
10 Examples Find the image of P under the reflection given by the line m. m 0.89 P 0.89 P
11 Examples Find the image of Q under the reflection given by the line m. m Q
12 Examples Find the image of Q under the reflection given by the line m. m Q Q
13 Examples Find the image of the quadrilateral ABCD under the reflection given by the line m. A m D B C
14 Examples Find the image of the quadrilateral ABCD under the reflection given by the line m. A m A D B B D C C
15 Examples Find the image of the triangle ABC under the reflection given by the line m. A m B C
16 Examples Find the image of the triangle ABC under the reflection given by the line m. A m A B B C C
17 Examples Find the axis of reflection, m, for the reflection that takes P to P. P P
18 Examples Given a point P and its image P, the axis of reflection is the perpendicular bisector of the line segment PP. P m P
19 Properties of Reflections 1. A reflection is completely determined by its axis of reflection.
20 Properties of Reflections 1. A reflection is completely determined by its axis of reflection....or A reflection is completely determined by a single point-image pair P and P (if P P ).
21 Properties of Reflections A fixed point of a motion is a point that is moved onto itself.
22 Properties of Reflections A fixed point of a motion is a point that is moved onto itself. For a reflection, any point on the axis of reflection is a fixed point.
23 Properties of Reflections A fixed point of a motion is a point that is moved onto itself. For a reflection, any point on the axis of reflection is a fixed point. 3. Therefore, a reflection has infinitely many fixed points (all points on the line m).
24 Properties of Reflections A m A D B B D C C The orientation of the original object is clockwise: read ABCDA going in the clockwise direction. The orientation of the image under the reflection is counterclockwise: A B C D A is read in the counterclockwise direction.
25 Properties of Reflections 4. A reflection is an improper motion because it reverses the orientation of objects.
26 Properties of Reflections What happens to an object if apply the same reflection twice?
27 Properties of Reflections What happens to an object if apply the same reflection twice? 5. Applying the same reflection twice is equivalent to not moving the object at all. So applying a reflection twice results in the identity motion.
28 11.3 Rotations
29 What is a Rotation? A rotation is a motion that swings an object around a fixed point. The fixed center point of the rotation is called the rotocenter. The amount of swing is given by the angle of rotation.
30 What is a Rotation? An example of a clockwise rotation with rotocenter O and angle of rotation α. It moves the point P to the point P.
31 What is a Rotation? Three different rotations of the same triangle: A 90 clockwise rotation with rotocenter O outside of the triangle.
32 What is a Rotation? Three different rotations of the same triangle: A 180 clockwise rotation with rotocenter O inside the triangle.
33 What is a Rotation? Three different rotations of the same triangle: A 360 clockwise rotation with rotocenter O inside the triangle.
34 What is a Rotation? Find the image of triangle ABC under a 90 counterclockwise rotation with rotocenter O. A O B C
35 What is a Rotation? Find the image of triangle ABC under a 90 counterclockwise rotation with rotocenter O. A O B C 90 C A B
36 What is a Rotation? Notice that the distance of each point from the rotocenter O does not change under the rotation: A O B C C A B
37 The Angle of Rotation As a convention, any angle in the counterclockwise direction has a positive angle measure. Any angle in the clockwise direction has a negative angle measure. A C F E B D
38 Examples Find the image of ABCD under the rotation with rotocenter O and angle of rotation 135. A B 135 O D C
39 Examples Find the image of ABCD under the rotation with rotocenter O and angle of rotation 135. A D B 135 O C D A C B
40 Examples Find the image of ABCD under the rotation with rotocenter O and angle of rotation 45. A B E D O
41 Examples Find the image of ABCD under the rotation with rotocenter O and angle of rotation 45. A B -45 O B E E A D D
42 Properties of Rotations What completely defines a specific rotation? Certainly a rotocenter and given angle define a specific rotation. Does one point-image pair?
43 Properties of Rotations Does one point-image pair completely define a rotation? No. For a given point-image pair, there are infinitely many possible rotocenters: P P
44 Properties of Rotations 1. A rotation is completely determined by two point-image pairs. The rotocenter is at the intersection of the two perpendicular bisectors: P 1 Q Q P
45 Properties of Rotations 1. A rotation is completely determined by two point-image pairs. The rotocenter is at the intersection of the two perpendicular bisectors: P 1 Q Q P What is the angle of rotation of this rotation?
46 Properties of Rotations 1. A rotation is completely determined by two point-image pairs. The rotocenter is at the intersection of the two perpendicular bisectors: P 1 Q Q P What is the angle of rotation of this rotation? 180.
47 Properties of Rotations Once we know the rotocenter, the angle of rotation is the measure of angle POP. This will be the same as the measure of angle QOQ.
48 Properties of Rotations What are the fixed points of a rotation?
49 Properties of Rotations What are the fixed points of a rotation? 2. A rotation has one fixed point, the rotocenter. Is a rotation a proper or improper motion?
50 Properties of Rotations What are the fixed points of a rotation? 2. A rotation has one fixed point, the rotocenter. Is a rotation a proper or improper motion? 3. A rotation is a proper motion. The orientation of the object is maintained.
51 Properties of Rotations What are the fixed points of a rotation? 2. A rotation has one fixed point, the rotocenter. Is a rotation a proper or improper motion? 3. A rotation is a proper motion. The orientation of the object is maintained. 4. A 360 rotation around any rotocenter is equivalent to the identity motion.
52 Properties of Rotations Any rotation is equivalent to one with angle of rotation between 0 and 360 : Notice that =90. A O A
53 The Angle of Rotation In each case, give an answer between 0 and 360. A clockwise rotation by an angle of 710 is equivalent to a clockwise rotation by an angle of...
54 The Angle of Rotation In each case, give an answer between 0 and 360. A clockwise rotation by an angle of 710 is equivalent to a clockwise rotation by an angle of
55 The Angle of Rotation In each case, give an answer between 0 and 360. A clockwise rotation by an angle of 710 is equivalent to a clockwise rotation by an angle of A clockwise rotation by an angle of 710 is equivalent to a counterclockwise rotation by an angle of...
56 The Angle of Rotation In each case, give an answer between 0 and 360. A clockwise rotation by an angle of 710 is equivalent to a clockwise rotation by an angle of A clockwise rotation by an angle of 710 is equivalent to a counterclockwise rotation by an angle of... 10
57 The Angle of Rotation In each case, give an answer between 0 and 360. A clockwise rotation by an angle of 710 is equivalent to a clockwise rotation by an angle of A clockwise rotation by an angle of 710 is equivalent to a counterclockwise rotation by an angle of A clockwise rotation by an angle of is equivalent to a clockwise rotation by an angle of...
58 The Angle of Rotation In each case, give an answer between 0 and 360. A clockwise rotation by an angle of 710 is equivalent to a clockwise rotation by an angle of A clockwise rotation by an angle of 710 is equivalent to a counterclockwise rotation by an angle of A clockwise rotation by an angle of is equivalent to a clockwise rotation by an angle of... 80
59 The Angle of Rotation In each case, give an answer between 0 and 360. A clockwise rotation by an angle of 710 is equivalent to a clockwise rotation by an angle of A clockwise rotation by an angle of 710 is equivalent to a counterclockwise rotation by an angle of A clockwise rotation by an angle of is equivalent to a clockwise rotation by an angle of A clockwise rotation by an angle of is equivalent to a counterclockwise rotation by an angle of...
60 The Angle of Rotation In each case, give an answer between 0 and 360. A clockwise rotation by an angle of 710 is equivalent to a clockwise rotation by an angle of A clockwise rotation by an angle of 710 is equivalent to a counterclockwise rotation by an angle of A clockwise rotation by an angle of is equivalent to a clockwise rotation by an angle of A clockwise rotation by an angle of is equivalent to a counterclockwise rotation by an angle of
61 Exercise: A rotation moves A to A and B to B. Find the rotocenter and the image of ABC under the rotation. A B C B A
62 Exercise: A rotation moves A to A and B to B. Find the rotocenter and the image of ABC under the rotation. A B O C B A C
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