§5.1 Geometric Transformations
Lesson 5 - "Rotations”
The next isometry we will look at are called rotations. An analogy we can use in describing a rotation would be that of a Ferris wheel . When the wheel moves each chair rotates to a new position. When the wheel stops, the position of a chair (P') can be referred to mathematically as the image of the initial position of the chair (P).
For two-dimensional figures, like the ones we will look at, a rotation is described by specifying a point called the center of rotation (or rotocenter for short), and an angle of rotation indicating the amount of rotation.
Figures can either be rotated clockwise (like the movement of the hands on your clock or watch), or counterclockwise (in the opposite direction). We must pick a convention for our rotations. Traditional texts would have us rotate in a counterclockwise fashion. And so it will be. The rotations we will describe will move counterclockwise about some rotocenter.
We need to now consider our mathematical notation for a rotation.
Picture of the original Ferris wheel.
The Ferris Wheel was the engineering highlight of the exposition and one of the most pervasive, lasting influences of the 1893 Chicago fair. The Ferris Wheel was Chicago's answer to the Eiffel Tower, the landmark of the 1889 Paris exhibition. The wheel was created by Pittsburgh, Pennsylvania bridge builder George W. Ferris.
If a counterclockwise rotation is to be made of 90º and it is to be made about a rotocenter called point O, then we will write RO, 90. If a clockwise rotation of 90º is to be made about O, we will write RO, −90.
To illustrate the notation for a rotation say that some point, P, was to be rotated about some rotocenter O 90 degrees. Then we would write
RO, 90 : P → P'.
This says that some point P gets mapped to P' by a rotation of 90 degrees about the rotocenter O.
Let's formalize a definition of a rotation.
A rotation about a point O through x° is a transformation such that:
(1) If point P is different from O, then OP = OP' and angle POP' = x°.
(2) If point P is the same as O, then O = P'.
We will look at a few examples next.
Find the image of the point A (4, 2) in the plane for the given rotations about the rotocenter, O, where O is the origin in the plane.
(a) RO, 90 : A → _____.
(b) RO, −90 : A → _____.
(c) RO, 180 : A → _____.
. . . to part (a)
Notice how I use the directions (coordinates) for getting to point A from the origin. This resulted in the (dotted line) blue L you see laying on its side.
Using this L-shape I can follow the rotation of the point A(4, 2) to its image location (−2, 4) as shown in the figure at the right.
I will use the same approach for finding the other images of the point A following the other two given rotations.
. . . to part (b) A negative 90 degree rotation means to rotate clockwise about the given rotocenter, the origin in this case. Again, notice how the letter-L is rotated to help determine the image of point A under this transformation.
. . . to part (c) A 180 degree rotation is considered a half-turn about the given rotocenter, the origin in this case. Again, notice how the letter-L is rotated to help determine the image of point A under this transformation. Because a 180 degree rotation is a half-turn it really does not matter in which direction the rotation occurs. You will still end up at the same place.
The next examples will move figures about a rotocenter.
What is the image of the non-square rectangle ABCD by the rotation RO, −90?
We are to rotate the rectangle about the point O in a 90 degrees clockwise fashion. (The negative 90 indicates the rotation is to occur clockwise. Positive 90 would mean to rotate counterclockwise.) The completed rotation is shown below with the pre-image figure in blue and the image figure in red.
What is the image of the triangle ABC by the rotation RO, 180.
We are to rotate the triangle 180 degrees about the rotocenter point O in a counterclockwise fashion this time. With the pre-image shown in blue and the image in red the completed rotation is shown below. The illustration below is only showing that point A has been rotated 180 degrees counterclockwise about the rotocenter point O to its image point A'. Note the same is true for each of the indicated points of the pre-image triangle.
In the special case of a 180 degree rotation two things are important to note. One is that a rotation of 180 degrees whether clockwise or counterclockwise gives the same result. And second we have a special name for this rotation - it is called a half-turn.
A rotation about a point O through 180°, clockwise or counterclockwise, is called a half-turn. We will designate a half-turn with the notation of HO instead of RO, 180.
Note: The subscripted letter indicated in the definition above can be any point. Usually when we use the letter O we are referring to the origin in the x, y plane. If a point different from the origin is being used as the rotocenter, a different letter will be used in the notation and that point defined for you. This is illustrated in the next example.
In the x, y-plane, a half-turn maps a point P to P' by rotating 180°. Find the images of the given points as they are rotated about their given rotocenters.
(a) HO:(3, 4) → _____, where O is the origin.
(b) HO:(−5, 2) → _____, where O is the origin.
(c) HA:(5, −1) → _____, where A is the point (2, 1).
The images of the given points are shown below. Refer to Example 5.1.10 part (c) for assistance in locating the image points in each case. For any half-turn rotation (even the one used in part (c) of this example) using the letter-L method described in Example 5.1.10 is recommended to help you find the image point.
(a) HO:(3, 4) → (−3, −4) (b) HO:(−5, 2) → (5, −2) (c) HA:(5, −1) → (−1, 3)
Observing the results of parts (a) and (b) only you should be able to make a conjecture as to the image of any point in the plane as it is rotated a half-turn about the origin. Make a conjecture and then ask your Instructor if it is correct.
A common mistake is to confuse a half-turn with a reflection. The two are in fact very different. A rotation, unlike a reflection, is what we call a proper rigid motion as it preserves left-right and clockwise-counterclockwise orientations within the rotated object.
What is the image of the triangle ABC by the rotation RO, 360.
The 360 degree rotation returns the triangle to its original position as the illustration below shows.
A rotation (clockwise or counterclockwise) of 360 degrees is called the identity motion since it returns the figure to its original position. The identity motion will always return the pre-image figure to its original position regardless of where the rotocenter is.
Now if we were simply given two points P and P', there are infinitely many possible rotations that move P to P' - all we have to do is choose a rotocenter on the perpendicular bisector of the line segment PP' [Figure 5.1.1 (a)]. Clearly then we need another pair of points, call them Q and Q', to nail down where the rotocenter actually is and to specify the rotation. The rotation will be the unique point where the perpendicular bisectors of PP' and QQ' meet [Figure 5.1.1 (b)]. In the special case where PP' and QQ' are parallel, the rotocenter is the intersection of PQ and P'Q' [Figure 5.1.1 (c)].
(a) (b) (c) Figure 5.1.1 Finding the rotocenter requires at least two pairs of points.
All materials © Leo Lusk 2003