Whenever we think about rotations, we always imagine an object moving in a circular form. Like the Earth rotating on its axis. We experience the change in days and nights due to this rotation motion of the earth. So, we know that rotation is a movement of an object around a center.
But what about when dealing with any graphical point or any geometrical object? How are we supposed to rotate these objects and find their image? In this section, we will understand the concept of rotation in the form of transformation and take a look at how to rotate any image.
Rotation meaning
Rotations are transformations where the object is rotated through some angles from a fixed point. Examples of rotations include the minute needle of a clock, merry-go-round, and so on.
In all cases of rotation, there will be a center point that is not affected by the transformation. In the clock, the point where the needle is fixed in the middle does not move at all. In other words, the needle rotates around the clock about this point.
Rotating an object \(\mathbf{\pm d^{\circ}}\) about a point \((a, b)\) is to rotate every point of the object such that the line joining the points in the object and the point \((a, b)\) rotates at an angle \(d^{\circ}\) either clockwise or counterclockwise depending on the sign of \(d\).
If \(d\) is positive, then it is counterclockwise; otherwise, it is negative. In both transformations, the size and shape of the figure stay exactly the same. We denote rotation by \(R_{\text{angle}}\).
Properties of Rotation
The pre-image and images have some interesting properties of rotation.
The mapping in the rotation is from line to line, segment to segment, and angle to angle.
A rotation is a transformation in which every point and its image have the same distance and same angle from the vertex.
There is a congruence between the pre-image and the image after rotation.
The same orientation is maintained by rotation.
Distance and angle are preserved in the rotation transformation.
Rotation formula
Rotations around an axis are usually clockwise. Since rotation in the clockwise direction is denoted by a negative magnitude, rotation done in the counterclockwise direction is denoted by a positive magnitude.
In general, rotation can occur at any point with an uncommon rotation angle, but we will focus on common rotation angles like \(90^{\circ}, 180^{\circ}, 270^{\circ}\)
The general rotation formula about the origin \((0, 0)\) is as follows:
| Type of rotation | Point on pre-image | Point after clockwise rotation | Point after counterclockwise rotation |
| Rotation to \(90^{\circ}\) | \((x, y)\) | \((y, -x)\) | \((-y, x)\) |
| Rotation to \(180^{\circ}\) | \((x, y)\) | \((-x, -y)\) | \((-x, -y)\) |
| Rotation to \(270^{\circ}\) | \((x, y)\) | \((-y, x)\) | \((y, -x)\) |
Rotation rules in geometry
There are some basic rotation rules in geometry that need to be followed when rotating an image. The following basic rules are followed by any preimage when rotating:
Generally, the center point for rotation is considered \((0,0)\) unless another fixed point is stated.
The angle of rotation should be specifically taken.
Take note of the direction of the rotation, as it makes a huge impact on the position of the image after rotation.
Rotation of \(90^{\circ}\)
The clockwise rotation of \(90^{\circ}\) will result in the image with \((y, -x)\). So, \(x\) and \(y\) coordinate will switch places and with the multiplication of \(-1\) by \(x\) coordinate. And for the counterclockwise rotation of \(90^{\circ}\), the image will have \((-y, x)\). Rotation of \(90^{\circ}\) is also considered as \(-270^{\circ}\).
Rotation of \(180^{\circ}\)
The image with rotation of \(180^{\circ}\) in either clockwise or counterclockwise will have the same coordinates points of \((-x, -y)\). Hence, \(-1\) will be multiplied to both coordinates without switching places. Here the rotation of \(180^{\circ}\) is also taken as \(-180^{\circ}\).
Rotation of \(270^{\circ}\)
The coordinate points of a pre-image are swapped and \(y\) coordinate is multiplied by \(-1\) when rotating \(270^{\circ}\) clockwise. Or multiplied by \(-1\) with \(x\) after swapping when rotating \(270^{\circ}\) counterclockwise.
Rotation examples
Here are some solved rotation examples.
Rotate figure \(ABC\) with coordinates \(A (2, 1), B (3, 1), C (3, 2)\) \(90^{\circ}\) clockwise.
Solution:
Here we need to rotate the image \(ABC\) \(90^{\circ}\) clockwise. According to the rule, we have our points \((x, y)\) which will be mapped to \((y, -x)\).
Hence, we will individually apply the rotation formula to all three given points.
\[A (2, 1) \rightarrow A' (1, -2)\]
\[B (3, 1) \rightarrow B' (1, -3)\]
\[C (3, 2) \rightarrow C' (2, -3)\]
Figure \(A'B'C'\) has coordinates \(A'(1, -2), B' (1, -3), C' (2, -3)\)
Let's now plot our Figures.
Fig. 1. Clockwise \(90^{\circ}\) rotation of image.
Rotate figure \(XYZ\) with coordinates \(X (1, 1), Y(5, 5), Z(-2, 4)\) \(270^{\circ}\) counterclockwise.
Solution:
From the rotation formula of rotating an image counterclockwise \(270^{\circ}\), we will have our points (x, y) mapped to (y, -x). So, applying the rotation formula to individual points, we get
\[X(1, 1) \rightarrow X'(1, -1)\]
\[Y(5, 5) \rightarrow Y'(5, -5)\]
\[Z(-2, 4) \rightarrow Z'(4, 2)\]
Figure \(X'Y'Z'\) has coordinates X'(1, -1), Y'(5, -5), Z'(4, 2).
Let us now plot our figures.
Fig. 2. Image rotated \(270^{\circ}\) counterclockwise.
Rotations - Key takeaways
- Rotating an object \(\mathbf{\pm d^{\circ}}\) about a point \((a, b)\) is to rotate every point of the object such that the line joining the points in the object and the point (a, b) rotates at an angle \(d^{\circ}\) either clockwise or counterclockwise depending on the sign of \(d\).
- Rotation is denoted by \(R_{\text{angle}}\).
- Rotation is done clockwise or counterclockwise.
- The image is mapped to points \((y, -x)\) or \((-y, x)\) when rotated \(90^{\circ}\) clockwise or counterclockwise respectively.
- \(180^{\circ}\) rotation is the same for either clockwise or counterclockwise and is mapped to \((-x, -y)\).
- The image is mapped to points \((-y, x)\) or \((y, -x)\) when rotated \(270^{\circ}\) clockwise or counterclockwise respectively.
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