Rigid Motion Transformations & Examples | What is Rigid Motion? - Video & Lesson Transcript | Study.com

Understanding Rigid Motion Transformation

A rigid transformation or isometry of an object is best understood when presented visually. There are usually two images: the pre-image and the post-image. The pre-image is the state of the object before any movement takes place. Post-image is its state after the movement is done. Figure 1 shows the pre-image and post-image of an object. The apostrophe on the corner points signifies post-image construction. These are named prime. {eq}C' {/eq} is a post-image point and referred to as C prime.

Fig. 1 Pre-Image and Post-Image of a Rigid Transformation

The pre-image and post-image of the object have to satisfy congruency to be described as rigid. Congruency means that they are the same shape, size, and form. If the post-image and pre-image are placed on top of each other, there should be no difference in the size or shape of the two objects.

Motion or movement can be made in three ways: translation (a slide), where the pre-image is moved horizontally, vertically, or a combination of both on a flat surface or plane. It can also be a reflection or flip, where the object's one side is flipped over to the other side of a line. This line that is crossed or flipped over is called the line of reflection. The third move is rotation, where the object is rotated from a fixed pivot point, called the rotocenter.

The rigid transformation has vast uses in geometry. Perhaps the one trending use of rigid changes in 3D rendering in programming. A buyer is no longer required to see merchandise to inspect it from all angles. Now, there is modern technology that consumers can rotate, flip, and move an item on the website. In fact, in the furniture and home furniture market, technology can capture the size of the room via camera lenses and put the furniture in the space. The potential buyer may slide, rotate, or flip to his preference. The technology and programming that went into this new capability in the marketing and sales sector would not have been made possible without the use of rigid transformations.

Types of Rigid Motion: Examples

Without changing their shapes and sizes, objects transforming show the changes in location and orientation by showing pre-images and post-images. For each of the three rigid motions, a comparison of the two images will help differentiate the transformations from each other. Note, however, that another type of transformation is not included in this lesson. Dilations are excluded because this transformation changes the shape and size of a pre-image. In this lesson, only rigid transformations are covered.

Translation

A translation moves the pre-image to another position by changing the horizontal, vertical, or horizontal and vertical positions. If the object being translated is a polygon, each vertex point is moved at the same distance and in the same direction. In coordinate geometry, the points of the pre-image are changed to the post-image following a formula:

For any point {eq}(x,y) {/eq} in the pre-image, {eq}(x,y)\to (x+h, y+k) {/eq}

where {eq}h {/eq} and {eq}k {/eq} represent the horizontal and vertical shift on the graph of the image.

Positive {eq}h {/eq} means a movement to the right and a positive {eq}k {/eq} means a movement upwards. The negative values will be towards the opposite directions of leftwards and downwards, respectively.

It is worth remembering that each point of a figure is moved at the same distance and in the same direction in a translation. There is no change in orientation or change in direction.

Example 1: Translation

A triangle has the following point coordinates for its vertex points : A: (2,3) B: (3,1) C:(1,2) Graph and label the pre-image and the post-image following the translation rule {eq}(x,y)\to (x+2, y-3) {/eq}

Solution:

Step 1: The points are plotted on a graph and connected.

Step 2: The translation formula is calculated for the post-image for each vertex point.

Step 3: Post-image points are plotted, connected, and labeled.

Fig. 2 Translation of a triangle showing pre-image and post-image

Reflection

In a reflection transformation, the object is flipped over a line. This line is called a line of reflection or a mirror line.

Lines of reflection may include the x-axis, y-axis, y=x, y=-x, vertical lines, and horizontal lines. When a translation follows a reflection, it is called a glide transformation.

• For reflections over the x-axis: (x,y) becomes ( x, -y)
• For reflections over the y- axis: (x,y) becomes (-x, y)
• For reflections over the diagonal y=x: (x,y) becomes (y,x)
• For reflections over the diagonal y = -x: (x,y) becomes (-y,-x)

Although the formula makes graphing a lot easier, a good strategy for plotting the points for the post-image is to check the distance of the points perpendicular to the mirror line and plot the post-image points directly across using the same distance.

Example 2: Reflection

Graph and label the pre-image and reflect it over the y-axis: A (3,4) B (0,5) C (1,1)

Solution:

Step 1: The points are plotted on a graph and connected.

Step 2: There is a post-image point across the y-axis of equal distance for each vertex point.

Step 3: Post-image points are plotted, connected, and labeled.

Fig. 3 Reflection transformation showing pre-image and post-image

Rotation

The third rigid motion is a rotation or a turn. In this rigid motion transformation, the image or object is rotated about a fixed point, called its rotocenter. Every rotation has this pivot point and an angle. Every point in the post-image rotates by the same amount around the rotocenter.

In most cases, the rotocenter is the origin point in a cartesian plane. The angles also are most commonly {eq}90^{\circ} {/eq} and {eq}180^{\circ} {/eq} described as either rotating clockwise or counterclockwise.

Here are formulas that may be used as a guide if no graphing tool is available:

• {eq}90^{\circ} {/eq} clockwise: (x,y) ----> (y, -x)

• {eq}90^{\circ} {/eq} counterclockwise: (x, y) ----> (-y, x)

• {eq}180^{\circ} {/eq} (x,y) ----> (-x, -y)

Example 3: Rotations

A polygon with the following vertices is graphed and rotated {eq}90^{\circ} {/eq} counterclockwise about the origin. Draw the post-image of the rigid transformation and write the coordinates of the new vertices.

A: (1, 4) B: (4,6) C: (7,4) D: (4,2)

Solution :

Step 1: The points are plotted on a graph and connected.

Step 2: For each vertex point (x,y), a post-image point is (-y,x).

Step 3: Post-image points are plotted, connected, and labeled.

Fig. 4 Pre-image and post-image of a polygon showing 90 degree rotation CC

Lesson Summary

Rigid motion changes an object's location, orientation, and position. To fully understand the changes on an object or image, a pre-image and post-image are drawn on the graph for comparison. The pre-image is the original figure before any transformation takes place. Post-image is after the changes take place. To identify post-image figures, an apostrophe is placed on the corner points. This is read as 'prime.'

The three rigid motions are translation, reflection, and rotation. The translation is a change in the position of the image without a change in direction and orientation. Reflection is a flip that folds over a line of reflection. The points of the pre-image and post-image are equidistant from this mirror line. The third rigid motion is the rotation, which rotates the points of the pre-image at a constant angle over a fixed point called the rotocenter. A transformation not included in this lesson is the dilation that stretches or compresses an image or object, thus changing shape and size. This type of transformation is not a rigid motion.

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