Dilation is a transformative process which produces an image that is of the same shape as the original but is different in size. This process has two important factors which are used for its description-scale factor and the center. The center is a fixed point in the plane about which all the points are expanded or contracted. It is the only point which does not vary. It can lie inside, outside, or on the figure.
The process can lead to either an enlargement or reduction. If it creates a larger image then it is known as enlargement. If the image is smaller than the original, then it is called as reduction. If the scale factor is greater than 1, then the image is larger than the original and it is an enlargement. Similarly, if the scale factor is less than 1, then it is a reduction. If the scale factor is 1, then the new and the original image are of the same size and they are congruent.
The following are the key features of dilation which must be remembered:
- It is not a rigid transformation or isometry. This is due to the fact that the final image is not always of the same size as the pre-image.
- In rigid transformation process, the distances are preserved. However, it is not the case here. Length of different segments of the new image are not the same as the pre-image, except in cases where the scale factor is 1.
- The angle measures of the new image and the pre-image are the same.
- The parallel lines of the new image and pre-image are preserved. In short, the parallel lines in the pre-image remain parallel in the new image.
- The collinearity is preserved. The points remain on the same line in both pre-image and new image.
- Orientation is also preserved. The lettering order in the pre-image and new image remains the same.
- In most cases, the center is fixed as the origin (0,0) in the coordinate plane.
Real-Life Applications of Dilation
The following are the real-life applications of the process:
The process is extensively used in computer graphics. In case if I want to zoom in on a text or graphic, this real-life application comes in handy. The computer will apply a matrix to proportionally zoom the image or the object. It allows me to fit modify the size of the text or graphic to make it fit into the space proportionally.
The process of enlargement is also widely used in architecture. When architects are trying to design a building, they make models to work with. The prototype is true to the actual features of the planned building and is a scaled-down version of it. It is a common practice among architects to enlarge the prototype for making it easier for them to work. They commonly use the enlargement process to enlarge the prototype developed by using a scale factor larger than 1.
The enlargement process is also widely used for crime investigations. The crime scene detectives use dilated images of the pre-image and other evidences to study finer prints. The scale can be set greater than 1 to enlarge the pre-image to the required scale. This helps them to study crime-related evidences like photographs, fingerprints, etc. in greater details.
Another common application of enlargement is found in medical sciences. Doctors use enlargement process to enlarge the patient’s pupil to examine it better. After some time, the pupil contracts back to its normal size. It is also used for creating larger anatomical models for studying. Enlargement process is also useful for performing surgeries by enlarging the visuals of the organs on the display.
Toys and Art
Enlargement and reduction are also used for manufacturing toys, art forms, and crafts. Nesting doll sets which are placed from decreasing to increasing size use enlargement or reduction process. Each of the dolls placed after the smallest doll in an enlargement of it inside doll. The toys made in this process are consistent and similar to each other with only size difference.
Logos need to be scaled to different sizes for using on packaging boxes of varying sizes. A pizza outlet generally uses boxes of different sizes and the logo needs to be upscaled or downscaled accordingly. The enlargement or reduction process is used to proportionally scale the logos for better branding.
Photography and marketing are another popular real-life application of enlargement and reduction. Photographs can be scaled to different sizes based on requirement using this application. Marketing professionals use enlargement and reduction to proportionally scale the pre-image for being used in different sized print advertisements, billboards, etc.
The food and beverage industry makes use of different sized containers. In some cases, like coffee mugs, soft drink cups, etc. the containers are of the same shape and only their sizes differ. The process of enlargement or reduction can be effectively used in the food service industry to design enlarged or reduced containers of the same shape. It ensures that the containers are similar and consistent with only size differences. The similarity of containers makes it easier for undertaking consistent and effective branding.
Dilation is a useful process of enlargement or reduction of a pre-image to a desired size. In defining the process, center and scale are two important factors. The scale factor decides if the pre-image will be enlarged, reduced, or be congruent. The process is not a rigid transformation as the size of the final image and pre-image is different and the measurement of the different segments are also not the same. The final image and pre-image are however similar and the angle measurement, parallel lines, collinearity, and orientation are maintained.
The process of enlargement and reduction has many real-life applications. They are very commonly used in photography, advertisement, architecture, crime scene investigations, medical sciences, toys and crafts, food service, logo designing, etc. It makes possible to upscale or downscale the pre-image in proportion to the space which has to be filled.