omotetics

Omotetics, also known as homothety or dilation, is a geometric transformation that enlarges or shrinks a figure by a scale factor from a fixed point, called the center of homothety. All points in the figure are mapped to new points such that the line segment connecting the original point to the center of homothety is parallel to the line segment connecting the image point to the center. The ratio of the distance from the center to the image point to the distance from the center to the original point is constant and equal to the scale factor.

If the scale factor is positive, the image and the original figure lie on the same side of the center of homothety. A scale factor greater than 1 results in an enlargement, while a scale factor between 0 and 1 results in a reduction. If the scale factor is negative, the image and the original figure lie on opposite sides of the center of homothety, and the transformation also involves a point reflection through the center. A scale factor of -1 is equivalent to a point reflection. A scale factor of 1 results in the identity transformation, where the figure remains unchanged.

Omotetics preserve the shape of figures, meaning that the image is similar to the original figure. Angles are preserved, and lengths are scaled by the absolute value of the scale factor. Parallel lines remain parallel, and lines passing through the center of homothety are invariant. Omotetics are fundamental in geometry and have applications in various fields, including art, architecture, and computer graphics, for tasks such as scaling and resizing images.