Homotetramere

Homotetramere, also known as a homothety, is a transformation in geometry that enlarges or reduces a figure by a scale factor relative to a point not on the figure, called the center of homothety. This point serves as the fixed point through which all lines in the figure pass after the transformation. Homothety preserves the shape and orientation of the figure, meaning that angles remain unchanged and parallel lines remain parallel. The scale factor can be positive or negative; a positive factor results in an enlargement, while a negative factor results in an enlargement followed by a reflection.

In two-dimensional space, homothety can be represented by a matrix equation where the scale factor is applied to both the x and y coordinates of each point in the figure. The center of homothety acts as the origin for this transformation. Homothety is a type of similarity transformation, which means it preserves the ratios of lengths and areas. This property makes homothety useful in various applications, such as map scaling and architectural design.

In three-dimensional space, homothety can be extended to include the z-coordinate, resulting in a three-dimensional homothety. The principles remain the same, with the scale factor applied uniformly to all coordinates and the center of homothety serving as the fixed point. Homothety in three dimensions is also a similarity transformation, preserving the ratios of lengths, areas, and volumes.

Homothety is a fundamental concept in geometry and has applications in various fields, including mathematics, physics, and engineering. Its ability to preserve shape and orientation while scaling figures makes it a valuable tool for understanding and manipulating geometric transformations.