Hello there! I'm a specialist in the field of geometry and spatial transformations. I'm here to provide you with a comprehensive understanding of the term "direct isometry".

In geometry, an isometry is a transformation that preserves the distance between any two points. It is a type of rigid motion that can be thought of as a way to move a shape without changing its size or shape. Isometries are fundamental in the study of geometrical properties and symmetries.

A direct isometry, specifically, is a type of isometry that does not involve any reflection. It is a transformation that moves a figure from one position to another without flipping it over. This means that the orientation of the figure remains the same after the transformation. Direct isometries include translations, rotations, and glide reflections.

Let's delve a bit deeper into the components you mentioned:


1. Composition of Two Opposite Isometries: When you combine two opposite isometries, you get a direct isometry. Opposite isometries are transformations that are the inverse of each other. For example, if you have a reflection in a line (an opposite isometry), and you apply another reflection in a line that is perpendicular to the first, the result is a translation, which is a direct isometry.


2. Reflection in a Line: A reflection in a line is an example of an opposite isometry. It flips a figure over a line, changing its orientation. If you were to reflect a figure in line R1 and then again in line R2, which is perpendicular to R1, you would end up with a direct isometry, assuming the figure is symmetrical with respect to both lines.


3. Translation (T): A translation is a direct isometry. It is a rigid motion that moves every point of a figure the same distance in the same direction. Translations do not change the shape or size of a figure, nor do they change its orientation.

To summarize, a direct isometry is a transformation that preserves the distance between points and does not involve flipping the figure over. It includes transformations such as translations, rotations, and certain glide reflections. The composition of two opposite isometries results in a direct isometry, and reflections in lines are examples of opposite isometries.

Now, let's move on to the translation of the response into Chinese.

read more >>

A composition of two opposite isometries is a direct isometry. A reflection in a line is an opposite isometry, like R 1 or R 2 on the image. Translation T is a direct isometry: a rigid motion.read more >>