As a geometric expert, I am well-versed in the field of geometry and its various transformations. Let's delve into the concept of isometries and their relation to reflections.

Isometries are transformations that preserve the distance between any two points. This means that the shape and size of a figure remain unchanged after the transformation. There are several types of isometries, including translations, rotations, reflections, and glide reflections.

Translations are isometries that involve moving a figure a certain distance in a specific direction without rotating or flipping it.

Rotations are isometries where a figure is turned around a fixed point by a certain angle. The key point here is that every single rotation is indeed a direct isometry. This means that it directly maps each point of the figure to another point without any intermediate steps.

A fixed point of an isometry \( f \) is a point \( P \) such that \( f(P) = P \). In simpler terms, it's a point that remains stationary during the transformation. This is an important concept because it helps us understand the nature of the isometry.

Now, let's consider reflections. A reflection is an isometry that flips a figure over a line, known as the line of reflection. This line can be vertical, horizontal, or at any other angle. The question at hand is whether reflections have direct isometry.

The answer is affirmative. Reflections are a type of direct isometry because they map each point of a figure directly to another point across the line of reflection. There is no intermediate step involved in this process. The figure is simply flipped over the line, and each point maintains its distance from every other point in the figure.

It's also worth noting that reflections have a unique property: they have lines of fixed points. These are the lines that lie on the plane of reflection. Every point on these lines remains stationary during the reflection, which is a characteristic feature of reflections as isometries.

In conclusion, reflections are indeed direct isometries. They preserve the distance between points, directly map each point of a figure to another point across the line of reflection, and have lines of fixed points that remain unchanged during the transformation.

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Every single rotation is a direct isometry. ... A fixed point of an isometry f is a point P such that f(P) = P -- in other words, a point which does not get moved by the isometry. Remember that we classified the isometries into four types -- translations, rotations, reflections and glide reflections.read more >>