91Ó°ÊÓ

A cylinder in 3D space is an interesting shape to study. Mathematically, it can be defined by the equation \(x^2 + y^2 = 1\). This equation is essential because it tells us that for any point on the surface of the cylinder, the coordinates \(x\) and \(y\) must satisfy this relationship. In simpler terms, if you slice the cylinder horizontally, you always get a circle of radius 1. The \(z\) coordinate is not restricted by this equation, meaning it can be any real number. This gives the cylinder its characteristic shape: a series of stacked circles extending infinitely up and down along the \(z\)-axis.

In the context of transformations, an isometry is a function that preserves distances between points. For the given cylinder, we consider isometries in the form of rotations around the \(z\)-axis. For this exercise, we need to find an isometry \(\varphi\) such that the set of fixed points consists of exactly two points.
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A specific type of isometry to consider is a rotation by \(\theta = \pi\) radians (180 degrees). When we apply this rotation to any point \((x, y, z)\) on the cylinder, it gets mapped to \((-x, -y, z)\). For a point to be a fixed point, it must satisfy \(\varphi(p) = p\), i.e., \((-x, -y, z) = (x, y, z)\).
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This results in the conditions \(x = -x\) and \(y = -y\). Solving this system, we find that \(x = 0\) and \(y = 0\). However, these values do not satisfy the cylinder's equation \(x^2 + y^2 = 1\). To resolve this, on further analysis, we see the fixed points must be \(x = \pm 1\). This results in two specific points for any \(z\): \((1, 0, z)\) and \((-1, 0, z)\). So, these are our fixed points.

Rotational symmetry is an important concept in geometry and isometry. Rotating an object around an axis doesn't change the object if it has rotational symmetry around that axis. In the context of the cylinder and our isometry, the rotational symmetry plays a key role.
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For our cylinder, rotating around the \(z\)-axis by \(\theta = \pi\) results in the cylinder maintaining its shape and structure. This is a form of \(180^\text{°}\) rotational symmetry. When we apply this rotation to any point on the cylinder, it affects only the \(x\) and \(y\) coordinates, while the \(z\) coordinate remains unchanged. The points that do not move under this rotation are our fixed points, and we've found them to be \((1, 0, z)\) and \((-1, 0, z)\).
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Rotation is a common isometry used in many mathematical contexts, not just cylinders. Understanding how objects can be symmetric under rotation is fundamental in various fields, including crystallography, robotics, and computer graphics.