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A cylinder is a three-dimensional geometric shape that consists of two parallel circular bases connected by a curved surface. The bases are congruent and lie in parallel planes, while the curved surface is formed by connecting the corresponding points on the bases with straight lines.
In Cartesian coordinates, a cylinder can be defined using equations that describe the positions of its points in three-dimensional space. The equations for a cylinder in Cartesian coordinates depend on whether the cylinder is oriented along the $x-$axis, $y-$axis, or $z-$axis. Here are examples of equations that define cylinders in each orientation:
1. Cylinder oriented along the $x-$axis:
Equation: $x^2 + y^2 = r^2$
This equation represents a cylinder with radius $'r'$ centered at the origin $(0, 0, 0)$ and oriented along the $x-$axis. The variable $'x'$ represents the $x-$coordinate of a point on the cylinder, and $'y'$ represents the $y-$coordinate. The equation states that the sum of the squares of the $x$ and $y$ coordinates is equal to the square of the radius.
2. Cylinder oriented along the $y-$axis:
Equation: $x^2 + z^2 = r^2$
This equation represents a cylinder with radius $'r'$ centered at the origin (0, 0, 0) and oriented along the $y-$axis. The variable $'x'$ represents the $x-$coordinate of a point on the cylinder, and $'z'$ represents the $z-$coordinate. The equation states that the sum of the squares of the $x$ and $z$ coordinates is equal to the square of the radius.
3. Cylinder oriented along the z-axis:
Equation: $y^2 + z^2 = r^2$
This equation represents a cylinder with radius $'r'$ centered at the origin $(0, 0, 0)$ and oriented along the $z-$axis. The variable $'y'$ represents the $y-$coordinate of a point on the cylinder, and $'z'$ represents the $z-$coordinate. The equation states that the sum of the squares of the $y$ and $z$ coordinates is equal to the square of the radius.
These equations define the geometric shape of a cylinder in Cartesian coordinates and can be used to represent various cylindrical objects in mathematical models and calculations.
