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r, θ, φ Cylindrical Coordinates
Cylindrical Coordinates
x, y, z Cylindrical Coordinates
In the simulation below you can see the cylindrical coordinate system which measures the distance between the head of the arrow and its ending point, starting at the origin (0, 0, 0). Notice how in this simulation you input the r, z, and φ values to create the vector. This is one way of plotting a point using cylindrical coordinates.
Cylindrical coordinates are a coordinate system that uses the radius r, an angle θ, and a z coordinate in three dimensions. Cylindrical coordinates are best used to measure where something is on a circular surface. For example, cylindrical coordinates can be used to measure the radius and angle of an object that is located at a certain height. Cylindrical coordinates measure where an object is located on the surface of the circle through measuring the distance between the origin and the radius of the circle as well as knowing the angle at which an object is oriented in. For example, you can measure points away from the origin (0, 0, 0) and with a given angle, you can know which direction an object is pointing towards. It is worth noting that in cylindrical coordinates, the radius r, ranges from 0 to infinity, in any direction. The angle θ however has a limited range. It ranges from 0 to 2π. The cylindrical coordinate system differs from the three dimensional cartesian coordinate system in that the radius and angle are provided, and the components are left to be calculated.
The simulation above shows the cylindrical coordinate system in three dimensions, over the x, y, and z axes. As you input the x, y, and z components in the simulation, you can notice a black line (representing a vector) be plotted and pointing to a specific point on the cylinder. The x, y, and z components that you input, when written inside parentheses, these are the coordinates of the point on the cylinder's surface that the vector is pointing to. For example if you input the x, y, and z components: 15, 19, and 7, in parentheses, this is written as (15, 19, 7). These components now face a single point that is defined at (15, 19, 7) from the origin (0, 0, 0). If you want to convert these coordinate points to figure out the radius r, of the cylinder along with the angle θ and z, these values can be calculated using a few of the formulas mentioned below.
Three dimensional cartesian coordinates can be converted to cylindrical coordinates using the following formulas:
\[ r = \sqrt{x^2 + y^2} \]
\[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \]
\[ z = z \]
Converting to cylindrical coordinates with the values given:
\[ r = \sqrt{x^2 + y^2} \]
\[ r = \sqrt{15^2 + 19^2} \]
\[ r = \sqrt{225 + 361} \]
\[ r = \sqrt{586} \]
\[ r = 24.21 \]
\[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \]
\[ \theta = \tan^{-1} \left( \frac{19}{15} \right) \]
\[ \theta = \tan^{-1} (1.2666) \]
\[ \theta = 51.71 \]
\[ z = z \]
\[ z = 7 \]
\[ z = 7 \]
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