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Spherical Coordinates
x, y, z Spherical Coordinates
In the simulation below you can see the spherical 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 x, y, and z components to create the vector. This is one way of plotting a point using spherical coordinates. The other way is by using a radius r, and two angles θ, and φ. These inputs will be explained in the r, θ, φ section of spherical coordinates. Usually the x, y, and z components are figured out using the r, θ, φ values. Here however, the x, y, and z components are provided, but converting them to r, θ, φ values will be discussed as well.
Spherical coordinates are a coordinate system that uses the radius r, and two angles θ, and φ in three dimensions. Spherical coordinates are best used to measure where something is on a spherical surface. For example, spherical coordinates can be used to measure the longitude and latitude of a certain city or place on Earth. Spherical coordinates measure where an object is located on the surface of the sphere through measuring the distance between the origin and the radius of the sphere as well as knowing the angles at which an object is oriented in. For example, you can measure points away from the origin (0, 0, 0) and with given angles, you can know which direction an object is pointing towards. It is worth noting that in spherical coordinates, the radius r, ranges from 0 to infinity, in any direction. The angles however have a limited range. The angle θ for example ranges from 0 to 2π, whereas the angle φ ranges from 0 to π. The spherical coordinate system differs from the three dimensional cartesian coordinate system in that the distance and two angles are provided, and the components are left to be calculated.
The simulation above shows the spherical 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 sphere. The x, y, and z components that you input, when written inside parentheses, these are the coordinates of the point on the sphere's surface that the vector is pointing to. For example if you input the x, y, and z components: 8, 11, and 16, in parentheses, this is written as (8, 11, 16). These components now face a single point that is defined at (8, 11, 16) from the origin (0, 0, 0). If you want to convert these coordinate points to figure out the radius r, of the sphere along with the angles θ and φ, these values can be calculated using a few of the formulas mentioned below.
Three dimensional cartesian coordinates can be converted to spherical coordinates using the following formulas:
\[ r = \sqrt{x^2 + y^2 + z^2} \]
\[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \]
\[ \phi = \tan^{-1} \left( \frac{\sqrt{x^2 + y^2}}{z} \right) \]
or
\[ \phi = \cos^{-1} \left( \frac{z}{r} \right) \]
Converting to spherical coordinates with the values given:
\[ r = \sqrt{x^2 + y^2 + z^2} \]
\[ r = \sqrt{8^2 + 11^2 + 16^2} \]
\[ r = \sqrt{64 + 121 + 256} \]
\[ r = \sqrt{441} \]
\[ r = 21 \]
\[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \]
\[ \theta = \tan^{-1} \left( \frac{11}{8} \right) \]
\[ \theta = \tan^{-1} (1.375) \]
\[ \theta = 53.97 \]
\[ \phi = \tan^{-1} \left( \frac{\sqrt{x^2 + y^2}}{z} \right) \]
\[ \phi = \tan^{-1} \left( \frac{\sqrt{8^2 + 11^2}}{16} \right) \]
\[ \phi = \tan^{-1} \left( \frac{\sqrt{64 + 121}}{16} \right) \]
\[ \phi = \tan^{-1} \left( \frac{\sqrt{185}}{16} \right) \]
\[ \phi = \tan^{-1} \left( \frac{13.6}{16} \right) \]
\[ \phi = \tan^{-1} (0.85) \]
\[ \phi = 40.37 \]
or
\[ \phi = \cos^{-1} \left( \frac{z}{r} \right) \]
\[ \phi = \cos^{-1} \left( \frac{16}{21} \right) \]
\[ \phi = \cos^{-1} (0.7619) \]
\[ \phi = 40.37 \]
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