Spherical Coordinates

In summary, the convention for calculating the angle \phi is to measure it as the angle between the +z axis and a position vector of a point projected onto the yz plane. This convention is used in polar and cylindrical coordinates, with \theta being measured from the +x axis to the +y axis. While there is no general agreement on this issue, it is common to use this convention in mathematical problems, such as computing the intersection between a sphere and a cone using spherical coordinates. Using this convention, the angle \phi can be calculated using a triple integral and simplifies the computation.
  • #1
bomba923
763
0
Just curious, why is [tex] \phi [/tex] calculated as the angle between the +z axis and a position vector of a point of a function, as projected onto the yz plane? Why this convention?

In polar & cylindrical, [tex] \theta [/tex] is calculated from the +x axis to the +y axis (counterclockwise) for position vectors.

*So, why not extrapolate alphabetically, to have the [tex] \phi [/tex] be the angle between the +y axis and the position vector of a point as projected onto the yz plane? :smile:
 
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  • #2
Use whatever convention you like, there is no general agreement on this issue.
 
  • #3
arildno said:
Use whatever convention you like, there is no general agreement on this issue.

But nowhere have I found [tex] \phi [/tex] calculated as the angle between the +y axis and the position vector of a point, projected onto the yz plane...
 
  • #4
You will see why once you start working with problems involving these. For example, take the intersection between a sphere and a cone (I posted about this before). This can be computed by a double integral, but if you use a triple integral and spherical coordinates, it becomes much simpler (thanks to HallsofIvy on this).
 

1. What are spherical coordinates?

Spherical coordinates are a type of coordinate system used to describe the position of a point in 3-dimensional space. It uses three coordinates - radius, inclination, and azimuth - to specify the location of a point relative to a fixed origin.

2. How are spherical coordinates different from Cartesian coordinates?

Spherical coordinates differ from Cartesian coordinates in that they use a radial distance and angles to specify a point's location, rather than just x, y, and z coordinates. This is useful for describing points in curved spaces or for certain physical applications.

3. What is the range of values for each coordinate in spherical coordinates?

The radius coordinate, also known as the radial distance, can range from 0 to infinity. The inclination angle ranges from 0 to 180 degrees, and the azimuth angle ranges from 0 to 360 degrees.

4. How are spherical coordinates used in real-world applications?

Spherical coordinates are used in many applications, including physics, engineering, and astronomy. They are particularly useful for describing the positions of objects in space, as well as for solving problems involving spherical symmetry.

5. What is the conversion between spherical and Cartesian coordinates?

The conversion from spherical to Cartesian coordinates is as follows:
x = r * sin(θ) * cos(φ)
y = r * sin(θ) * sin(φ)
z = r * cos(θ)
where r is the radius, θ is the inclination angle, and φ is the azimuth angle. The conversion from Cartesian to spherical coordinates can be found by solving for r, θ, and φ in the above equations.

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