Identifying Surfaces in Spherical Coordinates

In summary, the conversation is about converting a spherical equation to rectangular coordinates and finding the equation of a sphere with radius 1/2 and center (0,1/2,0). The equation \rho = sin\theta * sin\phi is related to the equation \rho^{2} = x^{2} + y^{2} + z^{2} through conversion equations.
  • #1
josh28
4
0

Homework Statement


[tex]\rho = sin\theta * sin\phi[/tex]


Homework Equations


I know that [tex]\rho^{2} = x^{2} + y^{2}+z^{2}[/tex]


The Attempt at a Solution


I tried converting it to cartesian coordinates but I can't seem to get a workable answer that way. I know that the answer is the sphere with radius 1/2 and center (0,1/2,0) but I have no idea how to get there.

Thank you!
 
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  • #2
josh28 said:

Homework Statement


[tex]\rho = sin\theta * sin\phi[/tex]


Homework Equations


I know that [tex]\rho^{2} = x^{2} + y^{2}+z^{2}[/tex]
What are the other equations that relate spherical to rectangular coordinates?
josh28 said:

The Attempt at a Solution


I tried converting it to cartesian coordinates but I can't seem to get a workable answer that way. I know that the answer is the sphere with radius 1/2 and center (0,1/2,0) but I have no idea how to get there.

Multiply each side of this equation by rho.
[tex]\rho = sin\theta * sin\phi[/tex]

After that, use the conversion equations to convert everything to rectangular coordinates.
 
  • #3
Thank you!
 
  • #4
Sure - you're welcome!
 

1. What are spherical coordinates?

Spherical coordinates are a type of coordinate system used to locate points on a sphere. They consist of three values: radius, inclination, and azimuth. The radius represents the distance from the origin to the point, the inclination is the angle between the radius and the positive z-axis, and the azimuth is the angle between the projection of the radius on the xy-plane and the positive x-axis.

2. How are spherical coordinates different from cartesian coordinates?

Spherical coordinates are different from cartesian coordinates in that they use a different set of values to locate a point. Cartesian coordinates use x, y, and z values to locate a point in 3-dimensional space, while spherical coordinates use radius, inclination, and azimuth values to locate a point on a sphere.

3. Why is it important to be able to identify surfaces in spherical coordinates?

Identifying surfaces in spherical coordinates is important because it allows us to describe and analyze objects that have a spherical shape, such as planets, stars, and other celestial bodies. It also allows us to solve problems in various fields, including physics, engineering, and geography.

4. How do you convert between spherical and cartesian coordinates?

To convert from spherical coordinates to cartesian coordinates, you can use the following equations:

x = r * sin(inclination) * cos(azimuth)

y = r * sin(inclination) * sin(azimuth)

z = r * cos(inclination)

To convert from cartesian coordinates to spherical coordinates, you can use these equations:

r = √(x² + y² + z²)

inclination = arccos(z / r)

azimuth = arctan(y / x)

5. How can you use spherical coordinates to solve real-world problems?

Spherical coordinates can be used to solve real-world problems in various fields, such as navigation, astronomy, and geology. For example, in navigation, spherical coordinates can be used to determine the coordinates of a specific location on Earth's surface. In astronomy, they can be used to locate and track the position of celestial objects. In geology, they can be used to map the surface of a spherical planet or to analyze geological features such as mountains and valleys.

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