How to visualize in spherical and cylindrical coordinates

In summary, visualizing in spherical and cylindrical coordinates can be difficult, but it is important to exercise your imagination through drawing and building models to better understand how varying different coordinates affects the resulting figure. It is helpful to start with the definition of each shape and consider the distance from the reference point, and to remember that in spherical coordinates, the distance from the origin is always constant, while in cylindrical coordinates, the distance from the z-axis is constant.
  • #1
queenstudy
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Homework Statement



i just want to know how to visualize in spherical and cylindrical coordinates I am really having a rough time doing that
for example why is that when we keep r constant we get a sphere and θ constant a cone why??

Homework Equations





The Attempt at a Solution

 
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  • #2


The geometry follows from their definitions.

In spherical coordinates, you are just imagining you are at the origin of the coordinate system and you locate a particular position by how much you must turn to look at it and how far away it is. This is actually how you would naturally look for things IRL.

By comparison, in rectangular coords you are figuring how far you have to go in each direction of an artificial grid to reach the position.

By definition - a sphere is the set of all points in 3D that are a fixed distance from a reference point. The reference point is called the center of the sphere and the fixed distance is called the radius of the sphere.

In spherical coords, therefore, if you are at the center, the distance to any point on the surface is going to be fixed at the radius of the sphere. (If we don't put the origin of the coordinate system at the center of the sphere, then the r component will vary.)

You'll find a similar thing when you consider other shapes.
Start with the definition of a cone, and figure out how far away the surface is for someone sitting at the apex.

Everyone pretty much gets used to these things by sketching the situation.
 
  • #3


Welcome to PF, queenstudy! :smile:

Here's a picture of cylindrical coordinates:
CylindricalCoordinates_1001.gif


If you keep r constant, you're tracing the cylinder that you're actually seeing (assuming you vary theta and z).

And here's a picture of spherical coordinates:
SphericalCoordinates_1201.gif


Keeping r constant, while varying the other coordinates traces the sphere.

Keeping phi constant, while varying theta, traces the upper circle.
If we also vary r, we get a cone.
 
  • #4


i understand what you people said , but the problem is that when i keep r constant for instance , if i change the angle or z like i would have to change it like 1000000000 times to get the figure
i memorized them but still having problems like how in the world would i know that when i vary r ill get a cone can a person actually imagine that?
 
  • #5


queenstudy said:
i understand what you people said , but the problem is that when i keep r constant for instance , if i change the angle or z like i would have to change it like 1000000000 times to get the figure

Yes.
queenstudy said:
i memorized them but still having problems like how in the world would i know that when i vary r ill get a cone can a person actually imagine that?

Well, I recommend you try to draw a picture.
For instance, start with the upper circle from the spherical coordinates (corresponds to varying theta while keeping r constant).
Place a dot where the origin is.
And draw a line to connect the origin to the left side of the circle (corresponds to varying r while keeping theta constant).
And another line to the right side of the circle.

Does the result look like a cone to you?
 
Last edited:
  • #6


+1 to the serenaphile.
The only way to overcome the limits of your imagination is to exercize it.
I'm afraid that means drawing the pictures, building models, that kind of thing.
 
  • #7


ahhhhhhhhh i get i just thought like its something easy , that i can do thank you
 

What are spherical and cylindrical coordinates?

Spherical and cylindrical coordinates are alternative ways of representing points in three-dimensional space. Spherical coordinates use two angles (θ and φ) and a distance (r) from the origin, while cylindrical coordinates use an angle (θ), a distance (ρ) from the origin, and a height (z) above the xy-plane.

When should I use spherical coordinates instead of Cartesian coordinates?

Spherical coordinates are often used to describe objects or phenomena with spherical symmetry, such as planets or electromagnetic fields. They can also be useful when solving certain types of integrals, particularly those involving spherical or cylindrical shapes.

How do I convert between spherical and cylindrical coordinates?

To convert from spherical coordinates to cylindrical coordinates, use the following equations: ρ = rsin(φ), z = rcos(φ), and θ = θ. To convert from cylindrical coordinates to spherical coordinates, use the equations: r = √(ρ² + z²), φ = arctan(ρ/z), and θ = θ.

How do I visualize points in spherical and cylindrical coordinates?

In spherical coordinates, the angle θ represents the rotation around the z-axis, and the angle φ represents the rotation from the z-axis. The distance r is the distance from the origin. In cylindrical coordinates, the angle θ represents the rotation around the z-axis, the distance ρ is the distance from the origin in the xy-plane, and the height z is the distance above or below the xy-plane.

Are there any real-world applications of spherical and cylindrical coordinates?

Yes, spherical and cylindrical coordinates are used in a variety of fields, including physics, engineering, and geography. In physics, they are used to describe the motion of particles and the behavior of electromagnetic fields. In engineering, they are used in the design of structures and machinery. In geography, they are used to map and navigate the Earth's surface.

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