How to visualize in spherical and cylindrical coordinates

Re: imagination

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.

 

Re: imagination

Welcome to PF, queenstudy!

Here's a picture of cylindrical coordinates:


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:


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.

 

Re: imagination

Yes.

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?

 

Re: imagination

+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.