why the theta angle in spherical coordinates don't go beyond 0-pi? means to 2*pi?

Answer 1

Suppose you have a point in regular Cartesian (x,y,z) coordinates. The z-coordinate gives the height of the point above (or below) the xy plane.

Now take a look at the point (x,y,0) - the point on the xy plane that's directly above (or below) the coordinate (x,y,z).

The location of (x,y,0) can be defined in terms of r and theta, where r is the distance from the origin. The possible range of theta would have to be anywhere from 0 to 360 degrees (or 0 to 2pi if you prefer), since (x,y,0) can be on any point on a full circle of radius r centered on the origin. Therefore, theta can go from 0 to 2pi.

Now draw a line from the origin to (x,y,0). Then draw a line from the origin to (x,y,z). The angle between those two lines is your spherical phi coordinate.

phi can only range from 0 to pi, since the angle phi can only go from 90 degrees (or pi/2) above the xy plane to 90 degrees below it.

If you tried going more than 90 degrees above (or below) the xy plane, then you would no longer have the same x and y coordinates as the point (x,y,0). You'd have the same x and y coordinates as (-x,-y,0). That would not be kosher, so we restrict phi to a range of half a circle, 0 to pi.

Hopefully that clears things up. :-)

Answer 2

The theta angle does vary from 0 to 2pi, just like in cyllindrical coordinates.

Phi only goes from 0 to pi because that value of phi has taken you from the positive z-axis to the negative z-axis.

Picture it in cartesian coordinates for a moment...

Start with your hand pointing straight up... that is phi = 0.
At phi = pi/2 you are on the x-y plane.
At phi = pi you are pointing straight down.

Aloha

Answer 3

When you say spherical do you mean 3D space?
In 2D space, consider a circle with centre O and a horizontal radius r(on the right). Then the angle theta is measured counter-clockwise from r. So when it goes an entire circle and comes back to r, we just have to measure how much it goes beyond that.

Answer 4

Huh? Don't the coordinates go in a circle and cycle back over themselves?