1. Coordinate Systems

d. Cylindrical and Spherical Coordinates - 3D and nD

1. 3D Cylindrical Coordinates

Rectangular Coordinates \((x,y,z)\) are one way to specify a point, \(P\), in the space, but they are not the only way. We can alternatively use Cylindrical Coordinates \((r,\theta,z)\). Cylindrical coordinates are useful when you want to study cylinders or other objects which have rotational symmetry about the \(z\)-axis like a paraboloid, \(z=x^2+y^2\) or a cone \(z=\sqrt{x^2+y^2}\).

Cylindrical Coordinates \((r,\theta,z)\) are just 2D polar coordinates \((r,\theta)\) with an extra coordinate \(z\) which measures the distance above or below the \(xy\)-plane as in rectangular coordinates.

If you think about a point on a cylinder, \(r\) gives the radius of the cylinder, \(\theta\) gives the angle of the point on the cylinder measured from the positive \(x\)-axis and \(z\) gives the height of the point above the \(xy\)-plane.

Converting between Rectangular and Cylindrical Coordinates

To convert between rectangular and cylindrical coordinates, we have the equations \[\begin{aligned} x&=r\cos\theta \qquad \qquad &&r=\sqrt{x^2+y^2} \\[3pt] y&=r\sin\theta &&\tan\theta=\dfrac{y}{x} \\[3pt] z&=z &&z=z \end{aligned}\] with the same conditions on solving for \(\theta\) as there were for polar coordinates

Coordinate Surfaces

When you hold one of the coordinates fixed and let the other two vary, the point \(P=(r,\theta,z)\) traces out a coordinate surface. The surfaces are named by the coordinate which is constant.

When \(r\) is constant, you get a cylinder, or \(r\)-surface. You specify where you are on the surface by giving \(\theta\) and \(z\).
When \(\theta\) is constant, you get a vertical half plane (assuming \(r \ge 0\)), called a \(\theta\)-surface. You specify where you are on the surface by giving \(r\) and \(z\).
When \(z\) is constant, you get a horizontal plane, or \(z\)-surface. You specify where you are on the surface by giving \(r\) and \(\theta\).

Here are the plots of the coordinate surfaces:

Constant \(r\) Surface

Constant \(\theta\) Surface

Constant \(z\) Surface

Rotate these plots with your mouse:

Coordinate Curves

When you hold two of the coordinates fixed and let the other one vary, the point \(P=(r,\theta,z)\) traces out a coordinate curve. The curves are named by the coordinate which is changing. The coordinate curves are the intersections of the coordinate surfaces.

When \(\theta\) and \(z\) are constant, you get a radial ray (assuming \(r \ge 0\)) perpendicular to the \(z\)-axis called an \(r\)-curve.
When \(r\) and \(z\) are constant, you get a horizontal circle called a \(\theta\)-curve.
When \(r\) and \(\theta\) are constant, you get a vertical line called a \(z\)-curve.

Here are the plots of the coordinate curves: Rotate with your mouse.

\(r\)-Curves

\(\theta\)-Curves

\(z\)-Curves

Coordinate Grid

When you draw several coordinate curves or surfaces of each type you get a coordinate grid. Here is the cylindrical coordinate grid as surfaces and as curves. Notice how all points in space can be reached with these coordinates. You can watch an animation of the cylindrical coordinate grid at the top of this page.

Cylindrical Coordinate Grid as Surfaces
The \(r\)-surfaces are red cylinders.
The \(\theta\)-surfaces are cyan half-planes.
The \(z\)-surfaces are blue horizontal planes.

Cylindrical Coordinate Grid as Curves
The \(r\)-curves are red radial lines.
The \(\theta\)-curves are cyan circles.
The \(z\)-curves are blue vertical lines.