Question

For constants \(a, b,\) and \(c,\) describe the graphs of the equations \(\rho=a, \theta=b\), and \(\phi=c\) in spherical coordinates.

Short answer

The graph of the equation \(\rho=a\) is a sphere of radius \(a\) centered at the origin. The graph of the equation \(\theta=b\) is a plane passing through the origin and perpendicular to the x-y plane making an angle \(b\) with the positive x-axis. The graph of the equation \(\phi=c\) is a cone with an angle \(c\) to the positive z-axis.

Step-by-step solution

Graph the equation \(\rho=a\)

Set \(\rho=a\), where \(a\) is a constant. This describes a sphere with radius \(a\) centered at the origin. The value of \(\rho\) is the distance from the origin to a point in the space, so taking \(\rho\) to be a constant means that the points that satisfy this equation are all at a fixed distance, \(a\), away from the origin. This is precisely the definition of a sphere with radius \(a\).

Graph the equation \(\theta=b\)

Set \(\theta=b\), where \(b\) is a constant. This describes a plane passing through the origin and is perpendicular to the xy-plane, making an angle of \(b\) with the positive x-axis. The angle \(\theta\) is measured in the xy-plane, so setting \(\theta=b\) forces all points satisfying the equation to lie on the line that makes an angle of \(b\) with the x-axis. Extending this line to a plane passing through the origin gives the graphical representation of this equation.

Graph the equation \(\phi=c\)

Set \(\phi=c\), where \(c\) is a constant. This describes a cone with an angle \(c\) to the z-axis. The angle \(\phi\) is measured from the z-axis, so taking \(\phi\) to be a constant means that the points that satisfy this equation are all at a fixed angle, \(c\), away from the z-axis. This is exactly the definition of a cone with vertex at the origin.