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Section5.2Other Coordinate Systems

Sometimes a problem can't be solved until the correct coordinate system is chosen. You have previously done problems which showed you how to graph the coordinate transformation given by polar coordinates. The following problem shows you how to graph in a different coordinate system.

Exercise5.2.1

Consider the coordinate transformation \(T(a,\omega)=(a\cos\omega,a^2\sin \omega)\text{.}\)

(a)

Let \(a=3\) and then graph the curve \(\vec T(3,\omega)=(3\cos\omega,9\sin\omega)\) for \(\omega\in[0,2\pi]\text{.}\)

You can use the following SageMath Cell to check your answer.

Hint

You can start by making a \(\omega,x,y\) table.
(b)

Let \(\omega=\frac{\pi}{4}\) and then, on the same axes as above, add the graph of \(\vec T\left(a,\frac{\pi}{4}\right)=\left(a\frac{\sqrt 2}{2},a^2 \frac{\sqrt 2}{2}\right)\) for \(a\in[0,4]\text{.}\)

Again, you can use the SageMath Cell to check your answer. Notice that you can add the two plots together to superimpose them on each other.

(c)

To the same axes as above, add the graphs of:

(i)

\(\vec T(1,\omega), \vec T(2,\omega), \vec T(4,\omega)\) for \(\omega\in[0,2\pi]\)

Use the SageMath Cells above to check your answer.

Hint

when you're done, you should have a bunch of parabolas and ellipses.
(ii)

\(\vec T(a,0), \vec T(a,\pi/2), \vec T(a,-\pi/6)\) for \(a\in[0,4]\text{.}\)

Subsection5.2.1Exploring Other Coordinates

In 3 dimensions, the most common coordinate systems are cylindrical and spherical. The equations for these coordinate systems are in the table below.

Cylindrical Coordinates Spherical Coordinates
\(\begin{array}{l} x=r\cos\theta\\ y=r\sin\theta\\ z=z \end{array}\) \(\begin{array}{l} x=\rho\sin\phi\cos\theta\\ y=\rho\sin\phi\sin\theta\\ z=\rho\cos\phi \end{array}\)
Table5.2.1Conversion between coordinate systems
Cylindrical Coordinates Spherical Coordinates(Physics/Engineering) Spherical (Math)
\(\begin{array}{l} x=r\cos\theta\\ y=r\sin\theta\\ z=z \end{array}\) \(\begin{array}{l} x=r\sin\theta\cos\phi\\ y=r\sin\theta\sin\phi\\ z=r\cos\theta \end{array}\) \(\begin{array}{l} x=\rho\sin\phi\cos\theta\\ y=\rho\sin\phi\sin\theta\\ z=\rho\cos\phi \end{array}\)
Table5.2.2
Exercise5.2.2

Let \(P=(x,y,z)\) be a point in space. This point lies on a cylinder of radius \(r\text{,}\) where the cylinder has the \(z\) axis as its axis of symmetry. The height of the point is \(z\) units up from the \(xy\) plane. The point casts a shadow in the \(xy\) plane at \(Q=(x,y,0)\text{.}\) The angle between the ray \(\vec{OQ}\) and the \(x\)-axis is \(\theta\text{.}\) Construct a graph in 3D of this information, and use it to develop the equations for cylindrical coordinates given above.

Exercise5.2.3

Let \(P=(x,y,z)\) be a point in space. This point lies on a sphere of radius \(\rho\) (“rho”), where the sphere's center is at the origin \(O=(0,0,0)\text{.}\) The point casts a shadow in the \(xy\) plane at \(Q=(x,y,0)\text{.}\) The angle between the ray \(\vec{OQ}\) and the \(x\)-axis is \(\theta\text{,}\) and is called the azimuth angle. The angle between the ray \(\vec{OP}\) and the \(z\) axis is \(\phi\) (“phi”), and is called the inclination angle, polar angle, or zenith angle. Construct a graph in 3D of this information, and use it to develop the equations for spherical coordinates given above.

See the articles on Wikipedia or MathWorld for a discussion of conventions in different disciplines.

There is some disagreement between different fields about the notation for spherical coordinates. In some fields (like physics), \(\phi\) represents the azimuth angle and \(\theta\) represents the inclination angle. In some fields, like geography, instead of the inclination angle, the elevation angle is given—the angle from the \(xy\)-plane (lines of lattitude are from the elevation angle). Additionally, sometimes the coordinates are written in a different order. You should always check the notation for spherical coordinates before communicating using them.

Subsection5.2.2Computational Practice

These are provided to help you achieve better skills in basic computational answers

1
WeBWorK Exercise
2
WeBWorK Exercise
3
WeBWorK Exercise
4
WeBWorK Exercise