Objectives
- To understand how cylindrical coordinates are measured
- To understand shapes created by fixing or varying cylindrical coordinates
Section 2.2 Cylindrical 3D Coordinates
In this section, cylindrical coordinates are introduced in terms of two linear measurements and one angular measurement.
Subsection 2.2.1 Cylindrical Coordinate Measurements
In the first lesson on cylindrical coordinates in three dimensions, the measurements of the \(r\text{,}\) \(\theta\text{,}\) and \(z\) coordinates are introduced. Several examples are given showing the geometric measurements for each coordinate. One example includes reminders of how there is the same ambiguity in coordinate measurement as in polar coordinates.
Subsection 2.2.2 Cylindrical Coordinates: Surfaces and Regions
In the second lesson on cylindrical coordinates in three dimensions, a wireframe grid of cylindrical coordinates is shown and examples of how the coordinates can be measured is demonstrated. The graph of \(z=r\) is explored by first looking at points on the surface, then generalizing according to geometric relationships. The lesson finishes with a few questions to solidify the geometric notions of cylindrical coordinates including specifying a region of space using inequalities.
Activity 2.2.1. Understanding the graph of \(z=r^2\).
In the second lesson on cylindrical coordinates the graph of \(z=r\) is explored in terms of its relationship to cylindrical coordinates. This activity is an extension of that example that will explore the graph of \(z=r^2\text{.}\)
(a)
We begin in two dimensions. What do you think the graph of \(r^2 = 1\text{,}\) \(r^2 = 4\text{,}\) and \(r^2 = 9\) look like as two-dimensional polar coordinate graphs? Confirm your results using the tool below.
Hint.
If you are stuck, convert \(r^2\) to an expression with \(x,y\) and that might give you a clue.
(b)
What happens to the graph for negative values? For example, \(r^2=-1\text{.}\) Confirm your results using the tool above.
(c)
Does the value of \(\theta\) affect whether or not a point will be on the graph of \(z=r^2\) or \(z=r\text{?}\) What impact does this have on the graph?
(d)
Based on what we saw in part a, make a guess at what you think the graph of \(z=r^2\) will look like. Be specific in your description of the graph. Do not plot the graph yet.
(e)
What happens to the graph of \(z=r^2\) for negative values of \(z\text{?}\) What does this tell you about the graph below the \(xy\)-plane?
(f)
Explain why negative \(r\) values will not add any points to the plot.
Hint.
Refer back to what you discovered in part b.
(g)
Use the tool below to see what \(z=r^2\) looks like for various fixed levels of \(z\text{.}\)
(h)
Describe the plot of the graph of \(z=r^2\) and talk about why it is different from the cone plot for the graph of \(z=r\text{.}\)
Worksheet Practice Graphs with Cylindrical Coordinates
1.
In each of the following cases, the cylindrical coordinates \((r,\theta,z)\) of a point in space are given. Draw the point \(P\) in three-dimensional space by identifying the angle \(\theta\) and the distance \(r\) in your drawing.
- \(\displaystyle (2,\frac{\pi}{3},4)\)
- \(\displaystyle (1,\pi,-1)\)
2.
In each of the following cases draw the region of three-dimensional space that is described by the given description in cylindrical coordinates.
- \(r=1\) (include a written description as well)
- \(\theta=\frac{\pi}{2}\) (include a written description as well)
- \(0\leq r \leq 2\text{,}\) \(0\leq \theta \leq \frac{\pi}{2}\)
- \(0\leq r \leq 2\text{,}\) \(z=4\)
- \(0\leq \theta \leq \frac{\pi}{2}\text{,}\) \(0\leq z \)
- \(2\leq r \leq 3\text{,}\) \(\frac{\pi}{2} \leq \theta \leq \pi \text{,}\) \(0\leq z \leq 1\)
