Cycloids and Other Parametric Curves

Learning Outcomes

  • Recognize the parametric equations of a cycloid

Imagine going on a bicycle ride through the country. The tires stay in contact with the road and rotate in a predictable pattern. Now suppose a very determined ant is tired after a long day and wants to get home. So he hangs onto the side of the tire and gets a free ride. The path that this ant travels down a straight road is called a cycloid (Figure 10). A cycloid generated by a circle (or bicycle wheel) of radius a is given by the parametric equations

[latex]x\left(t\right)=a\left(t-\sin{t}\right),y\left(t\right)=a\left(1-\cos{t}\right)[/latex].

 

To see why this is true, consider the path that the center of the wheel takes. The center moves along the x-axis at a constant height equal to the radius of the wheel. If the radius is a, then the coordinates of the center can be given by the equations

[latex]x\left(t\right)=at,y\left(t\right)=a[/latex]

 

for any value of [latex]t[/latex]. Next, consider the ant, which rotates around the center along a circular path. If the bicycle is moving from left to right then the wheels are rotating in a clockwise direction. A possible parameterization of the circular motion of the ant (relative to the center of the wheel) is given by

[latex]x\left(t\right)=\text{-}a\sin{t},y\left(t\right)=\text{-}a\cos{t}[/latex].

 

(The negative sign is needed to reverse the orientation of the curve. If the negative sign were not there, we would have to imagine the wheel rotating counterclockwise.) Adding these equations together gives the equations for the cycloid.

[latex]x\left(t\right)=a\left(t-\sin{t}\right),y\left(t\right)=a\left(1-\cos{t}\right)[/latex].

 

A series of circles with center marked and a point on the circle drawing out a curve as if the circle was rolling along a plane. The shape made seems to be half an ellipse with height the diameter of the original circle and with major axis the circumference of the circle.

Figure 1. A wheel traveling along a road without slipping; the point on the edge of the wheel traces out a cycloid.

Now suppose that the bicycle wheel doesn’t travel along a straight road but instead moves along the inside of a larger wheel, as in Figure 11. In this graph, the green circle is traveling around the blue circle in a counterclockwise direction. A point on the edge of the green circle traces out the red graph, which is called a hypocycloid.

Two circles are drawn both with center at the origin and with radii 3 and 4, respectively; the circle with radius 3 has an arrow pointing in the counterclockwise direction. There is a third circle drawn with center on the circle with radius 3 and touching the circle with radius 4 at one point. That is, this third circle has radius 1. A point is drawn on this third circle, and if it were to roll along the other two circles, it would draw out a four pointed star with points at (4, 0), (0, 4), (−4, 0), and (0, −4). On the graph there are also written two equations: x(t) = 3 cos(t) + cos(3t) and y(t) = 3 sin(t) – sin(3t).

Figure 2. Graph of the hypocycloid described by the parametric equations shown.

The general parametric equations for a hypocycloid are

[latex]\begin{array}{}\\ \\ x\left(t\right)=\left(a-b\right)\cos{t}+b\cos\left(\frac{a-b}{b}\right)t\hfill \\ y\left(t\right)=\left(a-b\right)\sin{t}-b\sin\left(\frac{a-b}{b}\right)t.\hfill \end{array}[/latex]

 

These equations are a bit more complicated, but the derivation is somewhat similar to the equations for the cycloid. In this case we assume the radius of the larger circle is a and the radius of the smaller circle is b. Then the center of the wheel travels along a circle of radius [latex]a-b[/latex]. This fact explains the first term in each equation above. The period of the second trigonometric function in both [latex]x\left(t\right)[/latex] and [latex]y\left(t\right)[/latex] is equal to [latex]\frac{2\pi b}{a-b}[/latex].

The ratio [latex]\frac{a}{b}[/latex] is related to the number of cusps on the graph (cusps are the corners or pointed ends of the graph), as illustrated in Figure 12. This ratio can lead to some very interesting graphs, depending on whether or not the ratio is rational. Figure 11 corresponds to [latex]a=4[/latex] and [latex]b=1[/latex]. The result is a hypocycloid with four cusps. Figure 12 shows some other possibilities. The last two hypocycloids have irrational values for [latex]\frac{a}{b}[/latex]. In these cases the hypocycloids have an infinite number of cusps, so they never return to their starting point. These are examples of what are known as space-filling curves.

A series of hypocycloids is given. The first is a three pointed star marked a/b = 3. The second is a four pointed star marked a/b = 4. The third is a five pointed star marked a/b = 5. None of these first three figures has lines that cross each other. The fourth figure is a five pointed star but this one has lines which cross each other and looks like the star that children first learn to draw; it is marked a/b = 5/3. A similar sort of star with seven points is next and is marked a/b = 7/3. Then a similar star with eight points is next and is marked a/b = 8/3. The next figure is a complicated series of curves that ultimately creates a small rosette in the middle; this is marked a/b = π. Lastly, there is an even more complicated series of curves that creates a large rosette with sharper florets marked a/b = the square root of 2.

Figure 3. Graph of various hypocycloids corresponding to different values of [latex]\frac{a}{b}[/latex].

Activity: The Witch of Agnesi

Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However, perhaps the strangest name for a curve is the witch of Agnesi. Why a witch?

Maria Gaetana Agnesi (1718–1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the “versoria,” a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, “versiera,” but in Latin, this same word means a “female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since.

The witch of Agnesi is a curve defined as follows: Start with a circle of radius a so that the points [latex]\left(0,0\right)[/latex] and [latex]\left(0,2a\right)[/latex] are points on the circle (Figure 13). Let O denote the origin. Choose any other point A on the circle, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through [latex]\left(0,2a\right)[/latex]. The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle.

Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution. In this project you will parameterize these curves.

A circle with bottom at point O (the origin) and top at point (0, 2a) is drawn. The x axis is drawn from point O, and the y axis is drawn up from point O through (0, 2a). Parallel to the x axis is a line drawn from (0, 2a); it has point B marked to the right. A line from point B to point O passes through the circle at point A. A line is drawn parallel to the x axis from point A, and it forms a right angle with a line drawn down from point B; these lines intersect at point P. There is a curve that is symmetric about the y axis that passes through the point P. This curve has its maximum at (0, 2a) and gently decreases through the point P.

Figure 4. As the point A moves around the circle, the point P traces out the witch of Agnesi curve for the given circle.

    1. C is the point on the x-axis with the same x-coordinate as A.
    2. x is the x-coordinate of P, and y is the y-coordinate of P.
    3. E is the point [latex]\left(0,a\right)[/latex].
    4. F is the point on the line segment OA such that the line segment EF is perpendicular to the line segment OA.
    5. b is the distance from O to F.
    6. c is the distance from F to A.
    7. d is the distance from O to B.
    8. [latex]\theta[/latex] is the measure of angle [latex]\text{\angle }COA[/latex].On the figure, label the following points, lengths, and angle:

    The goal of this project is to parameterize the witch using [latex]\theta[/latex] as a parameter. To do this, write equations for x and y in terms of only [latex]\theta[/latex].

  2. Show that [latex]d=\frac{2a}{\sin\theta }[/latex].
  3. Note that [latex]x=d\cos\theta[/latex]. Show that [latex]x=2a\cot\theta[/latex]. When you do this, you will have parameterized the x-coordinate of the curve with respect to [latex]\theta[/latex]. If you can get a similar equation for y, you will have parameterized the curve.
  4. In terms of [latex]\theta[/latex], what is the angle [latex]\text{\angle }EOA?[/latex]
  5. Show that [latex]b+c=2a\cos\left(\frac{\pi }{2}-\theta \right)[/latex].
  6. Show that [latex]y=2a\cos\left(\frac{\pi }{2}-\theta \right)\sin\theta[/latex].
  7. Show that [latex]y=2a{\sin}^{2}\theta[/latex]. You have now parameterized the y-coordinate of the curve with respect to [latex]\theta[/latex].
  8. Conclude that a parameterization of the given witch curve is

    [latex]x=2a\cot\theta ,y=2a{\sin}^{2}\theta ,-\infty <\theta <\infty[/latex].
  9. Use your parameterization to show that the given witch curve is the graph of the function [latex]f\left(x\right)=\frac{8{a}^{3}}{{x}^{2}+4{a}^{2}}[/latex].

Activity: Travels with My Ant: The Curtate and Prolate Cycloids

Earlier in this section, we looked at the parametric equations for a cycloid, which is the path a point on the edge of a wheel traces as the wheel rolls along a straight path. In this project we look at two different variations of the cycloid, called the curtate and prolate cycloids.

First, let’s revisit the derivation of the parametric equations for a cycloid. Recall that we considered a tenacious ant trying to get home by hanging onto the edge of a bicycle tire. We have assumed the ant climbed onto the tire at the very edge, where the tire touches the ground. As the wheel rolls, the ant moves with the edge of the tire (Figure 14).

As we have discussed, we have a lot of flexibility when parameterizing a curve. In this case we let our parameter t represent the angle the tire has rotated through. Looking at Figure 14, we see that after the tire has rotated through an angle of t, the position of the center of the wheel, [latex]C=\left({x}_{C},{y}_{C}\right)[/latex], is given by

[latex]{x}_{C}=at\text{and}{y}_{C}=a[/latex].

 

Furthermore, letting [latex]A=\left({x}_{A},{y}_{A}\right)[/latex] denote the position of the ant, we note that

[latex]{x}_{C}-{x}_{A}=a\sin{t}\text{and}{y}_{C}-{y}_{A}=a\cos{t}[/latex].

 

Then

[latex]\begin{array}{c}{x}_{A}={x}_{C}-a\sin{t}=at-a\sin{t}=a\left(t-\sin{t}\right)\hfill \\ {y}_{A}={y}_{C}-a\cos{t}=a-a\cos{t}=a\left(1-\cos{t}\right).\hfill \end{array}[/latex]

 

There are two figures marked (a) and (b). Figure a has a circle with point A on the circle at the origin. The circle has

Figure 5. (a) The ant clings to the edge of the bicycle tire as the tire rolls along the ground. (b) Using geometry to determine the position of the ant after the tire has rotated through an angle of t.

Note that these are the same parametric representations we had before, but we have now assigned a physical meaning to the parametric variable [latex]t[/latex].

After a while the ant is getting dizzy from going round and round on the edge of the tire. So he climbs up one of the spokes toward the center of the wheel. By climbing toward the center of the wheel, the ant has changed his path of motion. The new path has less up-and-down motion and is called a curtate cycloid (Figure 15). As shown in the figure, we let b denote the distance along the spoke from the center of the wheel to the ant. As before, we let t represent the angle the tire has rotated through. Additionally, we let [latex]C=\left({x}_{C},{y}_{C}\right)[/latex] represent the position of the center of the wheel and [latex]A=\left({x}_{A},{y}_{A}\right)[/latex] represent the position of the ant.

There are three figures marked (a), (b), and (c). Figure a has a circle with

Figure 6. (a) The ant climbs up one of the spokes toward the center of the wheel. (b) The ant’s path of motion after he climbs closer to the center of the wheel. This is called a curtate cycloid. (c) The new setup, now that the ant has moved closer to the center of the wheel.

  1. What is the position of the center of the wheel after the tire has rotated through an angle of t?
  2. Use geometry to find expressions for [latex]{x}_{C}-{x}_{A}[/latex] and for [latex]{y}_{C}-{y}_{A}[/latex].
  3. On the basis of your answers to parts 1 and 2, what are the parametric equations representing the curtate cycloid?

    Once the ant’s head clears, he realizes that the bicyclist has made a turn, and is now traveling away from his home. So he drops off the bicycle tire and looks around. Fortunately, there is a set of train tracks nearby, headed back in the right direction. So the ant heads over to the train tracks to wait. After a while, a train goes by, heading in the right direction, and he manages to jump up and just catch the edge of the train wheel (without getting squished!).

    The ant is still worried about getting dizzy, but the train wheel is slippery and has no spokes to climb, so he decides to just hang on to the edge of the wheel and hope for the best. Now, train wheels have a flange to keep the wheel running on the tracks. So, in this case, since the ant is hanging on to the very edge of the flange, the distance from the center of the wheel to the ant is actually greater than the radius of the wheel (Figure 16).

    The setup here is essentially the same as when the ant climbed up the spoke on the bicycle wheel. We let b denote the distance from the center of the wheel to the ant, and we let t represent the angle the tire has rotated through. Additionally, we let [latex]C=\left({x}_{C},{y}_{C}\right)[/latex] represent the position of the center of the wheel and [latex]A=\left({x}_{A},{y}_{A}\right)[/latex] represent the position of the ant (Figure 16).

    When the distance from the center of the wheel to the ant is greater than the radius of the wheel, his path of motion is called a prolate cycloid. A graph of a prolate cycloid is shown in the figure.

    There are three figures marked (a), (b), and (c). Figure a has a circle and a point A that is outside the circle on the y axis (below the origin). The circle is tangent to the x axis at the origin. The circle appears to be travelling to the right on the x axis, with point A being above the x axis in a second image of the circle drawn slightly to the right. Figure b has a circle in the first quadrant with center C. It touches the x axis at xc. A point A is drawn outside the circle and a right triangle is made from this point and point C. The hypotenuse is marked b, the angle at C between A and xc is marked t, and the distance from C to xc is marked a. Lines are drawn to give the x and y values of A as xA and yA, respectively. Similarly, a line is drawn to give the y value of C as yC. Figure c shows the curve that point A would trace out, as the circle travels to the right. It is vaguely sinusoidal with an extra loop at the bottom once per revolution.

    Figure 7. (a) The ant is hanging onto the flange of the train wheel. (b) The new setup, now that the ant has jumped onto the train wheel. (c) The ant travels along a prolate cycloid.

  4. Using the same approach you used in parts 1– 3, find the parametric equations for the path of motion of the ant.
  5. What do you notice about your answer to part 3 and your answer to part 4?

    Notice that the ant is actually traveling backward at times (the “loops” in the graph), even though the train continues to move forward. He is probably going to be really dizzy by the time he gets home!