Fundamentals of Parametric Equations: Learn It 3

Cycloids and Other Parametric Curves

Let’s explore a fascinating connection between everyday motion and parametric curves. Picture yourself riding a bicycle down a straight road. As your wheels roll forward, every point on the tire traces a specific path through space. The path traced by a point on the edge of a rolling wheel creates a special curve called a cycloid.

cycloid

The curve traced by a point on the edge of a circle as it rolls along a straight line without slipping. For a wheel of radius [latex]a[/latex], the parametric equations are:

  • [latex]x(t) = a(t - \sin t)[/latex]
  • [latex]y(t) = a(1 - \cos t)[/latex]

To understand where these equations come from, we’ll break down the motion into two components:

1. The wheel’s center motion: As the wheel rolls, its center moves horizontally at a constant height [latex]a[/latex] (the radius). This gives us:

  • [latex]x(t) = at[/latex]
  • [latex]y(t) = a[/latex]

2. The point’s rotation around the center: A point on the edge rotates clockwise around the center. Relative to the center, this motion is:

  • [latex]x(t) = -a\sin t[/latex]
  • [latex]y(t) = -a\cos t[/latex]

The negative signs account for the clockwise rotation (if the wheel moves left to right). If the negative sign were not there, we would have to imagine the wheel rotating counterclockwise.

Combining these motions gives us the cycloid equations. The point experiences both the forward motion of the center AND the circular rotation around it.

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 10. A wheel traveling along a road without slipping; the point on the edge of the wheel traces out a cycloid.
Visualizing the Motion: Think of a piece of gum stuck to your bicycle tire. As you ride forward, the gum moves in two ways simultaneously: forward with the wheel AND around the wheel’s center. The cycloid is the actual path the gum traces through space.

What happens if instead of rolling along a straight line, a circle rolls along the inside of a larger circle, 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. The resulting curve 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 11. Graph of the hypocycloid described by the parametric equations shown.

hypocycloid

The curve traced by a point on a circle of radius [latex]b[/latex] as it rolls inside a larger circle of radius [latex]a[/latex]. The parametric equations are:

  • [latex]x(t) = (a-b)\cos t + b\cos\left(\frac{a-b}{b}t\right)[/latex]
  • [latex]y(t) = (a-b)\sin t - b\sin\left(\frac{a-b}{b}t\right)[/latex]

These parametric equations might look complex, but they follow a pattern similar to what we saw with the cycloid. Let’s break down what’s happening.

For a hypocycloid, we have a small circle of radius [latex]b[/latex] rolling inside a larger circle of radius [latex]a[/latex]. The center of the rolling circle travels along a circular path of radius [latex]a - b[/latex], which explains the first term in both the [latex]x(t)[/latex] and [latex]y(t)[/latex] equations.

The period of the second trigonometric function in both equations is [latex]\frac{2\pi b}{a-b}[/latex]. This period, along with the ratio [latex]\frac{a}{b}[/latex], determines the shape of the resulting curve.

the ratio [latex]\frac{a}{b}[/latex] and cusps

The ratio [latex]\frac{a}{b}[/latex] directly controls the number of cusps (sharp points or corners) on the hypocycloid:

  • When [latex]\frac{a}{b} = 3[/latex]: You get a deltoid with [latex]3[/latex] cusps
  • When [latex]\frac{a}{b} = 4[/latex]: You get an astroid with [latex]4[/latex] cusps
  • When [latex]\frac{a}{b}[/latex] is rational: The curve has a finite number of cusps and closes
  • When [latex]\frac{a}{b}[/latex] is irrational: The curve has infinite cusps and never closes
 Figure 11 shows the classic example where [latex]a = 4[/latex] and [latex]b = 1[/latex], producing a four-cusped astroid. Figure 12 displays additional possibilities, including cases where [latex]\frac{a}{b}[/latex] is irrational.
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 12. Graph of various hypocycloids corresponding to different values of [latex]\frac{a}{b}[/latex].

When the ratio is irrational, something fascinating happens: the hypocycloid never returns to its starting point and creates infinitely many cusps. These curves are examples of space-filling curves—they wind around indefinitely, gradually filling more and more of the enclosed space.

Hypocycloids aren’t just mathematical curiosities. You’ll find them in:

  • Mechanical engineering: Gear tooth profiles and cam designs
  • Physics: Paths of particles in certain force fields
  • Architecture: Decorative patterns and structural designs