Problem 68 - Calculus Early Transcendentals: Pearson New International Edition

Question

Plot the Lissajous figure defined by \(x=\cos 2 t\), \(y=\sin 7 t, 0 \leq t \leq 2 \pi\). Explain why this is a closed curve even though its graph does not look closed.

Step-by-Step Solution

Answer

The curve is closed because the motions complete full cycles over the interval \(0 \leq t \leq 2\pi\).

Step 1: Understand the Parametric Equations

In this problem, we are given parametric equations to plot a Lissajous figure. The equations are \(x = \cos 2t\) and \(y = \sin 7t\). These equations describe how \(x\) and \(y\) change with respect to the parameter \(t\), a variable that ranges from 0 to \(2\pi\).

Step 2: Analyze the Frequencies

The coefficients of \(t\) in the equations are 2 and 7, which represent the frequencies of variation in the x and y directions, respectively. These differing frequencies will affect the appearance and periodicity of the graph.

Step 3: Calculate the Periods of Motion

The period of \(x = \cos 2t\) is \(\frac{2\pi}{2} = \pi\), and the period of \(y = \sin 7t\) is \(\frac{2\pi}{7}\). The overall pattern will repeat over the least common multiple of these periods.

Step 4: Find the Least Common Multiple

Find the least common multiple of the periods \(\pi\) and \(\frac{2\pi}{7}\). It is \(2\pi\) since that is the smallest value where both the cosine and sine functions complete an integer number of full cycles.

Step 5: Plot the Lissajous Figure

Plot the values of \(x = \cos 2t\) and \(y = \sin 7t\) for \(t\) ranging from 0 to \(2\pi\). This will graphically display the Lissajous figure. The curve will appear to have multiple loops or crossings.

Step 6: Explain the Curve's Closure

Although it may visually appear complex and open, the curve is closed because both x and y will return to their original positions for \(t = 2\pi\). This is due to the least common multiple of their periods aligning to \(2\pi\), ensuring the entire pattern completes a full loop back to the starting point.

Key Concepts

Parametric Equations

In mathematics, parametric equations are a pair of equations where both the x and y coordinates are defined in terms of a third variable, often denoted as \(t\). In the case of the Lissajous figure, the equations given are \(x = \cos 2t\) and \(y = \sin 7t\). These expressions show how both x and y change as the parameter \(t\) varies from 0 to \(2\pi\).

Why use parametric equations? They allow us to trace out complex paths in a plane, differing significantly from traditional y = f(x) functions. With parametric equations, you focus more on the path traced out by the changes in t, rather than a one-to-one correspondence between x and y.

To graph these equations, you substitute values into t and compute corresponding x and y coordinates. This methodically defines a path or shape—like the Lissajous figure—showcasing beautiful, looping designs that are the heart of parametric plots.

Frequencies

The term 'frequencies' in parametric equations refers to how rapidly the function oscillates. In the Lissajous figure, the coefficients of \(t\) in \(x = \cos 2t\) and \(y = \sin 7t\) are the frequencies for the x and y components, respectively.

The frequency affects how many times the sine or cosine pattern completes within a certain interval. Here:

• The frequency of \(2\) in \(x = \cos 2t\) means the cosine wave completes 2 full cycles over the interval \(0\) to \(2\pi\).
• The frequency of \(7\) in \(y = \sin 7t\) dictates that the sine wave finishes 7 cycles across the same interval.

These differing frequencies cause the path traced by the parametric equations to interweave, creating an intricate pattern characteristic of Lissajous figures.

Periodicity

Periodicity is an important concept to understand when dealing with harmonics in parametric equations. It refers to the distance needed for a function to repeat its pattern. For the Lissajous figure concerned here, it involves calculating the period of the sine and cosine components separately.

The period of \(x = \cos 2t\) is calculated as \(\frac{2\pi}{2} = \pi\). Meanwhile, the period of \(y = \sin 7t\) is \(\frac{2\pi}{7}\).

These differing periods mean each function will repeat its cycle at different intervals. The least common multiple (LCM) of these periods is essential here. In this scenario, both functions' periodicity aligns again at the interval \(2\pi\), meaning the Lissajous figure will complete one whole set of patterns by this point.

This repetitive nature of cycles greatly contributes to the beauty of Lissajous figures, as the blend of different frequency oscillations results in a seamless repeating pattern.

Closed Curve

A common question about Lissajous figures is their curve closure properties, despite appearing open or overlapping. A curve is considered closed if it returns to its starting point. For our Lissajous figure, the closure occurs due to the synchronization of cycle completions at \(t = 2\pi\), aligning with the LCM of the individual periods.

Here's what's happening explicitly:

• Initially, at \(t = 0\), the figure starts at a specific point based on initial values of \(\cos(0)\) and \(\sin(0)\).
• As \(t\) progresses to \(2\pi\), both \(x\) and \(y\) cyclically return to their original states, despite their complex paths in-between.

This alignment ensures that at the end of the full cycle, the endpoints join together to form a closed curve.

Even if visually it seems open or unaligned, the mathematical precision ensures closure through the set relationship of frequencies and periods.