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 ratio of frequencies (2:7) is rational.

Step 1: Understanding Lissajous Figures

Lissajous figures are complex and beautiful shapes resulting from the parametric equations of two intersecting sine waves. In this problem, the equations are defined by the cosine and sine functions of different frequencies: \(x = \cos 2t\) and \(y = \sin 7t\). These different frequencies will influence the looping and closure of the curve.

Step 2: Graphing the Functions

To graph the Lissajous figure, substitute various values of \(t\) ranging from 0 to \(2\pi\) into the equations \(x = \cos 2t\) and \(y = \sin 7t\). This generates the pairs \((x, y)\). Because of the nature of sine and cosine functions, all points will lie within the rectangle defined by \(-1 \leq x \leq 1\) and \(-1 \leq y \leq 1\).

Step 3: Analyzing Closure Criterion

For a Lissajous curve \(x = \cos(at)\) and \(y = \sin(bt)\), the curve is closed if \(\frac{a}{b}\) is rational. Here, \(\frac{a}{b} = \frac{2}{7}\), which is rational. Therefore, despite any visual gaps due to resolution or sampling, the theoretical curve is closed.

Step 4: Understanding Periodicity

The individual periodicities of the trigonometric functions influence the closure of the curve. \(\cos 2t\) completes a cycle every \(\pi\), and \(\sin 7t\) every \(\frac{2\pi}{7}\). Both functions realign their starting conditions every time \(t\) is a common multiple of these periods, namely every \(2\pi\) units, ensuring a closed loop.

Key Concepts

Parametric Equations

Parametric equations are a powerful tool in mathematics used to describe curves in the plane. These types of equations express coordinates \((x, y)\) as functions of a parameter, typically denoted as \(t\). By using parametric equations, we can generate curves that might be impossible to describe with a single-function equation such as the familiar \(y = f(x)\).
In the context of Lissajous figures, the parametric equations \(x = \cos 2t\) and \(y = \sin 7t\) are employed to create fascinating patterns. Here, the parameter \(t\) ranges from 0 to \(2\pi\) and dictates both the progression through the curve and the pace at which the curve is drawn.
One key aspect of parametric equations is their capability to represent complex and intricate paths, which often exhibit symmetries and patterns not easily visible in standard Cartesian coordinates. By changing the functions used for \(x(t)\) and \(y(t)\), or altering the range of \(t\), many different shapes can be drawn.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, play a crucial role in controlling the behavior of Lissajous figures. These functions are periodic, meaning they repeat their values in regular intervals. For the figure in our exercise:

• \(x = \cos 2t\)
• \(y = \sin 7t\)

Both the functions share a periodic nature. Specifically, \(\cos 2t\) completes a full cycle, returning to its starting point every \(\pi\) units, expressing a period of \(\pi\). Conversely, \(\sin 7t\) has a period of \(\frac{2\pi}{7}\).
These periodic functions influence how the points \((x, y)\) are plotted over time. As \(t\) increases, each function oscillates in its designated range, creating loops and intersections characteristic of Lissajous patterns.
The interaction of these different frequencies creates the intricate "dancing" patterns typical of Lissajous curves. Their harmonics and overtones form visual representations of the underlying mathematical beauty. By choosing different coefficients, such as the 2 and 7, various figures can be achieved.

Curve Closure

The concept of curve closure in Lissajous figures is closely tied to the ratio of the frequencies of the trigonometric functions. For a curve given by \(x = \cos(at)\) and \(y = \sin(bt)\) to close, the ratio \(\frac{a}{b}\) must be rational.
In this exercise, the ratio \(\frac{a}{b} = \frac{2}{7}\) is rational, meaning ultimately, both the cosine and sine functions will realign to their starting points after a certain interval of \(t\). This ensures that despite any discrepancies in the plotting due to insufficient resolution, the theoretical curve is indeed closed.
Furthermore, the alignment happens when \(t\) equals a common multiple of their periods, which is \(2\pi\) here because both functions will have just completed a whole number of oscillations.
Even if visually, continuity might appear broken due to the resolution of the graph, mathematically, the curve is a continuous, closed loop. Understanding this principle helps in appreciating the subtleties in Lissajous patterns and confirms the elegance hidden in mathematical expressions.