The parametric curve given by x=cosa t and y=sinb t is known...

The parametric curve given by $x=\cos a t$ and $y=\sin b t$ is known as a Lissajous figure. The $x$ -coordinate oscillates $a$ times between 1 and -1 as $t$ goes from 0 to 2 $\pi$, while the $y$ -coordinate oscillates $b$ times over the same $t$ interval. This behavior is repeated over every interval of length $2 \pi$. The entire motion takes place in a unit square. Plot the following Lissajous figures for a range of $t$ that ensures that the resulting figure is a closed curve. In each case, count the number of times that the curve touches the horizontal and vertical borders of the unit square. (a) $x=\sin t, y=\cos t$ (b) $x=\sin 3 t, y=\cos 5 t$ (c) $x=\cos 5 t, y=\sin 15 t$ (d) $x=\sin 2 t, y=\cos 9 t$

The parametric curve given by $x=\cos a t$ and $y=\sin b t$ is known as a Lissajous figure. The $x$ -coordinate oscillates $a$ times between 1 and -1 as $t$ goes from 0 to 2 $\pi$, while the $y$ -coordinate oscillates $b$ times over the same $t$ interval. This behavior is repeated over every interval of length $2 \pi$. The entire motion takes place in a unit square. Plot the following Lissajous figures for a range of $t$ that ensures that the resulting figure is a closed curve. In each case, count the number of times that the curve touches the horizontal and vertical borders of the unit square.
(a) $x=\sin t, y=\cos t$
(b) $x=\sin 3 t, y=\cos 5 t$
(c) $x=\cos 5 t, y=\sin 15 t$
(d) $x=\sin 2 t, y=\cos 9 t$

Key Concepts

Lissajous Figures

Lissajous figures are the graphical representations of trajectories defined by parametric equations where each coordinate is a sinusoidal function, typically of the form x = cos(a·t) and y = sin(b·t) or similar. They are used to study the relationship between different oscillatory motions and the effects of frequency ratios and phase differences on the resulting pattern.

Parametric Equations

Parametric equations express the coordinates of the points of a curve as functions of a variable, usually time. This approach allows for the description of complex curves where both x and y vary over time, enabling the analysis of movement and periodic properties in a two-dimensional plane.

Periodicity and Closed Curves

The periodic nature of sine and cosine functions means that a full cycle is completed over a fixed interval, typically 2?. When the frequency ratios in parametric equations are rational, the resulting curve is closed, meaning it eventually retraces its path exactly, completing a finite and repeated pattern.

Frequency Ratio

The frequency ratio, determined by the integers multiplying the parameter in the sine and cosine functions, controls the number of oscillations in the curve's x and y directions. This ratio is critical in determining the symmetry, complexity, and number of intersections or contacts with the boundaries of the region in which the curve lies.

Unit Square Constraints

Since sine and cosine functions have ranges of [-1, 1], the plots of these functions, in the context of Lissajous figures, fit naturally within a unit square. This bounded nature is important when analyzing how the curves interact with the borders—specifically, how many times they touch or cross the horizontal and vertical boundaries.

Intersection and Border Analysis

Analyzing the number of times a Lissajous curve touches or intersects the borders of its bounding box involves understanding the oscillatory behavior of the curve. This requires examining the points where the sine or cosine functions reach their maximum or minimum values, which correspond to the points of contact with the borders.

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