We have already stated that the ellipse is a Lissajous curve. Other well-known curves are Lissajous curves as well. We list some of them with their names [3]. Readers with graphing calculators can see their shapes, and those of other Lissajous curves not listed.

Parametric equation	Name	Implicit equation
	Circle
	Line segment
	Ellipse
	Lemniscate of Gerono
	Letter C (Parabola)
	Tschirnhausen cubic
	ABC’s logo
	Saddlebag

These examples are useful to point out some of the properties of the Lissajous curves. For instance, in the case of the segment, the parabola, or the Tchirhausen cubic the moving point turns around and runs the same course back. We will refer to those instances as degenerate cases. In the non-degenerate cases, the point never reverses direction. The terminology may not be the best but is useful to identify the cases readily. A Lissajous curve (1) is degenerate when either m is even and or m is odd and , where d = mq - np . In the degenerate case, the curve can be determined with t varying in an interval of length π. But not every interval of this length can describe the curve completely. This discussion about the parameter is important if in the definition of parametric curve the condition of being locally injective is included [1] [4].

In analytic geometry a plane curve can be defined by its implicit equation . If is a polynomial, it is called an algebraic curve and a transcendental curve if f is a transcendental function. Lissajous curves are algebraic curves.
The implicit equation of a Lissajous curve is obtained by eliminating the parameter t from the parametric equations. This process can be done as follows:

1. Express x and y in terms of cos mt, sin mt, cos nt, and sin nt using the sum and difference of angles formulas.
2. Express the trigonometric functions of multiple angles in terms of sin t and cos t.
3. Express the functions sin t and cos t in terms of using the familiar formulas

We then have a rational parametrization of the curve,
4. Eliminate the parameter u.

The above process can be done, sometimes simply, but most of the time in a complicated way that requires time and the use of tools of algebraic elimination [5]. This difficulty becomes apparent when one applies this process to some of the parametric equations listed above. We will prove that the implicit equations of Lissajous curves can be found using Chebyshev polynomials.

Lissajous Figures and Chebyshev Polynomials
Julio Castiñeira Merino

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