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In the case of a single parameter, parametric equations are commonly used to express the trajectory of a moving point, in which case, the parameter is often, but not necessarily, time, and the point describes a curve, called a parametric curve. In the case of two parameters, the point describes a surface, called a parametric surface.
A Lissajous figure, made by releasing sand from a container at the end of a Blackburn pendulum. A Lissajous curve / ˈ l ɪ s ə ʒ uː /, also known as Lissajous figure or Bowditch curve / ˈ b aʊ d ɪ tʃ /, is the graph of a system of parametric equations
Plane curves can be represented in Cartesian coordinates (x, y coordinates) by any of three methods, one of which is the implicit equation given above. The graph of a function is usually described by an equation = in which the functional form is explicitly stated; this is called an explicit representation.
In it, geometrical shapes can be made, as well as expressions from the normal graphing calculator, with extra features. [8] In September 2023, Desmos released a beta for a 3D calculator, which added features on top of the 2D calculator, including cross products, partial derivatives and double-variable parametric equations. [9]
The Fermat spiral with polar equation = can be converted to the Cartesian coordinates (x, y) by using the standard conversion formulas x = r cos φ and y = r sin φ.Using the polar equation for the spiral to eliminate r from these conversions produces parametric equations for one branch of the curve:
Such a parametric equation completely determines the curve, without the need of any interpretation of t as time, and is thus called a parametric equation of the curve (this is sometimes abbreviated by saying that one has a parametric curve). One similarly gets the parametric equation of a surface by considering functions of two parameters t and u.
The curve is given by the following parametric equations: [2] = ... or by the following polar equation: = ...
In topology and knot theory, the trefoil is usually defined using a knot diagram instead of an explicit parametric equation. In algebraic geometry, the trefoil can also be obtained as the intersection in C 2 of the unit 3-sphere S 3 with the complex plane curve of zeroes of the complex polynomial z 2 + w 3 (a cuspidal cubic).