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In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly x and y. For example, the unit circle is defined by the implicit equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1}. In general, every implicit curve is defined by an equation of the form
for some function F of two variables. Hence an implicit curve can be considered as the set of zeros of a function of two variables. Implicit means that the equation is not expressed as a solution for either x in terms of y or vice versa.
If F {\displaystyle F} is a polynomial in two variables, the corresponding curve is called an algebraic curve, and specific methods are available for studying it.
Plane curves can be represented in Cartesian 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 y = f {\displaystyle y=f} in which the functional form is explicitly stated; this is called an explicit representation. The third essential description of a curve is the parametric one, where the x- and y-coordinates of curve points are represented by two functions x, y both of whose functional forms are explicitly stated, and which are dependent on a common parameter t . {\displaystyle t.}