Implicit curve

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

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.

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(t), y(t) both of whose functional forms are explicitly stated, and which are dependent on a common parameter

The fifth example shows the possibly complicated geometric structure of an implicit curve.

The implicit function theorem describes conditions under which an equation

This theorem is the key for the computation of essential geometric features of the curve: tangents, normals, and curvature.

In practice implicit curves have an essential drawback: their visualization is difficult.

But there are computer programs enabling one to display an implicit curve.

Special properties of implicit curves make them essential tools in geometry and computer graphics.

In general, implicit curves fail the vertical line test (meaning that some values of x are associated with more than one value of y) and so are not necessarily graphs of functions.

If the defining relations are sufficiently smooth then, in such regions, implicit curves have well defined slopes, tangent lines, normal vectors, and curvature.

There are several possible ways to compute these quantities for a given implicit curve.

, one can express these formulas directly in terms of the partial derivatives of

into the formulas for a tangent and curvature of the graph of the explicit equation

yields The essential disadvantage of an implicit curve is the lack of an easy possibility to calculate single points which is necessary for visualization of an implicit curve (see next section).

A smooth approximation of a convex polygon can be achieved in the following way: Let

Then a subset of the implicit curve with suitable small parameter

For example, the curves contain smooth approximations of a polygon with 5 edges (see diagram).

In case of two lines one gets For example, the product of the coordinate axes variables yields the pencil of hyperbolas

For example, generates blending curves between the two circles The method guarantees the continuity of the tangents and curvatures at the points of contact (see diagram).

For an implicit curve one has to solve two subproblems: In both cases it is reasonable to assume

In practice this assumption is violated at single isolated points only.

For the solution of both tasks mentioned above it is essential to have a computer program (which we will call

that is exactly on the curve, up to the accuracy of computation: In order to generate a nearly equally spaced polygon on the implicit curve one chooses a step length

The algorithm traces only connected parts of the curve.

If the implicit curve consists of several or even unknown parts, it may be better to use a rasterisation algorithm.

If the net is dense enough, the result approximates the connected parts of the implicit curve.