Explicit/Implicit Equations

For equations of the form y = f(x), which are explicit equations,
there is only one y value for each x value. The explicit form cannot represent closed or multiple-valued curves.
This limitation can be overcome by using an implicit equation of the general form f(x, y) = 0.
Both explicit and implicit elements are axis dependent. Thus, the choice of the coordinate system affects the calculation of their properties.

Parametric Equations

Shapes required for geometric modeling cannot be expressed with ordinary, single- valued functions such as y = f(x). There are many reasons for this. First, the shapes of many objects we want to model are independent of any coordinate system. If we want to fit a curve or surface through a set of points, we quickly see that it is the relationship between these points themselves that determines the resulting shape and not the relationship between these points and some arbitrary coordinate system. Infact, most modeling applications require that the choice of a coordinate system does not affect the shape.

For these reasons and many others related to ease of programming and computability, the dominant means of representing shapes in geometric modeling is with parametric equations. For example, a 2 dimensional curve is expressed by a set of 2 functions