Nonparametric Curves:
Mathematically specking, either a parametric or a nonparametric form is utilized to represent a curve. A nonparametric representation is implicit or explicit. For a plane curve, an explicit, nonparametric form is indicated by :
y = f (x)
An instance is the equation of a straight line, y = mx + b. In this form, for each x-value only one y-value is get. As a result, closed or multiple-value curves, for instance a circle, may not be represented explicitly. Implicit representations of the form is following
f (x, y) = 0
do not have this limitation.
A general second-degree implicit equation written down as following
ax2 + 2bxy + cy2 + 2dx + 2ey + f = 0
provides a variety of two-dimensional curve forms called as conic sections. The three types of conic sections are the hyperbola, the parabola, & the ellipse.
Both implicit and explicit nonparametric curve representations are axis dependent. So, the choice of coordinate system affects the ease of use. For instance, if in the selected coordinate system, an infinite slope is needed as a boundary condition, either the chosen coordinate system should be changed or the infinite slope numerically represented by a large negative or positive value.
Further, while points on an axis-dependent nonparametric curve are calculated at equal increments in x or y, they are not distributed evenly along the curve length. This unequal distribution of points affects the quality and accuracy of graphical representation. Despite of these limitations, nonparametric representations are useful. Though, their limitations lead to an interest in parametric curve representations.