Nonparametric Curves:
Mathematically, either a parametric or a nonparametric form is utilized to represent a curve. A nonparametric representation is either implicit or explicit. For a plane curve, an explicit, nonparametric form is provided by:
y = f (x)
An instance is the equation of a straight line, y = mx + b. In this form, for every x-value just one y-value is attained. As a result, closed or multiple-value curves, for example a circle, may not be represented explicitly. Implicit representations of the form
f (x, y) = 0
do not contain this restriction.
A general second-degree implicit equation written as following
ax2 + 2bxy + cy2 + 2dx + 2ey + f = 0
gives a variety of two-dimensional curve forms known as conic sections. The three kinds of conic sections are the hyperbola, the parabola, and the ellipse.
Explicit and implicit both nonparametric curve representations are axis based. Therefore, the choice of coordinate system influences the ease of use. For instance, if in the selected coordinate system, an infinite slope is needed as a boundary condition, either the selected coordinate system ought to be changed or the infinite slope numerically defined by a big negative or positive value.
Additional, while points on an axis-dependent nonparametric curve are computed at alike increments in x or y, they are not distributed evenly along the curve length. This unequal distribution of points influences the accuracy and quality of graphical representation. Despite of these restrictions, nonparametric representations are useful. But, their restriction lead to an interest in parametric curve representations.