Singular Point of A Curve - Parametric Curves

Parametric Curves

A parameterized curve in R2 is defined as the image of a function g:R→R2, g(t) = (g₁(t),g₂(t)). The singular points are those points where

Many curves can be defined in either fashion, but the two definitions may not agree. For example the cusp can be defined as an algebraic curve, x3−y2 = 0, or as a parametrised curve, g(t) = (t2,t3). Both definitions give a singular point at the origin. However, a node such as that of y2−x3−x2 = 0 at the origin is a singularity of the curve considered as an algebraic curve, but if we parameterize it as g(t) = (t2−1,t(t2−1)), then g′(t) never vanishes, and hence the node is not a singularity of the parameterized curve as defined above.

Care needs to be taken when choosing a parameterization. For instance the straight line y = 0 can be parameterised by g(t) = (t3,0) which has a singularity at the origin. When parametrised by g(t) = (t,0) it is nonsingular. Hence, it is technically more correct to discuss singular points of a smooth mapping rather than a singular point of a curve.

The above definitions can be extended to cover implicit curves which are defined as the zero set f−1(0) of a smooth function, and it is not necessary just to consider algebraic varieties. The definitions can be extended to cover curves in higher dimensions.

A theorem of Hassler Whitney states

Theorem. Any closed set in Rn occurs as the solution set of f−1(0) for some smooth function f:Rn→R.

Any parameterized curve can also be defined as an implicit curve, and the classification of singular points of curves can be studied as a classification of singular point of an algebraic variety.