The   Condition

However, this magnified limit need not happen on a parametric graph when  .  Teh graph of

is a point, not a line.  For example, consider the 2D smooth function

We will show that this function does not smoothly parametrize its 2D image.  Notice that   has  .  Also notice that  , the parametric graph is the graph    with a right angle corner and no derivative at  .  

            with      and    

Zooming in on the explicit graph of    at   or X[0], we see tend to zero and the magnified increment of the curve tend to  .

Looking straight down the z-axis at the same curve, we see    and no matter how much we magnify the curve, the 2D appearance does not change.   

On a scale of    the increment  .  A sufficiently magnified 2D increment and derivative are both just a point.  They don't give us tangency and in this specific example, the 2D parametric curve does not have a tangent.  The fuction IS still smooth, but the GRAPH is NOT.

Created by Mathematica  (July 2, 2004)