A plane curve is defined (locally) as the locus of points where a
``good'' function *f* of two variables vanishes. In particular, let
*p* = (*a*, *b*) be a point where *f* vanishes, then we assume that
*f* (*x*, *y*) = *f*_{1, 0}(*x* - *a*) + *f*_{0, 1}(*y* - *b*) + *o*(**x**) is * continuously differentiable* at this point. The curve is said to be
*singular* at *p* if both the above coefficients are 0; otherwise
we call the curve *non-singular* or *smooth*. For a smooth
curve through *p* the line
*f*_{1, 0}(*x* - *a*) + *f*_{0, 1}(*y* - *b*) = 0 is called
the tangent line. It is the ``best'' linear approximation to the curve
in an obvious way:

A different way of representing curves is to think of a curve as a
``moving point''. A curve can be given in parametric form by writing a
pair of functions
(*x*(*t*), *y*(*t*)) so that as *t* varies we will trace
out a curve. As before we will insist on the two functions being
continuously differentiable. We say that our curve is non-singular
at ``time'' *t* = *t*_{0} we need at least of the pair
(*x'*(*t*_{0}), *y'*(*t*_{0})) to be
non-zero; otherwise we call the curve singular.

To go from the equation to the parametric form we need to show that
lines parallel to a line which is *not* tangent to the curve will
meet the curve in exactly one point near the given point. This done
through the implicit function theorem. (Note to author: Exercises to
be added here).

One of the advantanges of working with ``orders'' of vanishing is that
make these theorems ``explicit'' if we only need our equations to be
satisfied upto terms of some order. For example, we say that
(*x*(*t*), *y*(*t*)) is a parametrisation at *t* = *t*_{0} upto order *r* of the
curve
*f* (*x*(*t*), *y*(*t*)) = *o*((*t* - *t*_{0})^{r}). Similarly, two curves, *f* and *g*
are said to *osculate* upto order *r* if
*f* - *g* = *o*(**x**^{r}). In
particular, any curve osculates upto order 2 with a conic; thus it is
possible to write a parametrisation upto order two quite explicitly.

Finally, there is one distinguished parametrisatisation. Let
(*x*(*t*), *y*(*t*)) be a curve. Thinking of this as a moving point we have
not only a tangent line but a tangent vector (called the velocity
vector)
(*x'*(*t*), *y'*(*t*)). It is thus natural to define the *speed*
of the curve as length of this vector.
We can ask for a constant speed (or more accurately constant energy)
parametrisation. In other words, can we find *t* = *u*(*s*) so that

(*x'*(*t*)^{2} + *y'*(*t*)^{2})_{ | t = u(s)}*u'*(*s*)^{2} = constant

(*x*(*t*), *y*(*t*)) = ,