Conditions for a parametric curve to avoid self-intersection? Suppose a planar curve $C$ is defined by parametric
equations in $t$: $x(t)$ and $y(t)$.
Are there conditions on these two functions that guarantee
that $C$ does not self-intersect?
For example,
the Maclaurin trisectrix
can be defined by
$$x(t) = \frac{t^2-3}{t^2+1},
\;\;\;\;\;y(t)=\frac{t(t^2-3)}{t^2+1}$$
and it self-intersects:

 
 
 
 
 
 
 
 
 


So, rational functions do not suffice to imply non-self-intersection.
Pointers would be appreciated—Thanks!
 A: By its very nature, this question cannot expect a definitive answer but here are some suggestions.


*

*For a curve with parametrisation of the form $(\int^t(u)du,f(t))$ for a function $f$ of one variable it is the case if $f$ is injective or, better, if and only if $\int^s f(u)du\neq \int^t f(u)du$ whenever $f(s)=f(t)$.

*Many important curves have parametrisations of this form (circle, cycloid, catenary, ....).

*In a certain sense "every" curve has such a parametrisation.  More precisely, consider the curve with parametrisation $(x(s),y(s))$.  Since self-intersection is preserved under diffeomorphisns, we can suppose that the curve lies in the upper half plane.  Under the new parameter $t$ where the latter is, as a function of $s$, the primitive of $\frac{x'(s)}{y(s)}$, the parametrisation will have the above form.

*Of course, this will not work universally since this $t$ will not always be a 
reparametrisation but it will be in many concrete situations, e.g., if $x$ is strictly monotone.
