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 $$x(t) = \frac{t^3-3}{t^2+1}, \;\;\;\;\;y(t)=\frac{t(t^2-3)}{t^2+1}$$ and it self-intersects:

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                  ![MathWorld][1]

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So, rational functions do not suffice to imply non-self-intersection. Pointers would be appreciated—Thanks!

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:

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                  ![MathWorld][1]

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So, rational functions do not suffice to imply non-self-intersection. Pointers would be appreciated—Thanks!