How to tell that a parametric curve does not intersect itself?

Question

For example: if a curve is defined by $x=3\cos{t}-\cos{3t}$, $y=3\sin{t}-\sin{3t}$, $0\le t \le \pi$

then $\frac{dx}{dt}=-3\sin{t}+3\sin{3t}$, and $\frac{dy}{dt}=3\cos{t}-3\cos{3t}$

So, we can not say that $x(t)$ and $y(t)$ are monotonic function. So, it may attain the same ordered pair. how can I tell that it passes through each point only once? please help me to understand. thank you.

Answer 1

Try to rewrite the curve:

$(x_t,y_t)=(\cos{t}-\cos{3t},3\sin{t}-\sin{3t})=...=(\cos(t)(2-\cos(2t)),4\sin^3(t))$

Then, try to find $t_0\neq t_1$ such that $(x_{t_o},y_{t_0})=(x_{t_1},y_{t_1})$. Start with the second equation/coordinate and the solutions you obtain on this replace in the first equation/cordinate. You will get an answer to your question.

Answer 2

Intersection of a curve with itself can be found using $$ \{ (t_0,t_1) \in R \times R| x(t_0)=x(t_1), y(t_0)=y(t_1), t_0\neq t_1 \}$$