Eliminating time from parametric questions

Question

Is there a general technique to eliminate a parameter from two parametric equations? E.g. given the following two parametric equations dictating the motion of a point how can I eliminate parameter $t$ (standing for time)?

\begin{align} x_1(t)&=\sin(2\pi t)+2\cos(2\pi t)\\ x_2(t)&=\cos(2\pi t)+2\sin(2\pi t) \end{align}

Intuitively, I think that these are the parametric equations of a rotated ellipsis. Using Matlab I get the following figure

Answer 1

$$\begin{cases} x_1=\sin (2 \pi t)+2 \cos (2 \pi t)\\ x_2=2 \sin (2 \pi t)+\cos (2 \pi t)\\ \end{cases} $$ set $$\sin (2 \pi t)=s;\;\cos(2 \pi t)=c$$ we know that $$\cos^2(2 \pi t)+\sin^2(2 \pi t)=1\to c^2+s^2=1$$

the system becomes $$\begin{cases} x_1=s+2 c\\ x_2=2s+c\\ \end{cases} $$ whence $$\begin{cases} c=\frac{1}{3} (2 x_1-x_2)\\ s= \frac{1}{3} (2 x_2-x_1)\\ \end{cases} $$ now remember that $$c^2+s^2=1$$ $$\left[\frac{1}{3} (2 x_1-x_2)\right]^2+\left[ \frac{1}{3} (2 x_2-x_1)\right]^2$$ After some simplification we get $$5 x_1^2-8 x_1x_2+5 x_2^2=9$$

Answer 2

There isn't really a general method, I think. You have to judge each case separately. Here it looks like identities such as $\sin^2(a) + \cos^2(a)= 1$ are going to be helpful in eliminating $t$ terms.