Converting $(x,y)=(\cos t+2\cos4t+4\cos8t, \sin t+2\sin4t+4\sin8t)$ to a single Cartesian equation

Question

Is it possible to convert the parametric equations:

$$\begin{align} x &= \cos t + 2\cos 4t + 4\cos 8t \\ y &= \sin t + 2 \sin 4t + 4 \sin 8t \end{align}$$ into a single Cartesian equation?

More generally, suppose we have a parametric equation system where the $x$ variable is given as a linear combination of 3 $\cos$ functions and the $y$ variable is given as a linear combination of 3 $\sin$ functions. Is it possible to convert this parametric equation system into a Cartesian one?

The standard parametric equation technique of squaring both sides doesn't seem to work here, as we end up with cross products between trigonometric functions.

Answer 1

It can be done, but don't expect the results to be pretty. In general ...

• Expand the $x$ and $y$ expressions in terms of $\cos t$ and $\sin t$.
• Use, say, the Weierstrass substitution $$\sin t = \frac{2u}{1+u^2} \qquad \cos t = \frac{1-u^2}{1+u^2}$$ to get the sines and cosines into a common form. (This can save some trouble of multiple squarings and applications of $\cos^2t+\sin^2t=1$. )
• Eliminate $u$ using Groebner bases or the method of resultants; the latter is conveniently available via the Resultant command in Mathematica,

For your specific problem, the $u$-form of the expressions can be written $$\begin{align} (1+u^2)^8 x &= 7 - 522 u^2 + 7222 u^4 - 31842 u^6 + 51876 u^8 \\ &\quad - 31870 u^{10} + 7194 u^{12} - 534 u^{14} + 5 u^{16} \\[4pt] (1+u^2)^8 y &= 2u(41 - 1137 u^2 + 8637 u^4 - 22933 u^6 + 23003 u^8 \\ &\quad - 8595 u^{10} + 1151 u^{12} - 39 u^{14}) \end{align}$$ Then, eliminating $u$ yields ... (deep breath) ...

$$\begin{align} 0 &= 637553840 - 632073728 x - 433834448 x^2 + 403688768 x^3 + 164267808 x^4 \\ &- 95543936 x^5 - 35461280 x^6 + 10494808 x^7 + 4137356 x^8 - 552640 x^9\\ &- 256324 x^{10} + 13660 x^{11} + 8524 x^{12} - 128 x^{13} - 145 x^{14} + x^{16} \\ &- 649796048 y^2 + 412735552 x y^2 + 400148672 x^2 y^2 - 193975040 x^3 y^2 \\ &- 115104608 x^4 y^2 + 31555528 x^5 y^2 + 16926384 x^6 y^2 - 2209536 x^7 y^2 \\ &- 1286740 x^8 y^2 + 68316 x^9 y^2 + 51144 x^{10} y^2 - 768 x^{11} y^2 - 1015 x^{12} y^2 + 8 x^{14} y^2 \\ &+ 244834080 y^4 - 97869952 x y^4 - 124875616 x^2 y^4 + 31593224 x^3 y^4 \\ &+ 26007112 x^4 y^4 - 3312768 x^5 y^4 - 2584744 x^6 y^4 + 136664 x^7 y^4 \\ &+ 127860 x^8 y^4 - 1920 x^9 y^4 - 3045 x^{10} y^4 + 28 x^{12} y^4 \\ &- 45232288 y^6 + 10533016 x y^6 + 17784496 x^2 y^6 - 2207488 x^3 y^6 \\ &- 2597032 x^4 y^6 + 136696 x^5 y^6 + 170480 x^6 y^6 - 2560 x^7 y^6 - 5075 x^8 y^6 + 56 x^{10} y^6 \\ &+ 4566412 y^8 - 551616 x y^8 - 1305172 x^2 y^8 + 68364 x^3 y^8 + 127860 x^4 y^8 \\ &- 1920 x^5 y^8 - 5075 x^6 y^8 + 70 x^8 y^8 \\ &- 262468 y^{10} + 13676 x y^{10} + 51144 x^2 y^{10} - 768 x^3 y^{10} - 3045 x^4 y^{10} + 56 x^6 y^{10} \\ &+ 8524 y^{12} - 128 x y^{12} - 1015 x^2 y^{12} + 28 x^4 y^{12} \\ &- 145 y^{14} + 8 x^2 y^{14} \\ &+ y^{16} \end{align}$$

There may-or-may-not be a way to manipulate the equation into a more-concise form. I'll leave such considerations to the reader.