3
$\begingroup$

I am trying to find a parametrization for the curve defined implicitly by $$ F(x,y)=a_1\cos x+a_2\sin x+a_3\cos y+a_4\sin y-b=0, $$ where $a_1$, $a_2$, $a_3$, $a_4$ and $b$ are constants satisfying $a_1^2+a_2^2+a_3^2+a_4^2+b^2=1$, but I do not know what to try. I would like to be able to do it by myself, so I am looking mostly for hints on how to perform this: what methods I need to apply and what steps should I follow.

Thanks in advance!

$\endgroup$
1
$\begingroup$

As a first step, you can combine $$\begin{gathered} {a_1}\cos x + {a_2}\sin x = \sqrt {{a_1}^2 + {a_2}^2} \left( {\frac{{{a_1}}}{{\sqrt {{a_1}^2 + {a_2}^2} }}\cos x + \frac{{{a_2}}}{{\sqrt {{a_1}^2 + {a_2}^2} }}\sin x} \right) \\ = \sqrt {{a_1}^2 + {a_2}^2} \sin (\theta + x) \\ \end{gathered} $$ for some constant angle $\theta$ (where $\theta = {\text{atan}}2({a_1},{a_2})$). You can do similarly for ${a_3}\cos y + {a_4}\sin y$. This will give you just one variable $x$ and one variable $y$ and an easier equation.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.