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Let $C=\{(x,y)\in \Bbb R^2: (x-x_0)^2+(y-y_0)^2=r^2\}$ and let $\varphi :[0,2\pi]\to \mathbb{R}^2$, $\theta \mapsto (x_o+r\cos \theta, y_0+r\sin \theta)$, with $r>0$.

I'm trying to prove that $C\subseteq \varphi ([0,2\pi])$.

What I tried:

Let $(x,y)\in C$. Then $$\frac{y-y_0}{r}=\pm \sqrt {1-\left(\frac{x-x_0}{r}\right)^2}$$ and

$$-1\leq \frac{x-x_0}{r}\leq 1$$

So I set $$\theta=\arccos \frac{x-x_0}{r}$$

There are no issues with the first component of $\varphi (\theta)$.

The second component gives: $$y_0+r\sin \theta=y_0+r\cdot \left(\pm \sqrt{1-(\cos \theta)^2}\right)=y_0+r\cdot\left(\pm \sqrt{1-\left(\frac{x-x_0}{r}\right)^2}\right)$$

Now I'd like to replace $\pm \sqrt{1-\left(\frac{x-x_0}{r}\right)^2}$ with $\displaystyle \frac{y-y_0}{r}$, but I can't due to the possibility of different signs. I even say that they can have different signs. For instance set $x_0=3, y_0=5, x=2, y=5-\sqrt 3$. Then $\displaystyle \frac{y-y_0}{r}<0$, but $\displaystyle \sin \arccos \frac{x-x_0}{r}>0$.

Is there a way to fix what I tried?

If not, what $\theta$ should I take?

Please read $\textbf{carefully}$: I'm not interested in geometric proofs. I want an analytical proof in the fashion of what I tried.

Thank you for your time.

Can anyone help me regarding the comment on Abel's answer? Thanks.

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Let a point $(x,y)\in C$ be given. Then at least one of the following holds: $$({\rm i}) \ x-x_0>0,\quad ({\rm ii}) \ y-y_0>0,\quad ({\rm iii}) \ x-x_0<0,\quad ({\rm iv}) \ y-y_0<0\ .$$ These cases are not disjoint, but in each case a $\theta$ will be produced such that $(x,y)=\phi(\theta)$. Consider the two following examples: The point $(1,1)$ satisfies (i) and (ii), and a careful analysis would show that the two $\theta$ values obtained for this point coincide. On the other hand, the point $(1,-1)$ satisfies (i) and (iv), but the two $\theta$ values obtained for this point differ by $2\pi$.

In case (i) let $$\theta:=\arcsin{y-y_0\over r}\ \in\bigl]-{\pi\over2},{\pi\over 2}\bigr[\quad .$$ Then $y=y_0+r\sin\theta$, and $$(x-x_0)^2 =r^2-(y-y_0)^2= r^2(1-\sin^2\theta)=r^2\cos^2\theta\ .$$ As $x-x_0$, $\cos\theta$, and $r$ are all $>0$ it follows that $x-x_0=r\cos\theta$, or $x=x_0+r\cos\theta$. This proves $(x,y)=\phi(\theta)$ for a certain $\theta\in\ \bigl]-{\pi\over2},{\pi\over 2}\bigr[\ $.

In case (ii) let $$\theta':=-\arcsin{x-x_0\over r}\ \in\bigl]-{\pi\over2},{\pi\over 2}\bigr[\quad .$$ Then $x=x_0-r\sin\theta'$, and $$(y-y_0)^2 =r^2-(x-x_0)^2= r^2(1-\sin^2\theta')=r^2\cos^2\theta'\ .$$ As $y-y_0$, $\cos\theta'$, and $r$ are all $>0$ it follows that $y-y_0=r\cos\theta'$, or $y=y_0+r\cos\theta'$. If we now let $\theta:=\theta'+{\pi\over 2}$ then $(x,y)=\phi(\theta)$ for a certain $\theta \in\ \bigl]0,\pi[\ $.

In case (iii) let $$\theta':=-\arcsin{y-y_0\over r}\ \in\ \bigl]-{\pi\over2},{\pi\over 2}\bigr[\quad .$$ Then $y=y_0-r\sin\theta'$, and $$(x-x_0)^2 =r^2-(y-y_0)^2= r^2(1-\sin^2\theta')=r^2\cos^2\theta'\ .$$ As $x-x_0<0$, and $\cos\theta'$, $r$ are $>0$ it follows that $x-x_0=-r\cos\theta'$, or $x=x_0-r\cos\theta'$. If we now let $\theta:=\theta'+\pi$ then $(x,y)=\phi(\theta)$ for a certain $\theta\in\ \bigl]{\pi\over2},{3\pi\over 2}\bigr[\ $.

The case (iv) now may be safely left to you.

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  • $\begingroup$ @Christian Blatter On the second case: "it follows that $y-y_0=r\cos\theta'$, or $y=\color{red}{x_0}+r\cos\theta'$". Is there no danger with you taking non mutually exclusive cases? For instance the first and second one may happen at the same time. $\endgroup$ – Git Gud Apr 13 '13 at 13:07
  • $\begingroup$ @Git Gud: Thank you for signaling the typo. – Concerning your question, see my edit. $\endgroup$ – Christian Blatter Apr 13 '13 at 13:34
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Hint: If your sign comes out wrong, try picking $\phi = 2\pi-\theta$ and note that $\cos(\phi)=\cos(\theta)$, whereas $\sin(\phi) = -\sin(\theta)$.

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    $\begingroup$ I see what you mean, but I still fail to write something like: $$\theta=\begin{cases} something_1, \text{ if $P$} \\something_2, \text{ if $\neg P$} \end{cases}$$ Can you tell me what $P, something_1, something_2$ are? Thanks. $\endgroup$ – specialguest Apr 11 '13 at 12:47
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    $\begingroup$ You have to write something like $\theta = \begin{cases} \arccos\left(\frac{x-x_0}{r}\right) & y-y_0\geq 0 \\ 2\pi-\arccos\left(\frac{x-x_0}{r}\right) & y-y_0 < 0\end{cases}$, or the other way around depending on how exactly you define your $\arccos$. $\endgroup$ – Abel Apr 12 '13 at 13:37
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    $\begingroup$ I'm truly grateful for your answer. However Christian's answer doesn't have any credit given to it and I can't decide which answer I like the best, so I decided to accept his. Again: thank you. $\endgroup$ – specialguest Apr 12 '13 at 16:48
  • $\begingroup$ @specialguest That seems fair :). Glad I could help. $\endgroup$ – Abel Apr 12 '13 at 16:52

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