2
$\begingroup$

We can see that the angle of $$\frac{x^2}{a^2}+\frac{y^2}{1-a^2}=1\ \ \ (0\lt a\lt 1)$$ from every point on $$C : x^2+y^2=1$$ is $\pi/2$.

$\hspace1in$enter image description here

Then, here is my question.

Question : If the angle of a figure $F$ from every point on $C$ is constant, then can we say that $F$ is an ellipse?

Here, suppose that $F$ is a simple closed curve on the coordinate plane, and that $F$ is strictly inside of $C$.

For a fixed point $P$ on $C$, let $L_P$ be the set of the lines from $P$ to every point on $F$. Also, let $\theta_P$ be the set of the angles, which include $F$, between every two distinct elements of $L_P$.

Then, suppose that the angle of $F$ from a fixed point $P$ on $C$ is defined as the maximum element of $\theta_P$.

I've been interested in this question, but I don't have any good idea. Can anyone help?

$\endgroup$
  • $\begingroup$ Sorry, I misread the question. $\endgroup$ – Calvin Lin Apr 1 '14 at 23:35
  • 1
    $\begingroup$ Even if we stick with $\pi/2$ as our angle, I'm not sure that the only such curve is an ellipse. There might be something clever that you can construct from circular arcs, analogous to a curve of constant width. $\endgroup$ – TonyK Apr 4 '14 at 8:53
4
+50
$\begingroup$

The points from which a given curve are seen at a fixed angle are called isoptics. If the angle is a right angle, they are called orthoptics. These have been investigated by several authors. The title of a paper by A. Miernowski ("Parallelograms inscribed in a curve having a circle as $\frac \pi 2$-isoptic") in Ann. Univ. Mariae Curie 62 (2008) 105-111, suggests that the answer to your query is negative.

$\endgroup$
  • $\begingroup$ I suspected that it's impossible for this topic to be uncovered at all :) thanks for the reference! $\endgroup$ – Evgeny Apr 4 '14 at 15:39
  • $\begingroup$ The paper is freely available here (PDF file). But unfortunately there aren't any pictures. In fact I found it impossible to follow, but that's probably just me... $\endgroup$ – TonyK Apr 4 '14 at 18:24
  • $\begingroup$ I found the following which are related to your problem. "Conjectured isoptic characterisation of a circle (Klamkin, Amer. Math. Monthly 95 (1988) 845) and "Isoptic characterisation of a circle" by Nitsche (same journal, 97 (1990) 45-47. It seems that you need to know that two isoptics are an ellipse to deduce that the curve is a ellipse. $\endgroup$ – jena Apr 5 '14 at 6:28
  • $\begingroup$ @jena: Thank you very much for your answer and comment! $\endgroup$ – mathlove Apr 5 '14 at 6:51
  • 1
    $\begingroup$ @TonyK: I found the following PDF which I found from the references of the paper jena wrote in his/her answer. This includes some pictures. kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/0766-01.pdf $\endgroup$ – mathlove Apr 5 '14 at 6:54

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.