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 $\pi/2$.

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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?

  • $\begingroup$ Sorry, I misread the question. $\endgroup$ – Calvin Lin Apr 1 '14 at 23:35
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    $\begingroup$ 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

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.

  • $\begingroup$ 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
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    $\begingroup$ @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

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