Finding a curve which satisfies a special condition about angle

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 $\pi/2$.

$\hspace1in$ 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?

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