Is every curved line a part of a circle?

Question

The title says it all.Does every curved line represent a part of a circle?Is there any formal proof for this?

@UmbertoP.-It is a line which is not straight (I mean the angle in it is not $180^0$).That's all I know.. — Soham, Sep 29 '15 at 16:36
No, but at every point of a path you can associate a circle to it based on the curvature of the path. In general, the circle radius will change from one point to the next. — Cameron Williams, Sep 29 '15 at 16:36
You can see: https://en.wikipedia.org/wiki/Osculating_circle. — Emilio Novati, Sep 29 '15 at 16:41
@CameronWilliams-I have two queries.1.If every curve represents a circle then what is an ellipse?2.If every point on a line represents a circle then a straight line also represents a circle... — Soham, Sep 29 '15 at 16:41
Just in case you were confused by all these helpful answers: No, in general a curved line does not represent a part of a circle — TonyK, Sep 29 '15 at 17:35

score 2 · Answer 1 · answered Sep 29 '15 at 16:38

Every smooth enough ($C^2$ maybe? I forget the exact condition) has, for each point, a radius of curvature. Intuitively the radius of the best fitting circle which is tangent at that point. If the radius of curvature is constant, then good things happen. (Try to prove something in the constant radius of curvature case.)

In general, the entire curve need not lie on a circle. Just about anything other than a circle will serve as a counter example. $x^2$ as an easy one.

$C^2$ is basically required: you need to take the derivative of the tangent vector. — Chappers, Sep 29 '15 at 16:43

score 2 · Answer 2 · answered Sep 29 '15 at 16:47

Suppose $P$ is some point on a curve (curved line). It’s definitely possible, and in fact, quite typical, for the part of the curve near $P$ to not be shaped exactly like a circle, even for a short bit. In other words, no matter how you try to draw a circle through $P$ that tries to “match” part of the curve, the circle won’t coincide exactly with the curve close to $P$ except at the one point $P$.

You can get a circle to “hug” the curve at $P$, but if you zoomed in to view the curve and the circle up close, the circle and $P$ would be touching only at a single point, in much the same way that a curve and a straight line can be close (tangent), but only touch at one point.

score 1 · Answer 3 · answered Mar 11 '17 at 19:29

Take $f(x)=x^2$ and suppose that there are $a,b\in \mathbb{R}$ such that
$f(x)=y$ for every $x\in (a,b)$.
Here $y=y_0+\sqrt{r^2-(x-x_0)^2}$ for every $x\in (a,b)$ or $y=y_0-\sqrt{r^2-(x-x_0)^2}$ for every $x\in (a,b)$.
(both of the sign are not possible otherwise $f(x)$ wouldn't be a function)
So, $f(x)=y\Rightarrow x^2=y_0+\sqrt{r^2-(x-x_0)^2}$ for every $x\in (a,b)$ which s a contradiction.
I had the very same question as yours the last two days.
I am surprised it is not true.

Is every curved line a part of a circle?

3 Answers3