Does a circle have a constant rate of change of it self that defines

[MotoPayton]

Does a circle have a constant rate of change of it self that defines it as a circle.
I have never heard of such a thing but I am curious. It must right??

Since every circle is the same no matter the radius(the arch will change by some constant amount per radian)

If that doesn't make sense.. I am asking by how much does some line segment need to fold onto itself to form a perfect circle. What will be the rate of change of that line segment to form a perfect circle and with respect to what variable??

 

[Char. Limit]

I'm not quite sure what you're asking, but the curvature of a circle of radius r is the constant r^-1.

 

[Hyperbolful]

A differential geometry perspective.

Sooo... I'm not entirely sure if this will help, but I think that the answer to the "constant speed question" is not necesarily. Idealy we could make the circle be traced out at constant speed.

Here is how i think of it.
C:[0,1] --> S1
S1 will be the circle as a subset of the plane.
We can make
C=(cos(t),sin(t))

this takes the first point of the line, and then puts it at the (1,0) point on the circle. Every other point is then curled to follow the tangent vector (-sin(t), cos(t) ).
that gives us a "constant rate".

Now it may be shown that there are probably infinitely many other paramaterizations of a circle, and some analytic geometry will show that you could define circles that don't have constant speed paramaterizations, in fact you could paramaterize it to not be "smooth".

I hope this helps, but I now think maybe you were asking if there is a way to characterize the circle as a "curling" of the line?

 

[Mute]

Does a circle have a constant rate of change of it self that defines it as a circle.
I have never heard of such a thing but I am curious. It must right??

Since every circle is the same no matter the radius(the arch will change by some constant amount per radian)

If that doesn't make sense.. I am asking by how much does some line segment need to fold onto itself to form a perfect circle. What will be the rate of change of that line segment to form a perfect circle and with respect to what variable??

You can define a circle in polar coordinates by

$$\frac{dr(\theta)}{d\theta} = 0$$
with $r(\theta_0) = R \neq 0$. i.e., the radius of the curve does not change with theta, and you know that r for some value of theta is a non-zero number.

 

[CCErejer]

To get the equation of the circle, requirements are radius and center point right..??
(x-h)^2-(y-k)^2=r^2