Why Does the Osculating Circle Formula Involve Derivatives?

[Ele38]

Hi guys!
I learned yesterday what an osculating circle is and I am learning how to find the radius of curvature of some curves. For example I have found that for y=x^2 the radius of the osculating circle for the point [0,0] is 0.5 (That's why circular mirror works similarly to parabolic mirror, with the focus equal to radius/2, right?)
I found that result using non standard analysis, but I know that there is a formula that is used to find the radius of curvature.
\frac{(1+y'^2(x))^{3/2}}{|y''(x)|}
What I can figure out is why this formula can calcuate the radius, i do not understand why there are the first and the second derivatives of the function. Do you know how to demonstrate this formula?

Thanks,
Ele38

 

[LCKurtz]

The problem is that radius is defined for circles. The notion of "radius of curvature" for a general curve has to be defined. As you might expect, the greater the concavity of a curve, the quicker it is turning, much as a circle with a small radius turns sharply. Since concavity is tied to the second derivative, it is not surprising that the notion of radius of curvature involves first and second derivatives. Once you have that notion defined, the osculating circle is the circle that best fits the curve at a point. Google osculating circle to find more details and some nice animations.

 

[Ele38]

Thank you, I did not think about concavity. What is "obscure" to me is what doest "circle that best fits the curve at a point" means in math language...

 

[LCKurtz]

You might find this wikipedia article helpful.

http://en.wikipedia.org/wiki/Curvature

 

[CellCoree]

Hi Ele38,

I can explain the reasoning behind the osculating circle formula. The osculating circle is a circle that best approximates a curve at a given point. In order to find the radius of this circle, we need to consider the curvature of the curve at that point. Curvature is a measure of how much a curve deviates from being a straight line. It is calculated using the first and second derivatives of the function at that point.

The first derivative, or slope of the curve, tells us the rate of change of the function at that point. The second derivative, or the rate of change of the slope, tells us how quickly the curve is changing direction. These two values are essential in determining the curvature of the curve at a given point.

The formula you mentioned, \frac{(1+y'^2(x))^{3/2}}{|y''(x)|}, combines the first and second derivatives in a way that allows us to calculate the curvature at a given point. The numerator takes into account the slope of the curve and the denominator takes into account the rate of change of the slope. When we use this formula, we are essentially finding the radius of a circle that has the same curvature as the curve at that point.

I hope this explanation helps you understand the purpose of the osculating circle formula and why it involves the first and second derivatives of the function. Keep exploring and learning about this interesting mathematical concept!