Circle of Best Approximation and Curvature Recently I have been studying the curvature formula. I understood the change in direction with respect to distance definition $||\frac{dT}{ds}||$, but was bothered when introduced to the radius of curvature definition. Since most explanations use the differential definition to explain the derivation of the formula, they only gloss over the radius form.
On Wikipedia it this is what it states about Radius of Curvature: "For a curve, it equals the radius of the circular arc which best approximates the curve at that point."
https://en.wikipedia.org/wiki/Curvature
What my question is, is how can the circle of best approximation be defined? in Taylor Series, we learn about the best possible polynomial approximation of a certain degree, so how do we define the best circle approximation?
thank you for reading :)
 A: You are indeed right when you mention the analogy with Taylor series.
Consider a plane curve, namely the locus $f(x,y)=0$ in the $(x,y)$-plane, with $f$ a sufficiently smooth function in two variables.
Since we can focus on a single (smooth) point, let us assume by a translation that $f(x,y)$ passes through the origin. Up to rotating the plane we can also assume that the normal to the curve at the origin is the $y$-axis. With these assumptions the curve near the origin is described by the parametrization
$$
x=t,\qquad y=h(t),
$$
for some smooth function $h(t)$ defined in a neighborhood of $t=0$, satisfying $h(0)=0=h'(0)$.
Any circle with center on the $y$-axis (the normal to the curve) and passing through the origin is given by the equation
$$
c_r(x,y)=x^2+(y-r)^2-r^2=0,
$$
for some real number $r$.
To express the osculating condition means that the intersection between the curve and the circle at $0$ has multiplicity two; namely, the function of $t$
$$
c_r(x=t,y=h(t))=t^2+(h(t)-r)^2-r^2
$$
always vanish to first order at $t=0$, as you can check by the Taylor series. However, the condition that it vanishes to higher degree (i.e. at least to second order) at $t=0$ selects a special value for $r$:
$$
r=1/h''(0).
$$
That is how one defines the circle of best approximation ("osculating circle"), and as a consequence, how one shows that its radius is the inverse of the curvature at that point. (It is easy to show that $h''(0)$ is the curvature at the origin of the plane curve $f(x,y)=0$.)
A: The tangent to a curve is the analogue of the first order Taylor development of a function. It is the best local approximation in the sense that the remainder is a sublinear function of the abscissa. (For any other lines, it would be a linear function.)
A tangent is the limit of a chord when the two endpoints tend to each other.
Likewise, the osculating circle to a curve is the analogue of the second order Taylor development. The approximation is such that the remainder is subquadratic (when for other quadratic polynomials, it would be quadratic).
An osculating circle is the limit of a circle by three points on the curve, when these points tend to a single.
[The next approximation levels could be the osculating parabola (third order development) and osculating conic (fourth order).]
