Center and radius of the osculating circle - The limiting position of a circle trough three points

Question

I am stucked on problem 1.7.2.b of Differential Geometry of Curves and Surfaces by Manfredo do Carmo. The problem is similar as this topic, but here the exercise defines the osculator circle, ie, this circle of exercise is whose we call osculator circle, and is in $\mathbb{R}^3$. My difficult is another of that topic.

(Adaptated exercise) Let $\alpha(s) \colon I \to \mathbb{R}^3$ be the parametrization by arc length of a curve. Consider a circle that passes through the three points $\alpha(0)$, $\alpha(h_1)$, and $\alpha(h_2)$. Prove that as $(h_1, h_2) \to (0, 0)$, the limiting position of this circle has center at the line that contains $n(0)$ and radius $1/k(0)$.

We can consider the canonical form of the curve at $0$.

As some similar exercises of the book, I beginned witring $P: ax+by+cz=0$ as the plane containing $\alpha(0),\alpha(h_1),\alpha(h_2)$ (by item a) of the question we know that its limiting position is the osculating plane). The center $C$ of this circle $\Pi$ is given by the intersection of mediatrixs at $P$, with directions

$$v_1=(a,b,c)\wedge \alpha(h_1)\quad,\quad v_2=(a,b,c)\wedge\alpha(h_2)$$

Am I correct...?

Well, considering $|(a,b,c)|=1$, we obtain:

$$v_1\cdot v_2=\begin{vmatrix}|(a,b,c)|^2&0\\ 0&\alpha(h_1)\cdot \alpha(h_2) \end{vmatrix}=\alpha(h_1)\cdot \alpha(h_2)$$

We ask for $k_1,k_2$ s.t.

$$\alpha(h_1)+v_1k_1=\alpha(h_2)+v_2k_2$$

I wrote some calculus from here but they are confused and I could not finish...

Many thanks in advance.

Answer 1

I think I get the answer by taking circle

$$C:(x-a)^2+(y-b)^2+(z-c)^2=a^2+b^2+c^2=r^2$$

instead of plane $P$,

defining

$$F(s)=(x-a)^2+(y-b)^2+(z-c)^2-r^2$$

and doing some derivatives and calculus of these kind ( Derivatives of first/second order going to zero).

Many thanks for attention on topic and I am open to suggestions and corrections and still offer the bounty.

Answer 2

If $\alpha(0)$, $\alpha(h_1)$, and $\alpha(h_2)$ are three distinct non-aligned points, then there exist a unique circle passing through them. If $h_1,h_2$ are distinct and close enough to $0$, this is granted if $\kappa(0) \neq 0$.

This circle lies in the plane $\alpha(0) + span\{\alpha(h_1)-\alpha(0), \alpha(h_2)-\alpha(h_0)\}$. Call $c$ the center of the circle and set: \begin{align} C:=&c-\alpha(0)\\ U:=&\alpha(h_1)-\alpha(0)\\ V:=&\alpha(h_2)-\alpha(0)\\ W:=&V-\dfrac{\langle V, U \rangle}{\langle U,U \rangle}U \end{align} Notice $W\neq 0$, since the three points $\alpha(0), \alpha(h_1), \alpha(h_2)$ are not aligned. The fact $c$ is equidistant from $\alpha(0), \alpha(h_1), \alpha(h_2)$ corresponds to the following to the equations: \begin{equation} \Vert C \Vert^2 = \Vert C-U \Vert^2 \end{equation} \begin{equation} \Vert C \Vert^2 = \Vert C-V \Vert^2 \end{equation} Setting $C=aU+bW$ you can solve these equations to find $a,b$ and, therefore, the center of the circle (use the fact $U\perp W$). All you have to do now is taking the limit as $h_1,h_2\to 0$ with $h_1\neq h_2$. The Frenet formulas should be helpful. After long calculation you should be able to show what is claimed.

Answer 3

Taking $F(s)=(x(s)-a)^2+(y(s)-b)^2+(z(s)-c)^2-r^2$ where $(a,b,c)$ is the center of a circle crossing $\alpha(0),\alpha(h_1),\alpha(h_2)$. Consider $F'(s)=2(x'(x-a)+y'(y-b)+z'(z-c))$ using local canonical form of $\alpha$ and taking the limit as $s$ approaches $0$ gives $F'(0)=-2a$. Similarly for $F''(s)=2(x''(x-a)+x'^2)+2(y''(y-b)+y'^2)+2(z''(z-c)+z'^2)$ we get $F''(0)=2-kc$. Using the mean value theorem twice and noticing $F(0)=F(h_1)=F(h_2)=0$ (and consequently points $F'(k_1)=F'(k_2)=0$ and $F''(j)=0$). If $h_1,h_2 \rightarrow 0$ then $a\rightarrow 0$ and $b\rightarrow \frac{1}{k}$ we know from part a $c\rightarrow 0$ so we know the center is $\frac{n(0)}{k}$ as desired