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I'm struggling with the construction of the osculating circle of a curve $\gamma$ as the limit circle passing threw the points $\gamma(s), \gamma(s-h_1), \gamma(s+h_2)$, and finding the expressions for the radius and center of it, since every proof I've seen so far assumes some kind of convergence (often uniform) of functions involved in the proof, that is not proved nor assumed. For example, in the answer of Ted Shifrin in this post Limit definition of the osculating circle , he says that the function $g_h$ converges for $h \to 0^+$, and he means uniformly, as he state it in the comments, but I can't manage to prove it nor to find a reference who does it. Another example is the answer of Mars to this question Center and radius of the osculating circle - The limiting position of a circle trough three points , where he supposes (I think) uniform convergence as $(h_1, h_2) \to 0$, in order to say (for example) that $F'_{h_1,h_2}(k(h_1)) \to F'(0)$.

Could someone help me filling these steps?

Many thanks in advance.

Lorenzo Vanni
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  • I don’t have it on me now, but I recall the first few pages of Spivak’s differential geometry Vol 2 has a discussion of osculatung circles. – peek-a-boo Dec 29 '23 at 20:05
  • The second-differentiability of $\gamma$ at the point should be sufficient to prove uniform convergence of the function $g$ on a compact neighborhood of the origin (you can interpret that to mean a closed and bounded interval). – Paul Sinclair Dec 30 '23 at 12:54
  • @TedShifrin - FYI, since this question involves one of your answers. – Paul Sinclair Dec 30 '23 at 12:55

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