Osculating Circle

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\documentclass{article}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usetikzlibrary{math}
\usepackage{ifthen}
\usepackage[active,tightpage]{preview}
\PreviewEnvironment{tikzpicture}
\setlength\PreviewBorder{1pt}
%
% File name: truncated-cylinder.tex
% Description: 
% A geometric representation of a truncated cylinder is shown.
% 
% Date of creation: September, 19th, 2021.
% Date of last modification: October, 9th, 2022.
% Author: Efraín Soto Apolinar.
% https://www.aprendematematicas.org.mx/author/efrain-soto-apolinar/instructing-courses/
% Source: page 48 of the 
% Glosario Ilustrado de Matem\'aticas Escolares.
% https://tinyurl.com/5udm2ufy
%
% Terms of use:
% According to TikZ.net
% https://creativecommons.org/licenses/by-nc-sa/4.0/
% Your commitment to the terms of use is greatly appreciated.
%
\begin{document}
%
\begin{center}
\tdplotsetmaincoords{70}{120}
%
\begin{tikzpicture}[tdplot_main_coords]
    % Components of the vector function
    \tikzmath{function equis(\t) {return cos((\t) r);};}
    \tikzmath{function ye(\t) {return sin((\t) r);};}
    \tikzmath{function zeta(\t) {return 0.5+0.25*sqrt(\t);};}
    % Components of the derivative f the vector function
    \tikzmath{function dequis(\t) {return -sin((\t) r);};}
    \tikzmath{function dye(\t) {return cos((\t) r);};}
    \tikzmath{function dzeta(\t) {return 0.125/sqrt(\t);};}
    % Components of the normal vector (second derivative of $\vec{r}(t)$
 
 
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