Geodesic
Osculating quadric of the spatial curve
Journal of the Belarusian State University. Mathematics and Informatics, Tome 1 (2017), pp. 11-15.

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In the investigation of local properties of a space curve assotiated objects which have good approximation characteristics are often used. The main ones – the osculating plane and the osculating sphere. As known, the osculating plane has tangency of at least $2^{nd}$ degree with the curve, while the osculating sphere – at least $3^{rd}$ degree. In the paper a problem of finding of $2^{nd}$ degree surface (the osculating quadric) which has tangency of at least $6^{th}$ degree is considered. It is proved the osculating quadric exists and a method of its construction is described. Also existence of osculating quadric of any basic type of $2^{nd}$ degree surface is pointed out.
Keywords: space curve; osculating sphere; osculating quadric. 
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V. Lysenko; V. L. Timokhovich. Osculating quadric of the spatial curve. Journal of the Belarusian State University. Mathematics and Informatics, Tome 1 (2017), pp. 11-15. http://geodesic.mathdoc.fr/item/BGUMI_2017_1_a1/

[1] M. Ya. Vygodskii, “Differentsialnaya geometriya”, 1949

[2] S. P. Finikov, “Kurs differentsialnoi geometrii”, 1952