Osculating quadric of the spatial curve

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.