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In the work some relations between three techniques, Hopf's bundle,Kustaanheimo-Stiefel's bundle, 3-space with spinor structure have beenexamined. The spinor space is viewed as a real space that is minimally (twiceas much) extended in comparison with an ordinary vector 3-space: at thisinstead of 2\pi-rotation now only 4\pi-rotation is taken to be the identitytransformation in the geometrical space. With respect to a given P-orientationof an initial unextended manyfold, vector or pseudovector one, there may beconstructed two different spatial spinors, $\xi$ and $\eta$, respectively. Bydefinition, those spinors provide us with points of the extended space odels,each spinor is in the correspondence $2 \longrightarrow 1 with points of avector space. For both models an explicit parametrization of the spinors \xiand \eta by spherical and parabolic coordinates is given, the parabolic systemturns out to be the most convenient for simple defining spacial spinors. Foursof real-valued coordinates by Kustaanheimo-Stiefel, U_{a} and V_{a}, real andimaginary parts of complex spinors \xi and \eta respectively, obey twoquadratic constraints. So that in both cases, there exists a Hopf's mappingfrom the part of 3-sphere S_{3} into the entire 2-sphere S_{2}. Relationbetween two spacial spinor is found: \eta = (\xi - i\sigma ^{2}\xi^{*})/\sqrt{2}, in terms of Kustaanheimo-Stiefel variables U_{a} and V_{a} itis a linear transformation from SO(4.R), which does not enter its sub-groupgenerated by SU(2)-rotation over spinors.

Spinor Structure of P-Oriented Space, Kustaanheimo-Stifel and Hopf Bundle - Connection between Formalisms