Abstract. In this paper, we study on spinors with two hyperboliccomponents. Firstly, we express the hyperbolic spinor representa-tion of a spacelike curve de ned on an oriented (spacelike or time-like) surface in Minkowski space R 31 . Then, we obtain the relationbetween the hyperbolic spinor representation of the Frenet frameof the spacelike curve on oriented surface and Darboux frame ofthe surface on the same points. Finally, we give one example aboutthese hyperbolic spinors. 1. IntroductionSpinors used to expand space vector concepts in orthogonal grouptheory in particular such as rotation or Lorentzian groups in mathemat-ics and physics are the elements of the complex vector space. For the rst time, in the geometrical meaning, spinors has been studied by theFrench mathematician E. Cartan. Cartan obtained these spinors consist-ing of two complex components in terms of vectors in three-dimensionalEuclidean space, [3].The word \spinor was coined by P. Ehrenfest in his work on quan-tum physics, [19]. Moreover, W. Pauli introduced spin matrices and rst applied spinors to mathematical physics, [17]. Then P. A. M. Diracshowed the connection between spinors and the Lorentz groups. More-over, Dirac discovered the fully relativistic theory of electron spin, [7].The other uses of the spinors are also available. For example, in themechanics of solids, the components of unit spinors have been used forover 50 years as \Cayley-Klein parameters. Also spinor theory is alsoassociated with electricity transmission line, [6].