Dual Bézi̇er curves and ruled surfaces

Let a dual Bézier curve \hat{B}(t)=B(t)+\varepsilon B^\ast(t) , where B(t) and B^\ast(t) are the Bézier curves of degree n with control points {P_i} and {{P^\ast}_i} respectively be given. And also \widetilde{B}(t) be a projection curve of \hat{B}(t) to a unit dual sphere. In this study the ruled surface corresponding the closed spherical curve \widetilde{B}(t); the moving frame {U_1,U_2=U_1'/||U_1'||,U_3=U_1×U_2} , where \vec{U_1 }. is a unit vector of director curve of ruled surface corresponding closed spherical curve \widetilde{B}(t); the equation of the derivatives of the moving frame {\vec{U_1},\vec{U_2},\vec{U_3}} and geodezic curvatures are stated in terms of the real and dual parts of given dual Bézier curve \hat{B}(t). In addition the instantaneous Pfaffian vector and the pole and the steiner vectors of the motion are stated in terms of the real and dual parts of given dual Bézier curve \hat{B}(t). Finally the integral invariants of the \vec{U_1}- , \vec{U_2}- , ve \vec{U_3}- closed trajectory surfaces corresponding the closed spherical curve \widetilde{B}(t) are obtained.