Afr. Mat. DOI 10.1007/s13370-017-0519-3 On special type-2 Bishop motion and Bishop Darboux rotation axes of the space curve Ay¸ se Zeynep Azak 1 · Melek Masal 1 Received: 30 April 2016 / Accepted: 27 July 2017 © African Mathematical Union and Springer-Verlag GmbH Deutschland 2017 Abstract We have given a generalization of one parameter special Frenet motion to type-2 Bishop motion in Euclidean 3-space E 3 . Type-2 Bishop motion have been defined for space curve β and then type-2 Bishop Darboux vector of this motion has been calculated for fixed and moving spaces in E 3 . Also, we have showed that type-2 Bishop rotation for space curves is decomposed into two simultaneous rotations. One of the axes of this simultaneous rotations is a parallel of the binormal vector of the curve, the other is the direction of the type-2 Bishop Darboux vector of the curve. Keywords Bishop frame · Bishop Darboux vector · Bishop Darboux rotation axes Mathematics Subject Classification 53A04 Curves in the Euclidean Space · 53A17 Kinematics 1 Introduction Firstly, Yılmaz and Turgut have introduced a new version of Bishop frame using a common vector field as binormal vector field of a regular frame and called this frame type-2 Bishop frame. Additionally, they have given the type-2 Bishop spherical indicatrices, [13]. Later, Kızıltu˜ g and et al have defined slant helices and obtained some characterizations of slant helices according to type-2 Bishop frame in the Euclidean 3-space [9]. In addition to these studies, Bükcü and Karacan have given the generalization of one parameter Frenet motion to Bishop motion in the Euclidean 3-space which is defined by Bottema. They have also inspired by the article of Barthel and defined Darboux vector of this motion and obtained some B Ay¸ se Zeynep Azak apirdal@sakarya.edu.tr Melek Masal mmasal@sakarya.edu.tr 1 Department of Mathematics and Science Education, Faculty of Education, Sakarya University, 54300 Hendek, Sakarya, Turkey 123