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Fig. 2. Illustration of the coordinate systems and their transformation. (A) The helical swimming trajectory is shown on a cylinder. V_X is the net displacement speed of the cell, where _c is the angular speed of the cell, R_p is the radius of the cylinder and t is time. e_para, e_tan and e_rad are the unit direction vectors where e_para is parallel to the axis of the cylinder, e_rad is radial to a circular transections of the cylinder and e_tan is tangential to the circular transection whose plane is perpendicular to the cylinder's axis. (X_I,Y_I,Z_I) is the inertial frame fixed relative to the laboratory and (X',Y',Z') is a coordinate translating with the cell and rotating with the cell and is rotating about X' axis (identical to X_I axis) at the angular speed of _c. e_para, e_tan and e_rad are the components parallel to the net displacement, tangential and radial to the swimming trajectory. (X',Y',Z')^T=R_c+T₁·(X_I,Y_I,Z_I)^T, where superscript T indicates the transposed vector, R_c is the position of the cell in the inertial frame (see Equation 1) and T₁ is a matrix indicating the rotational movement of the cell defined in Equation 3, respectively. (B) Transformations between the coordinates , where , and are the Eulerian angles indicating cell orientation and T₂-T₄ are defined in Equations 4-7, to define the cell frame (x,y,z). (C) Model of the P. minimum cell and flagella in the cell frame. a₁, amplitude of longitudinal wave; a_t, amplitude of transverse wave; r_t radius of the circle along which the transverse wave is propagating; x_bt, x coordinate of the circle along which the transverse wave is propagating.

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