Analysis of the influence of "axis rotation" (sometimes called "side rotation") is more complex: There is a degree of axis rotation—generally 25° to 35° and varying with ball speed and rev rate—that may be considered optimal in that hook is maximized; however, this optimum axis rotation also causes minimal length. Specifically, Freeman & Hatfield (2018) report optimal axis rotation to be arcsin(ωr/v) where "ω" is rev rate (radians/sec), "r" is ball radius (m), and "v" is ball speed (m/s). Below and above optimal axis rotation, more length and less hook are encountered, with greater-than-optimal axis rotation causing a sharper hook.