Optimization of linkage of rotating surfaces of screw pumps (Q2780394)

Consider two contours on the complex plane centered at 1 and \(-1\): NEWLINE\[NEWLINE\begin{aligned} & z_1=1+F^*(\varphi)=1+f^*(\varphi)e^{-i\varphi},\quad 1-h\leq f^*(t)\leq 1+h;\\ & z_2=-1+F(\psi)=-1+f(\psi)e^{i\psi},\quad 1-h\leq f(t)\leq 1+h;\end{aligned}NEWLINE\]NEWLINE \(\varphi,\psi\in [0,2\pi]\). We call this pair of contours admissible if they admit a simultaneous uniform rotation with an equal angular speed in opposite directions, i.e. the interiors of those contours never intersect. Contours are linked if one of them stays fixed, the second rotates and the interiors do intersect at some moment of time. The authors study the behavior of such pairs of contours using singular points of smooth self-mappings of the plane.