The researches in the transmission mechanisms of the displacement engine, the displacement pumps and the displacement compressors are always mutually supporting and supplementary.
In the past, we classified those mechanisms by moving style, e. g. reciprocating or rotating generally, so that our innovative idea was limited or minimized. The author submitted a new idea that we should classify pumping mechanisms by closd volume type, e. g. “plane type” or “space type”, and at the same time, submitted a point of view about pumping mechanism evolution from plane to space. From all these, the author points out two clear lines of innovative thoughts of “spiral evolution” and “precession evolution” of mechanism, and then the law of two evolutions, e. g. “phase-display” and “phase-overlap” are explained as follows: in the process of periodical change of a closed volume, at a certain instant, the change of the closed area shape in the different sections along the axis or the circumference is the same as the change of the closed area shape at different instant in the same sections. So that it is possible to create a new sort of spiral pumping mechanism or precession pumping mechanism on the base of plane pumping mechanism along the axis or the circumference through evolution.
Finally, Some new sorts of spiral pumping mechanism and precession pumping mechanism are introduced.