Above is the link diagram of Joy valve gear. Red crosses are anchor point to the frame.

Very basic function of a valve gear is to move the valve in 90 degree phase difference against the piston. If efficiency is not a matter this arrangement only can run a steam engine. However to improve the efficiency valves should have "Lap" to allow cut-off. Also "Lead" is necessary to admit the steam slightly before the piston reaches its dead-points. So Lap+Lead will be an advance angle of the valve.

It is easy in case of Stephenson type. Just rotate the eccentrics by required advance angle from 90 degrees of the con-rod pin. In case of Walschaerts type, the return crank makes 90 degree phase shift and the combination lever combines this and movement of crosshead (no phase shift) to make required valve angle, say 110 degrees (90 deg. + 20 deg. Advance).

Now discuss how Joy type works. 90 degrees phase sift is taken from up-down movement of the con-rod. How the movement without phase shift is taken? It is easy by observing motion of the con-rod. Note the movement of point "C"! This point moves in larger oval shown in yellow having two components 0 degree (back and forth) and 90 degree (up and down) shifts. These components are combined together by "Vibrating Link" in the same manner as combination lever of Walschaerts. I wondered if so why "vibrating link" was not connected directly to the con-rod at point "C". If you can make the "vibrating link" very long it may be possible, but the length is quite limited. Large back-forth movement of con-rod makes this idea impractical making the operation angle of "vibrating link" too large. So "Jack Link" and Anchor Link" are provided to reduce the back-forth movement to manageable range. The movement of point "A" is shown by small yellow oval. With this arrangement up-down movement is not affected, but back-forth movement is reduced.