Equations for Four-Bar Line-Envelopes

A line-equation (in tangential coordinates) is derived for the envelope of a general straight line attached to the coupler of a planar hinged four-bar linkage. Since a line can be identified with an axis of relative translation parallel to a sliding joint, the study of how lines move in a mechanism has practical potential. Nevertheless the emphasis here is on the geometry of the line-envelopes, and some envelopes are plotted both as samples of what can be obtained and to exemplify some of the properties which they possess. Towards the end of the paper all the other forms of planar four-bar linkage, namely those in which one or two sliding joints replace hinges, are examined, and their envelope-equations are presented.