Real inflexions of the four-bar coupler curve

Abstract

A 4-bar is a simple engineering mechanism comprising four bars smoothly jointed together to form a movable quadrilateral with one fixed side. The locus of a point rigidly attached to the opposite side is a coupler curve. This work investigates methods of determining how many real inflexions are possible in a 4-bar coupler curve. We develop a means of identifying the coupler points needed to generate a coupler curve with a given number of real inflexions for a given 4-bar. We indicate conditions necessary to allow the maximum number of inflexions to occur, and demonstrate coupler curves where the maximum number occurs. Techniques associated with inflexions from differential geometry, analytic kinematics, algebraic geometry and computer graphics are reviewed.