Mechanisms for Generating Some Plane Mathematical Curves

Abstract

Based on geometrical problems, the synthesis of original mechanisms that trace different mathematical curves is made. Then the structure is analyzed and the calculations are made for the positions of the mechanisms, successive positions, diagrams of the curve generating point coordinates, diagrams of the displacements for some sliders, and the traced curves. Further curves generated by these mechanisms are also studied for different initial input data. In this way, with two original mechanisms, the cissoids (of the circle and of the straight line) are traced, in some cases in particular generating ellipses, parabolas and hyperbolas. Next Berard’s curve is studied, where a point of the mechanism traces a branch of the curve, and another point, the other branch. By modifying some dimensions, other curves generated by the mechanism are obtained. We also made the synthesis of a new mechanism that generates egg-shaped curve. Analysis of the mechanism led to the generation of this curve. We changed some data of the mechanism resulting in similar or modified curves, but positioned in different trigonometric quadrants. Another original mechanism, also based on a geometry problem, describes double egg curve. By changing the initial data, similar curves are obtained, some of them being incomplete. Another original mechanism, based on a geometry problem, describes Bernoulli quartic. In this case, an additional kinematic chain is used to provide the midpoint of a straight segment of variable length at the movement of the mechanism. The chapter continues with a mechanism based also on geometrical principles, which traces Maclaurin’s trisectrix. By changing some initial data, the mechanism traces completely different curves from the initial ones. There has also been a synthesis of some original mechanisms that trace ophiuride. In this case, other curves are obtained by modifying some dimensions of the mechanism. Finally, an original mechanism that traces the Pascal’s snail is studied. Different snails are obtained by changing some dimensions of the mechanism.