Trisectrix

Abstract: IN THE study of the strophoid, the fo lium of Descartes, and the witch of Agnesi that appear in every classical analytic geometry course, two questions often asked are why such curves evolved and how they were obtained. When one visualizes the mathematicians of the Middle Ages at work, one might see Merlin stirring a pot and periodically pulling out witches and folia, and although these might be inter esting to a few people, they will surely re main isolated, cold, and useless problems to others. And so I thought, until I brought to life two such curves, the trisectrix and the cissoid, for my class. Once your students are aware that it is impossible to trisect a general angle by us ing an unmarked straightedge and a collap sible compass, they will be interested to know that they can create a curve whose graph accomplishes exactly that. The trisectrix is the locus of points result ing from the intersection of two lines whose angles of inclination (the counterclockwise angle formed by a line and the positive x axis) are in a ratio of 3 :1. In figure 1, lines lx and l2 are sketched in a particular manner, and they have equations (1) y = tan a and (2) y = ( 2a) tan 3a,

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.

Abstract: In 1928 Henry Scudder described how to use a carpenter's square to trisect an angle. We use the ideas behind Scudder's technique to define a trisectrix---a curve that can be used to trisect an angle. We also describe a compass that could be used to draw the curve.