In geometry, a tractrix (; plural: tractrices) is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a pulling point (the tractor) that moves at a right angle to the initial line between the object and the puller at an infinitesimal speed. It is therefore a curve of pursuit. It was first introduced by Claude Perrault in 1670, and later studied by Isaac Newton (1676) and Christiaan Huygens (1693). Mathematical derivation Suppose the object is placed at (or in the example shown at right), and the puller at the origin, so is the length of the pulling thread (4 in the example at right). Then the puller starts to move along the axis in the positive direction. At every moment, the thread will be tangent to the curve described by the object, so that it becomes completely determined by the movement of the puller. Mathematically, if the coordinates of the object are , the of the puller is by the Pythagorean theorem. Writing that the slope of thread equals that of the tangent to the curve leads to the differential equation with the initial condition . Its solution is where the sign depends on the direction (positive or negative) of the movement of the puller. The first term of this solution can also be written where is the inverse hyperbolic secant function. The sign before the solution depends whether the puller moves upward or downward. Both branches belong to the tractrix, meeting at the cusp point . Basis of the tractrix The essential property of the tractrix is constancy of the distance between a point on the curve and the intersection of the tangent line at with the asymptote of the curve. The tractrix might be regarded in a multitude of ways: It is the locus of the center of a hyperbolic spiral rolling (without skidding) on a straight line. It is the involute of the catenary function, which describes a fully flexible, inelastic, homogeneous string attached to two points that is subjected to a gravitational field. The catenary has the equation . The trajectory determined by the middle of the back axle of a car pulled by a rope at a constant speed and with a constant direction (initially perpendicular to the vehicle). It is a (non-linear) curve which a circle rolling on a