straight line intersects perpendicularly at all times. The function admits a horizontal asymptote. The curve is symmetrical with respect to the -axis. The curvature radius is . A great implication that the tractrix had was the study of its surface of revolution about its asymptote: the pseudosphere. Studied by Eugenio Beltrami in 1868, as a surface of constant negative Gaussian curvature, the pseudosphere is a local model of hyperbolic geometry. The idea was carried further by Kasner and Newman in their book Mathematics and the Imagination, where they show a toy train dragging a pocket watch to generate the tractrix. Properties The curve can be parameterised by the equation . Due to the geometrical way it was defined, the tractrix has the property that the segment of its tangent, between the asymptote and the point of tangency, has constant length . The arc length of one branch between and is . The area between the tractrix and its asymptote is , which can be found using integration or Mamikon's theorem. The envelope of the normals of the tractrix (that is, the evolute of the tractrix) is the catenary (or chain curve) given by . The surface of revolution created by revolving a tractrix about its asymptote is a pseudosphere. Practical application In 1927, P. G. A. H. Voigt patented a horn loudspeaker design based on the assumption that a wave front traveling through the horn is spherical of a constant radius. The idea is to minimize distortion caused by internal reflection of sound within the horn. The resulting shape is the surface of revolution of a tractrix.An important application is in the forming technology for sheet metal. In particular a tractrix profile is used for the corner of the die on which the sheet metal is bent during deep drawing. A toothed belt-pulley design provides improved efficiency for mechanical power transmission using a tractrix catenary shape for its teeth. This shape minimizes the friction of the belt teeth engaging the pulley, because the moving teeth engage and disengage with minimal sliding contact. Original timing belt designs used simpler trapezoidal or circular tooth shapes, which cause significant sliding and friction. Drawing machines In October–November 1692, Christiaan Huygens described three tractrix-drawing machines. In 1693 Gottfried Wilhelm Leibniz devised a "universal tractional machine" which, in theory, could integrate any first order differential equation. The concept was an analog computing mechanism implementing the tractional