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Tractrix
In geometry, a tractrix (from Latin trahere 'to pull, drag'; 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).
Suppose the object is placed at (a, 0) and the puller at the origin, so that a is the length of the pulling thread. (In the example shown to the right, the value of a is 4.) Suppose the puller starts to move along the y 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 (x, y), then by the Pythagorean theorem the y-coordinate of the puller is . Writing that the slope of thread equals that of the tangent to the curve leads to the differential equation
with the initial condition y(a) = 0. Its solution is
If instead the puller moves downward from the origin, then the sign should be removed from the differential equation and therefore inserted into the solution. Each of the two solutions defines a branch of the tractrix; they meet at the cusp point (a, 0).
The first term of this solution can also be written
where arsech is the inverse hyperbolic secant function.
The essential property of the tractrix is constancy of the distance between a point P on the curve and the intersection of the tangent line at P with the asymptote of the curve.
The tractrix might be regarded in a multitude of ways:
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Tractrix AI simulator
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Tractrix
In geometry, a tractrix (from Latin trahere 'to pull, drag'; 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).
Suppose the object is placed at (a, 0) and the puller at the origin, so that a is the length of the pulling thread. (In the example shown to the right, the value of a is 4.) Suppose the puller starts to move along the y 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 (x, y), then by the Pythagorean theorem the y-coordinate of the puller is . Writing that the slope of thread equals that of the tangent to the curve leads to the differential equation
with the initial condition y(a) = 0. Its solution is
If instead the puller moves downward from the origin, then the sign should be removed from the differential equation and therefore inserted into the solution. Each of the two solutions defines a branch of the tractrix; they meet at the cusp point (a, 0).
The first term of this solution can also be written
where arsech is the inverse hyperbolic secant function.
The essential property of the tractrix is constancy of the distance between a point P on the curve and the intersection of the tangent line at P with the asymptote of the curve.
The tractrix might be regarded in a multitude of ways: