Tractrix

Tractrix ( from the Latin trahere " grind, drag " ), also swept path, drawing curve, tensile curve, Treidelkurve, is a special plane curve. The name derives from the fact that this curve is described by a point mass, which is drawn on a pole.

The actual (even) is the tractrix curve, wherein for each of the tangent of the portion between the contact point and the coordinate axis is constant. It is also called Huygens tractrix, after Christiaan Huygens, who solved the underlying problem in 1693 after it was described by Claude Perrault. She is one of the curves, which are called by the common name dogs curve. This curve has an important role in the hyperbolic geometry.

Already Leonhard Euler and others employed soon after the general tractrix, which allows any Rails. It plays an important role in the modeling of the driving behavior, namely, the reverse drive and the behavior when passing through a curve. The knowledge gained will be used in the design of roads, to check their navigability.

Actual tractrix

An explicit representation according to X (Y ) is in this case not possible.

• With t = arcosh d / y results in an elegant form (with z = sech 1/cosh z):

• With ω, sin ω = y / d, 0 ≤ ω ≤ π / 2, the angle between x - axis and tangent - does not require a hyperbolic function:

• With λ = tan ω / 2, a representation that facilitates working with tabulated values ​​:

Derivation

Below the curve is seen in the main layer 1: A0 ( 0 | 0) P 0 (0 | d ), A is traveling along the x-axis, with a = AA0:

Integration gives x = ± d · (t - tanh t) and back substitution:

Properties

• Obviously, 0 <| y | ≤ | d |. D <0, the graph is a mirror image to the y- axis.
• For P0 (0 | d ) fall both possible tangents together with the y- axis, the point P0 is thus an actual tip.
• The length of the curve between x = x1 and x = x2 is calculated as:
• The area under the tractrix:
• The evolute of the tractrix is the catenoid
• This curve is rotated about the x-axis, the result is the pseudo- sphere, which assumes the role of the ball in the hyperbolic geometry. For instance, the area under the tractrix the same as the semi-circle. The tractrix is here as the equivalent of the geodesic lines in the "normal" ( Euclidean ) space.

General tractrix

The concept of tractrix can be generalized:

The tractrix is therefore a general Radiodrome with the function

Application in road construction

With the help of the drag curve, the driving behavior of vehicles can be modeled, in particular the space required when cornering, but also the behavior of reverse rides as well as when towing a second vehicle.

When steering operation of a vehicle, the axes run behind the steering axle " off track ": You do not track exactly the same way, so for cornering a larger area is scanned, as the track stand pretending.

The size and nature of the swept area is dependent upon several factors:

• The driving behavior of the driver
• The design speed, that is, the maximum speed for which the road is designed.
• The radius of curvature: a curve is moved more closely, the wider is the area swept.
• The length of the vehicle: the longer a vehicle, the greater the swept area.
• The position of axles: Depending on where the axes are relative to the length of the vehicle, either a larger area swept to the inside of the curve or curve outside ( warning " rear swings out ").
• The outline of the vehicle: Trailer, semi-trailer, etc.
• The number of steered or rotary axes.

The swept area can be with the help of ready-made drag curves that were created for specific design vehicles, determine. For special vehicles, the drag curves can be calculated with the help of computer simulations. Basis for each calculation is the so -called driving line. The vehicle is moved ( in the middle of the steering axis in the rule) along said line with its guide point.

The swept areas clear of traffic signals, signs, etc. and fix accordingly. For lane extensions are necessary, for example in narrow radii. To pay for the training of curves and intersections that the oncoming traffic is not impeded and endangered Furthermore.

This moving geometric design of transportation facilities is contrary to the dynamic driving design as it is in particular applied to roads and highways.

Norms and Standards

• FSGV -Verlag: Rated vehicles and towing curves to verify the practicability of traffic areas, 2001