Pursuit curve

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The Radiodrome ( " Leitstrahlkurve ", from the Latin radius " beam " and the Greek dromos "Run, Running " ), or tracking curve is a special plane curve. It describes the motion of a point which has a different point. Both points move it at a constant, but not necessarily the same speed.

The straight Radiodrome describes the simple case in which the persecuted moves on a straight line. Pierre Bouguer described it in 1732 for the first time. She is one of the curves, which are called by the common name dogs curve, according to a formulation of the question with a dog who follows his master. Pierre- Louis Moreau de Maupertuis extended the problem soon after on any Rails. This led to the definition of the general Radiodrome.

The curve typically occurs in tracking problems in robotics and dynamic simulations on (Tracking problems).

Special Radiodrome

Just Radiodrome

Derivation

Hereinafter, the first main layer is considered: A0 ( 0 | 0) P 0 ( 0 | 1 ), A moves along the X- axis, k> 0:

An explicit representation for y (x) is here not possible.

EW Weisstein are in (3) a closed parametric representation.

• For any distance d0 = A0P0 ≠ 0 we replace x → x/d0, y → y/d0 (normalized variables).
• If the starting direction is not normal to the directrix, we obtain other constraints. The turning point is calculated from x '= 0, d0 = y | x' = 0
• For a general location of the directrix an appropriate coordinate transformation is carried out.

Properties

• The connecting line of the respective A and B is tangent to the Radiodrome.
• Clearly, x > 0 for all y: The start point of the tracker is of the turning point of the curve ( with respect to the x -axis).
• The side branch, ie y > d0 (t → - ∞) diverges always, without asymptote.
• The root is given by the expression K / (1-k ²).

Analysis of the velocity parameter k

K> = 1: The zero point is obviously outside of the domain. Consequently, the curve asymptotically approaches the x -axis: The tracker is slower and does not reach the persecuted, yet he crosses his path.

• With the same rate (k = 1) the pursuer runs in increasingly equal distance behind the persecuted: The curve shows the limit behavior of a tractrix.

K <1: There is exactly one zero. The tracker is faster than the persecuted, and reaches those in finite time. We call the zero point meeting point or snap point, the curve is in the catch point to actually end.

• For k > = ½ the curvature approaches 0, the tracker converges achieved his offering exactly from behind.
• For k = ½ the curvature converges generally to d0 / 2
• For k < ½ cuts in meeting the tangent of the Radiodrome the x-axis at an angle ≠ 0, the tracker reaches its destination so from the side.

If k <0 (ie w of the persecuted oriented away, or A migrates towards the negative x - values) is the graph of a mirror image to the y- axis.

The case k = 0 is trivial, namely a straight line. The tracker is "infinitely " fast, or the persecuted stands still.

For rational k ≠ 1, the function degenerates to an algebraic curve - are, for example, this curve is the degrees.

Circle Radiodrome

Moves the " persecuted " in a circle and starts the " pursuer " in the center, so there is a further version. Have persecuted and persecutor the same speed, then the persecuted " after infinite time " caught up.

General Radiodrome

The concept of Radiodrome can be generalized:

The general Radiodrome heard as a special case of the class of squint angle curves.