Tractrix-like curves

Generalized tractrices (tractories) exist, see this or this or this. (the last two are in French, but with slightly more detail than the first one.)

This old book ought to be of interest as well.

This book describes Euler's treatment of the problem of the tractory.

As a note, the problem of finding the generating curve, given the tractory, is a much easier problem (hint: use the tangent vector of a curve) than finding the tractory corresponding to a generating curve.

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For those who have difficulty reading French, the third link I mentioned gives the prescription for generating the corresponding tractory from a generating curve with parametric equations $(f(t)\quad g(t))$.

The parametric equations for the tractory of $(f(t)\quad g(t))$ (in vector form) is

$$\begin{pmatrix}f(t)\\g(t)\end{pmatrix}-\frac{a}{\sqrt{f^{\prime}(t)^2+g^{\prime}(t)^2}}\begin{pmatrix}\cos\;\alpha(t)&\sin\;\alpha(t)\\-\sin\;\alpha(t)&\cos\;\alpha(t)\end{pmatrix}\cdot\begin{pmatrix}f^{\prime}(t)\\g^{\prime}(t)\end{pmatrix}$$

or explicitly

$$\begin{align*}x&=f(t)-\frac{a}{\sqrt{f^{\prime}(t)^2+g^{\prime}(t)^2}}(f^{\prime}(t)\cos\;\alpha(t)+g^{\prime}(t)\sin\;\alpha(t))\\y&=g(t)-\frac{a}{\sqrt{f^{\prime}(t)^2+g^{\prime}(t)^2}}(g^{\prime}(t)\cos\;\alpha(t)-f^{\prime}(t)\sin\;\alpha(t))\end{align*}$$

where the function $\alpha(t)$ satisfies the differential equation

$$\frac{\mathrm d\alpha}{\mathrm dt}=\frac{f^{\prime}(t)g^{\prime\prime}(t)-g^{\prime}(t)f^{\prime\prime}(t)}{f^{\prime}(t)^2+g^{\prime}(t)^2}-\frac{\sin\alpha}{a}\sqrt{f^{\prime}(t)^2+g^{\prime}(t)^2}$$

and $a$ is the length of the segment running through the generating curve.

As an example, here is an animation showing the curve $(3\cos\;t-2\cos^3 t+\cos 2t\quad 2\sin^3 t+\sin 2t)$ and its tractory with segment length 1:

(This is a less fancy version of the last bicycle animation in the third French link.)