Tractrix

This is a gallery page containing specially selected image and media files. They have been chosen as highlights of a particular topic, but do not represent the full range of files that are available on Commons. For a wider selection of files connected with Tractrix, see Category:Tractrix.

Tractrix is the curve along which a small object moves when pulled with a pole by a puller.

formulas

Cartesian Coordinates (en)

$y(x)=\pm d\cdot \mathrm{arcosh}\frac{d}{x}\mp \sqrt{d^2-x^2}$

---

$y(x)=\pm d\cdot \ln \left|\frac{d+\sqrt{d^2-x^2}}{x}\right|\mp \sqrt{d^2-x^2}$

---

$x(y)=\pm d\cdot (\mathrm{arcosh}\frac{d}{y}-\sqrt{1-{\left(\frac{y}{d}\right)}^2})$

Parametric equations (en)

$x(t)=\pm d\cdot (t-\tanh t);$   $y(t)=d\cdot \mathrm{sech}t$

$t=\mathrm{arcosh}\frac{d}{y}$

---

$x(\omega )=\pm d\cdot (\cos \omega +\ln \tan \frac{\omega }{2});$   $y(\omega )=d\cdot \sin \omega$

$\omega ,\sin \omega =\frac{y}{d},0\le \omega \le \frac{\pi }{2}$

---

$x(\lambda )=\pm d\cdot (\frac{{\lambda }^2-1}{{\lambda }^2+1}-\ln \lambda );$   $y(\lambda )=2d\cdot \frac{\lambda }{{\lambda }^2+1}$

$\lambda =\tan \frac{\omega }{2}$

(all formulas from w:de:Traktrix)

graphs y=y(x)

graphs x=x(y)

derivation

$y^{\prime }=\frac{\mathrm{d}y}{\mathrm{d}x}$$=\pm \frac{y}{A-x}$$=\pm \frac{y}{\sqrt{d^2-y^2}}$

General Tractrix

• $\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{v}\mathrm{e}k:A_0\in k$
• $\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}{P}_0$
• $\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}t$
• $d=\overline{A_0{P}_0}$
• $𝐀(t):A(0)=A_0$
• $𝐏(t):P(0)=P_0$

$𝐀(t)=𝐏(t)+d\cdot \frac{\dot{𝐏}(t)}{|\dot{𝐏}(t)|}$

$t:\dot{𝐏}(t)\ne 0$