Tractrix

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

formulas

Cartesian Coordinates (en)

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

---

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

---

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

Parametric equations (en)

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

$t=\operatorname {arcosh} {d \over y}$

---

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

$\omega ,\ \sin \omega ={y \over d}\ ,\ 0\leq \omega \leq {\pi \over 2}$

---

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

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

(all formulas from w:de:Traktrix)

graphs y=y(x)

graphs x=x(y)

derivation

$y'={\mathrm {d} y \over \mathrm {d} x}$$=\pm \ {y \over {A-x}}$$=\pm \ {y \over {\sqrt {d^{2}-y^{2}}}}$

General Tractrix

• $\mathrm {curve} \,k:A_{0}\in k$
• $\mathrm {point} \,P_{0}$
• $\mathrm {parameter} \,t$
• $d={\overline {A_{0}P_{0}}}$
• ${\mathbf {A}}(t):A(0)=A_{0}$
• ${{\mathbf {P}}(t)}:P(0)=P_{0}$

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

$t:{{\mathbf {\dot {P}}}(t)}\neq 0$