Tractrix on a line below the x-axis

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In summary, the cusp of the tractrix is on the y-axis and is located at $\left(\frac{mL}{\sqrt{1+m^2}}, \frac{L}{\sqrt{1+m^2}}\right)$. The equation for the tractrix with the hitch point not on the x-axis but on a line below (y=-mx) can be found using either the parametrized or rotated versions of the equation.
  • #1
Dr Edward
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The cusp of the tractrix is on the y-axis.
I have been studying Sreenavisan et al (Mechanism and Machine Theory 45 (2010) 454–466)
Email me for copy, if needed.
What I need is the equation of a tractrix with the hitch point not on the x-axis but on a line below (y=-mx).
Any help would be gratefully received.
Dr Edward
 
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  • #2
Dr Edward said:
The cusp of the tractrix is on the y-axis.
I have been studying Sreenavisan et al (Mechanism and Machine Theory 45 (2010) 454–466)
Email me for copy, if needed.
What I need is the equation of a tractrix with the hitch point not on the x-axis but on a line below (y=-mx).
Any help would be gratefully received.
Dr Edward

Hello Dr Edward and welcome to MHB! ;)

We can move a graph vertically by replacing every occurrence of $y$ by $y-b$.
The result is shifting the graphs upwards by $b$.
Similarly we can shift a graph horizontally by replacing every occurrence of $x$ by $x-a$.
 
  • #3
Thank you very much. That is indeed helpful.
Here is an excerpt from SreenavisanView attachment 8556
In all his brilliance he makes the integrating task like getting an ice cream.
I tried MathCad and the result was kilometres long.
My knowledge of maths is not that good to simplify.
I need an equation y=f(x) or y=f(t).
Hope you can help
Cheers
Edward
 

Attachments

  • Sreenavisan.JPG
    Sreenavisan.JPG
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  • #4
Dr Edward said:
Thank you very much. That is indeed helpful.
Here is an excerpt from Sreenavisan
In all his brilliance he makes the integrating task like getting an ice cream.
I tried MathCad and the result was kilometres long.
My knowledge of maths is not that good to simplify.
I need an equation y=f(x) or y=f(t).
Hope you can help
Cheers
Edward

I kind of doubt that Sreenavisan has a nice simple formula in the form $y=f(x)$.
Instead I think he integrated and drew the graph with a numerical algorithm.
Note that the graph has 3 possible values for $y$ at $x=-0.1$.
That doesn't sound like a simple function of $x$ does it?

Anyway, if the hitch point travels along the x-axis, the known solution is (see Tractrix on MathWorld)
$$x=L \arsech \frac y L -\sqrt{L^2-y^2}$$
Finding a rotated form of this is bound to be horrible.

Alternatively, we have the parametrized version:
$$\begin{cases}x(t)=L(t-\tanh t) \\ y(t)=L \sech t\end{cases}$$
We can rotate and shift this as desired.
Let me skip the steps for now and jump to the result:
$$\begin{cases}X(t)=x_{e,0} + \frac L{\sqrt{1+m^2}}({(t-\tanh t) - m \sech t})\\
Y(t)=y_{e,0} + \frac L{\sqrt{1+m^2}}({m(t-\tanh t) + \sech t})\end{cases}$$
In your case we have approximately $(x_{e,0},y_{e,0})=(6,6)$, $m=1$, and $L=10$.
If we fill that in, we get the following graph from Wolfram|Alpha:

View attachment 8557

Looks the same as yours doesn't it?
 

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  • angled_tractrix.jpg
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  • #5
Wow
Thank you very much
Edward
 
  • #6
Sorry to trouble you again.
I graphed your equations and got the same result as you. (exciting for me).
Then I played with the coefficients and could not get the position I am looking for.
Here's the MathCad result with different numbers.View attachment 8558
What am i doing wrong?
Here's where I want to be:View attachment 8559
Hope you can help.
Edward
 

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  • Graph 2.JPG
    Graph 2.JPG
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  • #7
Looks like $m=-1$ was not substituted correctly...
 
  • #9
Nice curves - thank you very much.
I have to add a value (shown at the red dot in the picture) to align the cusp to P(0,5).
Also when I change the slope the cusp needs adjustment.
Any idea why this should be?
View attachment 8562
Once again, thanks for bringing me this far.
Edward
 

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  • Graph.JPG
    Graph.JPG
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  • #10
Hello Klaas,
Hate to be a nuisance but I had a thought.
Is it possible that the sloped hitch line has a y-intercept?
That might explain why the tractrix cusp does not stay on the y-axis.
Would love an answer. A solution? Even better.
Thanks for your input so far.
Edward
 
  • #11
We rotate the graph around the initial hitch end point $(x_{e,0},y_{e,0})$.
So that point always remains the same, and the sloping hitch line intersects it.
The cusp rotates around it just like everything else.

The location of the cusp is $\left(\frac{mL}{\sqrt{1+m^2}}, \frac{L}{\sqrt{1+m^2}}\right)$.
To keep it in the same spot at $(x_{cusp},y_{cusp})$ we can use:
\begin{cases}X(t)=x_{cusp} + \frac L{\sqrt{1+m^2}}({(t-\tanh t) - m(\sech t - 1)})\\
Y(t)=y_{cusp} + \frac L{\sqrt{1+m^2}}({m(t-\tanh t) + (\sech t} - 1))\end{cases}

If we pick $(x_{cusp},y_{cusp})=(0,L)$, the cusp will stay at $(0,L)$.
 
  • #12
Thanks Heaps.
Works beautifully
Edward
 

Related to Tractrix on a line below the x-axis

What is a Tractrix?

A Tractrix is a mathematical curve that represents the path traced by a point on a line that is constrained to move along a straight line while being pulled towards a fixed point. It is also known as the pursuit curve or catenary curve.

How is a Tractrix formed?

A Tractrix is formed by a combination of two mathematical concepts: a straight line and a circle. The fixed point is located at the center of the circle, while the line is used as a guide to keep the point moving along a straight path.

Why is the Tractrix important in mathematics?

The Tractrix has important applications in calculus, geometry, and physics. It is also used in the design of various mechanical systems, such as gears and pulleys, and in the study of projectile motion.

What are the properties of a Tractrix?

The Tractrix is a non-linear curve with a constant curvature. It has a horizontal asymptote at the fixed point, and a vertical tangent at the point where it intersects with the line. It is also a self-similar curve, meaning that a scaled-down version of the curve can be found within itself.

How is the Tractrix used in real-life applications?

The Tractrix has various real-life applications, including the design of ship hulls, airplane wings, and roller coaster tracks. It is also used in the construction of suspension bridges and the design of satellite orbits. In addition, the Tractrix is used in mathematical models to predict the movement of objects in space.

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