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I have come across a problem where I need to generate values that when plotted look like the blade of a handheld saw:

enter image description here

I am sorry, I was not able to understand how to plot this (because I do not have the equation), so had to draw this by hand. It would be great if a standard graph/equation for this already exists or if someone can point me to how to get started on this.

What I tried?

$y = \frac{1}{\sqrt{100}} \times {min}(\frac{1}{\sqrt{x}}, \frac{x}{4000^{1.5}})$

The above equation gives me a plot that looks like:

enter image description here

So, I have one point of the saw blade, but, how to repeat it with decreasing height on y-axis?

anurag
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    You'll probably have to define it piecewise. For example, you use one formula when $0\leq x\leq 1000$, another when $1000<x\leq 2000$, and so on. Will that suit your needs? – saulspatz May 21 '21 at 20:53
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    I'd suggest starting with a sawtooth function like arccot(tan(x)) or floor(x)-x+1 and applying the desired transformations. For instance you could apply an $x^2$ like function to get the curved shape, and add something like $\frac{1}{x}$ to get it to slope downward. – TomKern May 21 '21 at 21:27
  • @TomKern, I am not able to put the idea as an equation... – anurag May 22 '21 at 07:49
  • @saulspatz, piecewise would work, but, how to control/update the parameters to obtain the gradual decay? – anurag May 22 '21 at 07:51
  • Draw each tooth separately and figure out a formula like the one you have. To move the tooth $a$ units to the right, substitute $x-a$ for $x$. To move the tooth $b$ units up, add $b$ to the formula. – saulspatz May 22 '21 at 14:18

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