What is g(x) in Terms of f(x): Distance Path Path

[wheepep]

Path

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g(x) is a function which contains all points that are at a constant distance from f(x).

What is g(x) in terms of f(x)?

 

[MarkFL]

Let's begin by using a trivial parametrization of $f$ as follows:

\(\displaystyle x=t\)

\(\displaystyle y=f(t)\)

Now, a point on $g$ can be found to lie on the line:

\(\displaystyle y=-\frac{1}{f'(t)}(x-t)+f(t)\)

And the points on this line a distance $D$ from $(t,f(t))$ can be found from:

\(\displaystyle D=\sqrt{(x-t)^2+\left(y-f(t)\right)^2}=\left|y-f(t)\right|\sqrt{1+\left(f'(t)\right)^2}\)

Hence:

\(\displaystyle y=f(t)\pm\frac{D}{\sqrt{1+\left(f'(t)\right)^2}}\)

This then implies

\(\displaystyle -\frac{1}{f'(t)}(x-t)+f(t)=f(t)\pm\frac{D}{\sqrt{1+\left(f'(t)\right)^2}}\)

Which reduces to:

\(\displaystyle x=t\mp\frac{Df'(t)}{\sqrt{1+\left(f'(t)\right)^2}}\)

Thus, $g$ can be described parametrically by:

\(\displaystyle X=t\mp\frac{Df'(t)}{\sqrt{1+\left(f'(t)\right)^2}}\)

\(\displaystyle Y=f(t)\pm\frac{D}{\sqrt{1+\left(f'(t)\right)^2}}\)

 

[I like Serena]

Re: Path

g(x) is a function which contains all points that are at a constant distance from f(x).

What is g(x) in terms of f(x)?

Hi wheepep! ;)

Suppose we look at the parametrized function $\gamma(t) = (t, f(t))$.
The tangent of a parametrized function is $\gamma'(t)$.
The normal is $\gamma'^\perp(t)$.
And the normal at unit length is $N(t) = \frac{\gamma'^\perp(t)}{\| \gamma'(t) \|}$.
So the function that is at constant distance $c$ is:
$$\gamma_c(t) = \gamma(t) + cN(t) = \gamma(t) + \frac{c\gamma'^\perp(t)}{\| \gamma'(t) \|}$$In your problem statement we have:
$$\gamma(x) = (x,f(x)) \\
\gamma'(x) = (1,f'(x)) \\
\gamma'^\perp(x) = (-f'(x),1) \\
N(t) = \left(-\frac{f'(x)}{\sqrt{1+f'(x)^2}}, \frac{1}{\sqrt{1+f'(x)^2}}\right) \\
\gamma_c(x) = \left(x-\frac{cf'(x)}{\sqrt{1+f'(x)^2}}, f(x) + \frac{c}{\sqrt{1+f'(x)^2}}\right)
$$

EDIT: Which is the same result MarkFL just gave! (Wink)If we (can) reparametrize the function $\gamma$ to $\tilde\gamma$ such that it has unit speed, that is, has $\| \tilde\gamma'(s) \| = 1$, we have:
$$\tilde N(s) = \tilde\gamma'^\perp(s) \\
\tilde\gamma_c(s)=\tilde\gamma(s) + c\tilde\gamma'^\perp(s)
$$Let's pick an example.
Suppose we pick $\tilde\gamma(s) = (\cos s, \sin s)$, which corresponds to $f(x)=\sqrt{1-x^2}$, then it simplifies a bit:
$$\tilde \gamma'(s) = (-\sin s, \cos s) \\
\tilde N(s) = \tilde\gamma'^\perp(s) = (\cos s, \sin s)\\
\tilde\gamma_c(s) = \tilde\gamma s + c\tilde N(s) = (\cos s + c \cos s, \sin s + c \sin s) = (1+c)(\cos s, \sin s)
$$
which is a circle with radius $(1+c)$ as expected.

If we do the same thing with $f(x)=\sqrt{1-x^2}$ and $\gamma_c(x)=\left(x-\frac{cf'(x)}{\sqrt{1+f'(x)^2}}, f(x) + \frac{c}{\sqrt{1+f'(x)^2}}\right)$, we'll find the same result - it will just take much longer.As you can see, the problem becomes much simpler if the function is a parametrized function at unit speed.
Of course it all depends on the function we want to shift.
Did you have any particular function in mind? (Wondering)

 

[wheepep]

y = x²

 

[I like Serena]

y = x²

Let's see what it looks like when filling in the formula:
[LATEXS]
\begin{tikzpicture}
%preamble \usepackage{pgfplots}
\begin{axis}[ymax=10, samples=51]
\addplot[blue, ultra thick] (x,x^2);
\addplot[red, ultra thick] ({x - 2*x/(sqrt(1+4*x^2))},{x^2 + 1/(sqrt(1+4*x^2))});
\end{axis}
\end{tikzpicture}
[/LATEXS]
[TIKZ]
%preamble \usepackage{pgfplots}
\begin{axis}[ymax=10, samples=51]
\addplot[blue, ultra thick] (x,x^2);
\addplot[red, ultra thick] ({x - 2*x/(sqrt(1+4*x^2))},{x^2 + 1/(sqrt(1+4*x^2))});
\end{axis}
[/TIKZ]

 

[wheepep]

Interesting

 

[I like Serena]

Here's another plot with a couple more graphs at different distances:
\begin{tikzpicture}
%preamble \usepackage{pgfplots}
\begin{axis}[ymax=10, samples=151, grid=both,
axis lines=middle,
axis equal,
minor tick num=4,
xtick={-8,-6,...,8},
ytick={-4,-2,...,10},
grid style={line width=.1pt, draw=gray!30},
major grid style={line width=.2pt,draw=gray!70}
]
\addplot[green, ultra thick] ({x - 8*x/(sqrt(1+4*x^2))}, {x^2 + 4/(sqrt(1+4*x^2))});
\addplot[red, ultra thick] ({x - 4*x/(sqrt(1+4*x^2))}, {x^2 + 2/(sqrt(1+4*x^2))});
\addplot[blue, ultra thick] (x,x^2);
\addplot[red, ultra thick] ({x + 4*x/(sqrt(1+4*x^2))}, {x^2 - 2/(sqrt(1+4*x^2))});
\addplot[green, ultra thick] ({x + 8*x/(sqrt(1+4*x^2))}, {x^2 - 4/(sqrt(1+4*x^2))});
\end{axis}
\end{tikzpicture}

 

[wheepep]

Thanks