I recently took a liking to parallel curves and tried to find the area between them. Possible applications could be for making geometric swimming pools or some other area/volume based problem. The issue I ran into is something feels off about the equation and I need help proving that it is correct.
First, any equation is represented by $f(x)$. And the parallel curve is represented by the $x$ component $x(t)$, and the $y$ component $y(t)$. And so given any function $f(x)$ the parallel curve is...
\begin{align} x(t) &= t+\frac{f'(t)}{\sqrt{1+f'(t)^2}} \\ y(t) &= f(t)-\frac{1}{\sqrt{1+f'(t)^2}} \end{align} So let me try and explain how I came up with the equation below.
I thought of how the arc length equation is similar to the parallel parametric equation. The arc length equation being $$\int_{a}^{b}\sqrt{1+f'(t)^2}dt~.$$ The arc length can also be used with a parametric equation of $$\int_{a}^{b}\sqrt{x'(t)^2+y'(t)^2}dt~.$$
I attempted to create a visualization of what this looks like in my head. That a small trapezoid is formed with both legs having a distance of $1$. The two bases would have distances given by $\sqrt{1+f'(t)^2}$ for $f(x)$ and $\sqrt{x'(t)^2+y'(t)^2}$ for the parallel parametric function. Since the formula for a trapezoid is $A=\frac{b_1+b_2}{2}h$. Here $b_1$ and $b_2$ meaning base_1 and base_2 respectively, and $h$ means height.
Since These small segments are parallel and are joined by one of the legs perpendicularly, this means that $h = 1$. You can probable see where I'm going with this by using the trapezoid formula with the small lengths we have we get the area of with small trapezoid $$\frac{\sqrt{1+f'(t)^2}+\sqrt{x'(t)^2+y'(t)^2}}{2}~.$$ By using integration we do this like how integration was first explained using infinitely small rectangles. But in this case it's trapezoids.
$$A=\frac{1}{2}\int_{a}^{b}(\sqrt{1+f'(t)^2}+\sqrt{x'(t)^2+y'(t)^2})dt$$
Now, I have alreday tested this equation with a line and a semicircle. Since both of these functions are special and we already know the area between there parallel curve and the function its self.
Is this the correct approach? It feels correct. Thank you for your help!