I am studying the following problem:
Consider a closed contour $\mathcal{C}$ in $\mathbb{R}^2$ defined by $r(\theta)$ where $\theta\in[0,2\pi)$ and $r(0)=r(2\pi)$ (let the center to be zero for simplicity). We know a point $\mathbf{p}$, and from $\mathbf{p}$ we can draw the two lines that are tangential to $\mathcal{C}$ at points $\mathbf{a}$ and $\mathbf{b}$.
Now, along the section of the contour that is 'behind' the tangent points, there is a force acting normally to the contour and uniform in magnitude throughout. If we integrate this force, it turns out that, independently of the contour, the resultant (i) acts on the mid-point of $\mathbf{a}\mathbf{b}$ and (ii) is proportional to the length of $\mathbf{a}\mathbf{b}$.
I would like to generalize this result to 3 dimensions.
I have started thinking about the problem as follows. We now have a surface $\mathcal{S}$ defined by $r(\alpha,\beta)$, $\alpha,\beta\in[0,2\pi)$. From $p$, we now have a contour of tangential points on $\mathcal{S}$, but this contour does not necessarily lie in a plane. My hunch is that perhaps, as the resultant in 2 dimensions depended on a line, which is the minimal distance between two points, in 3 dimensions it might depend on the minimal surface corresponding the contour of tangential points on $\mathcal{S}$.
Any ideas or relevant references will be greatly appreciated.
PS: To help visualize the problem, you can think of $\mathcal{C}$ and $\mathcal{S}$ as physical objects, and $\mathbf{p}$ as a point light source. The force acts along the object where light does not reach it.