I have fucnion $f:\Bbb{R}^n \to \Bbb{R}^2$ and two vectors $\mathbf{v}_{min} = (v_{1_{min}}...v_{n_{min}})$, $\mathbf{v}_{max} = (v_{1_{max}}...v_{n_{max}})$ that defines a subset $S=\{\mathbf{v} \in \Bbb{R}^n | \mathbf{v}_{min} < \mathbf{v} < \mathbf{v}_{min}\}$. ($S$ is all points inside n-dimensional hypercube).

How to get a boundary line of the image $I = \{f(\mathbf{v}) | \mathbf{v} \in S\}$?

Image $I$ is compact, as well as set $S$. Function $f$ is continuous on the $S$.

  • I'm asking either for analytical or numeric solutions for the general case.

  • I'm glad to know what areas of mathematics study problems like that? I think it can be more than one approach.

Example with my result

Let $Y(v_1, v_2, v_3) = \frac{1}{v_1} + \frac{1}{v_3 + i100\pi v_2}$

$f:\Bbb{R}^3 \to \Bbb{R}^2$ defined as $(v_1, v_2, v_3) \to (Re(Y(v_1, v_2, v_3)), Im(Y(v_1, v_2, v_3)))$

I solved it in Python numerically and stochastically in the most naive way.

I defined the subset $S=\{\mathbf{v} \in \Bbb{R}^3 | \mathbf{v}_{min} < \mathbf{v} < \mathbf{v}_{min}\}$, where $\mathbf{v}_{min} = (10,0.01,0.2),\mathbf{v}_{max} = (100,0.1,2)$, by setting the limits of random number generator.

I sampled a lot of random vectors $\mathbf{v}$ to $f$ and kept dots that don't have neighbors in at least one direction.

Here is the boundary line of the image $I$: Image of the desired result that I got in Python


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