# Boundary line of the image of a subset under function

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 $$I$$: