I'm looking to 'visualize' higher dimensional functions in the best way possible (using plotting software,i.e., we are given a range of values in the domain to plot to the image of a function).
Starting with $\mathbb{R} \longrightarrow \mathbb{R}$, we know we can embed both the domain and image of this function in $\mathbb{R^2}$. For example, the classic $y = x$.
Restricting ourselves to $n+m \leq 3$ we can embed the functions $f :\mathbb{R^n} \longrightarrow \mathbb{R^m}$ in $\mathbb{R^3}$ also without problem.
For the cases where $m+n >3$, so far I've considered:
- For $\mathbb{R^2} \longrightarrow \mathbb{R^3}$ and $\mathbb{R^3} \longrightarrow \mathbb{R^3}$ using two separate graphs embedded in $\mathbb{R^2}$ or $\mathbb{R^3}$ representing the domain and image.
- For $\mathbb{R^2} \longrightarrow \mathbb{R^2}$ and $\mathbb{R^3} \longrightarrow \mathbb{R^2}$ using either separate graphs or using contour maps with the z-value being the euclidean norm of $(x,y)$ or $(x,y,z)$ depending on the domain. Using contour maps in this way seems appropriate for visualizing these functions since the general topology on $\mathbb{R^k}$, $ k \in \mathbb{N}$ is induced by the euclidean norm.
However for the cases where:
- $\mathbb{R} \longrightarrow \mathbb{R^3}$, is the only reasonable way to graph this function to graph the image embedded in $\mathbb{R^3}$ ? Or is there another "better" way to visualize such functions?
- $\mathbb{R^n} \longrightarrow \mathbb{R^m}$ where $m = 3k$ , $k \in \mathbb{N}$ and $n = 1,2,3$ , I'm thinking the only way to even attempt visualizing this is to separate the image into $k$ spaces in $\mathbb{R^3}$, and use one of the appropriate techniques above. Is this the closest way to visualizing such functions?
- For $\mathbb{R^n} \longrightarrow \mathbb{R^m}$ where $n>4$, is the only option to disregard the domain completely and separate the image into appropriate spaces embedded in $\mathbb{R}$ , $\mathbb{R^2}$ or $\mathbb{R^3}$?
Are there any other cases I'm not mentioning that are feasible to plot, at least in some way? Is there some field of mathematics associated with these problems? If so, where can I find the source material?