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There are methods to convert quantities to certain scales and plot them on a graph (e.g., logarithmic scale graph). I wanted to plot quantities that belong to range $(0, \infty)$ on a graph of a fixed size (let's say 1). I came up with using the parabolic function: $$x' = 1 - \frac 1 {x + 1}$$ where $x$ is the original quantity and $x'$ is the converted length on the scale. Using this, $(0, \infty)$ will be mapped to $(0, 1)$. Is this kind of graph in common use? Is there a name for such graph?

Is there an alternative way to fulfill this purpose?

sawa
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    You could also use $x'=\frac{2}{\pi}\tan^{-1}x$. – vadim123 Dec 25 '15 at 22:07
  • Just asked a similar question, did you ever find an example of what you are looking for.this is what i just asked https://math.stackexchange.com/questions/3233823/how-do-normal-shapes-translate-from-a-confined-infinite-graph-to-our-normal-coor – Joe May 23 '19 at 02:37

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The specific function certainly has the formal property you seek, but it's probably not in wide use.

Non-Cartesian coordinate systems, such as log coordinates, are used to highlight mathematical features. For example, power laws become linear in suitable combinations of log and linear coordinates.

Depending on your intent for your graph, there might well be "better" choices of coordinates. For example, you could:

  • Embed the first quadrant $(0, \infty) \times (0, \infty)$ in three-dimensional space and perform point projection as if looking from a point "above" the third quadrant, toward the origin (diagram).

  • Construct a conformal map from $(0, \infty) \times (0, \infty)$ to the unit square $(0, 1) \times (0, 1)$ or the quarter-disk.

Offhand, I expect you'll get visually-nicer results by not using a map of the form $(x, y) \mapsto \bigl(f(x), f(y)\bigr)$, but if you do go that route, I'd suggest the hyperbolic tangent $$ f(x) = \tanh x = \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}. $$

Perspective mapping a planar quadrant into a bounded box

Andrew D. Hwang
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