Formula for a "weighted" scale?

Question

I'm trying to find the formula (or even just the area of math) that I need to achieve the following concept.

I need a scale (X) from 0-100, that wraps around another scale (Y) where the Y values are "weighted" and become more and more compressed the further up scale X they sit.

Ideally I'd like the option to adjust the "compression" rate of scale Y, as hopefully demonstrated in the image below.

Example A features regular spacing on scale Y (ie. zero compression), so it naturally lines up perfectly with scale X.

Example B has a small amount of positive compression on scale Y, so that values on Y will be shifted more to the upper end of scale X.

Example C is a more extreme example of B, where even more compression is applied to scale Y and the values really start to bunch up at the top end of the X scale.

Using example C, if I wanted the 50% value the formula would return ~18%.

I hope I've explained this properly. Thanks in advance for any help.

scale examples

Andrei · Accepted Answer · 2018-09-08T18:33:13.997

1

It is much easier to think in terms of scale from $0$ to $1$. You can then multiply by $100$ if you want. You should draw the lines on the scale as $x^p$. Depending on the value of $p$, you can get lines more compressed towards $0$ or towards $1$. Here is an image with $100\cdot x^p$, with $p=0.5$ and $p=0.25$

edited Sep 08 '18 at 18:33

answered Sep 08 '18 at 17:26

Andrei

37,370

Thanks very much, it makes a lot more sense now I realise it's an exponential sequence. I think I've got a grasp on this now. – Andrew Sep 08 '18 at 19:08
It can also be something like $\sin \frac{\pi x}{2}$ – Andrei Sep 08 '18 at 19:18

Formula for a "weighted" scale?

1 Answers1