I'm trying to come up with a formula that I can use to represent distance from a central point, and will return a boost value in a range, eg 0..10.

This formula can accept two (or maybe 3) variables:

  1. importance: this controls the upper limit of the range, eg 0..10, 0..100
  2. radius: ie any distance within this inner radius should be pretty much as important as any other point within this radius
  3. dropoff: (optional) how quickly the boost should drop off as distance increases

My basic formula is:


    boost = importance / ((radius + distance) ** dropoff)

[download]
For instance, an importance of 100, a dropoff of 0.5 and a radius of 0 gives me a curve like: 

    16   : @@@@@@@@@@@@@@@@@@@@@@@@@
    32   : @@@@@@@@@@@@@@@@@
    64   : @@@@@@@@@@@@
    128  : @@@@@@@@
    256  : @@@@@@
    512  : @@@@
    1024 : @@@
    2048 : @@
    4096 : @
    8192 : @

[download]
But with a radius of 1000, I get a much smaller range: 
    16   : @@@
    32   : @@@
    64   : @@@
    128  : @@
    256  : @@
    512  : @@
    1024 : @@
    2048 : @
    4096 : @
    8192 : @

[download]

I would like to change the formula to make the upper limit of the range dependent only on importance and not on radius. For the life of me I can't figure out how to do that.

PS: distance can be considered to have an upper limit of, eg, 10,000km if that helps

PPS: As an optional extra, it would be nice to be able to configure the rate of dropoff without affecting the upper limit of the range, but I can live without it

In reply to [OT] Normalizing the return result of an exponential formula by clinton

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