Exponential Growth

ID:152015
 
May 23 2008, 4:55 am
I need some help with coming up with a balanced and intuitive system based in math. I know what I want to do, just not sure how to set it up.

I have a system that has a given rating (say 0-5) that represents a level of control over surrounding areas. As the rank of control grows, the area controlled or influenced increases exponentially.

In this scenario, the actual area is represented in tiles, the tiles representing a kilometer of area.

I've tried basic exponential growth (base^rating or rating^power). I also thought of using 10^rating for a magnitude based growth. This works well at lower ratings, but at rating 5 I'm looking at a control area of 100,000 km/tiles!

The progression I'm looking for would be something like a rating of 0 controls a small area, such as the source or up to 1 km from that source. At rating 5 (current limit), the area would be up to 100 km.

I can tweak the formula so that values fall within that range, but I prefer more discrete values. For example, 2.5^rating gives a range of 1 to 98 (rounded values). Rating 3 only gives a value of 16 km. This is not only an odd value (generally preferring numbers ending in 0 or 5), but it also is rather smaller than what I had thought a rating 3 should be. Thoughts?
May 23 2008, 5:04 am
If you want to preserve balance, increasing power exponentially is the wrong way to go. Increasing it linearly (or semi-linearly, like increasing radius of effect linearly) or logarithmically would be better. If the rank is indeed limited from 0 to 5, then I would say a linear radius is best. You could also try something based on a triangular progression, i.e. 1, 3, 6, 10, 15, 21.

Lummox JR
May 23 2008, 5:48 am (Edited on May 23 2008, 7:35 am)
In response to Lummox JR
Lummox JR wrote:
If you want to preserve balance, increasing power exponentially is the wrong way to go. Increasing it linearly (or semi-linearly, like increasing radius of effect linearly) or logarithmically would be better. If the rank is indeed limited from 0 to 5, then I would say a linear radius is best. You could also try something based on a triangular progression, i.e. 1, 3, 6, 10, 15, 21.

Lummox JR

I was looking at a logarithmic progression, but I'm not entirely sure how to set one up. Sadly, my math skills are currently capped at a marginal point somewhere in high school Alg II :( I'm working on fixing that, but for the immediate needs I'm a tad bit lost :)

EDIT:
Actually, I think I'll go with a modified triangular progression with a base of 25: Distance = (Rating*(Rating+1)/2)*5+25
That gives a line up of
  • 0 - 25
  • 1 - 30
  • 2 - 40
  • 3 - 55
  • 4 - 75
  • 5 - 100

Alternatively, I can take out the +25 for a regular progression
  • 0 - 0
  • 1 - 5
  • 2 - 15
  • 3 - 30
  • 4 - 50
  • 5 - 75


The first hits my max pretty nicely, but also starts out pretty big (25km). One alternative use would be as a percentage of a value, making it scalable based on other attributes.

The second one doesn't reach the 100 mark but does have a more even progression from start to finish.
May 23 2008, 1:41 pm
In response to Lummox JR
Arithmetic, geometric, and triangular sequences would make for a great Dream Tutor article...
May 24 2008, 11:21 am
In response to CriticalBotch
If you own a graphing calcuator, you could probably do this fairly easily. It could generate you either an exponential, linear, or quadratic equation hitting the points you want. It might not be very clean though.