Fractal Generation System

Post by **mkruer** » Tue Mar 09, 2004 10:18 am

OK I need the brain trust. I was digging through some of the data NASA has on Near Earth Objects (NEOâ€™s) can found some interesting trends for NEOâ€™s

Here are two graphs, the first one is Inclination, this is how inclined the objects are in their orbits. 0 being the same are the orbital plane, and 90 being perpendicular to the orbital plain. The other one Eccentricity, and how circular the orbits are, 0 being a perfect circle and 1 being a straight line. All the other data is liner. About the only thing I have to add is that the number vs size of ob there are about 2600 know NEOâ€™s and 700 of them are over 1 kilometer in diameter.

What I am looking for is a mathematical equation that approximately follows the listed data for the Eccentricity and Inclination. If someone can help me with the math, I would appreciate it. Also the information could be extremely helpful for any fractal generator that comes out.

peteyg · Post by **peteyg** » Tue Mar 09, 2004 7:39 pm

the first one looks like f(x)=x^2

The second one..... maybe f(x)=.5*(x^5-.1*(x-1)^4+.1) from x = -1 to 1. Though I couldn't get the middle part to be as linear as it is in your graph.

Try plugging that into a math plotting program, see how you like it.

pincushionman · Post by **pincushionman** » Tue Mar 09, 2004 8:52 pm

To avoid confusion, eccentricities are as follows:

e = 0: Circle.
0 < e < 1: Ellipse.
e = 1: Parabola.
e > 1: Hyperbola.

Circles and ellpses can describe closed orbits, hyperbolas describe paths of non-captured objects (enter along asymptote A, exit along aymptote B). Parabola is just the limit between orbit and escape.

As for your second plot, it looks like some variant of ax + b*tan(cx + d) + e will work. That is, if you choose c and d so that the asymptotes lie just a little outside of your domain. It's fairly linear in the middle, and goes up sharply on the ends. But i'm not sure which math functions you have at your disposal.

-pincushionman

peteyg · Post by **peteyg** » Wed Mar 10, 2004 3:02 am

Oh, tan. Duh.

Average Earthman · Post by **Average Earthman** » Wed Mar 10, 2004 1:37 pm

Could the second one be related to a normal distribution?

http://mathworld.wolfram.com/NormalDistribution.html

oligo · Post by **oligo** » Wed Mar 10, 2004 1:55 pm

For the second one: f(x)=0.5(1-sqrt(1-x) + sqrt(x))
for 0 <= x <= 1

oligo

Linknoid · Post by **Linknoid** » Wed Mar 10, 2004 6:53 pm

Excellent fit for second graph:

y=-ln(1/(100*x) - 0.01) / 10 + .1

The first graph looks more like a higher order power (not x squared), but I'm still playing around trying to get a good fit.

Post by **mkruer** » Wed Mar 10, 2004 8:52 pm

So far for Eccentricity f(x)=0.25(1-sqrt(1-x) + sqrt(x)) works the best however it still needs to be tweaked there needs to be more of an incline in the beginning and towards the end.

For Inclination f(x)=1-((1-x)^(1/8 )) is the closest I could come to the curve. There is a slight difference towards the end, but I would say that it is acceptable.

oligo · Post by **oligo** » Wed Mar 10, 2004 10:33 pm

or f(x)=0.25(1-(1-x)^.35 + x^.35)
tweak the .35 values to change inclination, one is for the beginning the other for the end

oligo

Post by **mkruer** » Wed Mar 10, 2004 10:34 pm

OK I think I found a good one for the Eccentricity

f(x)=0.5*(1-((1-x)^(1/3))+((x)^(1/3)))

This comes out the closets, however because it is using the cube root, it takes too sharp a downwards angel and upwards angle near the end. If there is a way to make it more of a curve let me know. In addition, I will post some other info that I found on planets, moons, comets etc... in order to better define the fractal algorithm.

I think with all this information we should be able to define a very close approximation for randomly generated systems

Post by **mkruer** » Thu Mar 11, 2004 11:15 am

OK this graph is a little complex to understand. Basically what I did was I took all the known moon for all gas giants in the entire system (131 of them) standardized them around the same mass, in this case 1 Jupiter Mass, and adjusted the orbit distance appropriately. Next I plotted all the Moons by distance and its orbit Standard or Retrograde. What I noticed was that all the Planets have a void gap between the inner and outer moons, in roughly the same spot, in this case between 12,000,000,000m and 18,000,000,000m mark. Also only 5% of the moons on the inside group are retrograde, and 95% are standard where as for the outside group 12% are standard and 88% are retrograde in orbit.

Again I hope to come up with a simple equation the illustrate the probability of moon at any give distance. In addition I need an equation that illustrate the likely hood of the moon being in a retrograde in it orbit at any give distance.

Next too come will be the Eccentricities, and Inclination for a moon at any give distance. Have fun

pincushionman · Post by **pincushionman** » Thu Mar 11, 2004 7:49 pm

You've got distance there on the y-axis, but what are you comparing it against?

-pincushionman

richard · Post by **richard** » Thu Mar 11, 2004 11:44 pm

mkruer: this is geat stuff, BTW!

Post by **mkruer** » Fri Mar 12, 2004 12:06 am

pincushionman wrote:You've got distance there on the y-axis, but what are you comparing it against?

-pincushionman

For the across all it says is the probability so it could be 0-1, or however may moons you would like to place within that distance. If you had 10 moons or 100 moons, they should all follow that general curve. Thatâ€™s all it says.

Post by **mkruer** » Fri Mar 12, 2004 5:10 am

OK here is the final Data on NEOâ€™s for systems. This should be all he basic that you need for generation of all the orbits of any give NEOâ€™s, and its size. Mass is almost irreverent, however using some other asteroids the mass is between 3 (g/cmÂ³) and 8 (g/cmÂ³) with a standard bell curve.

Eccentricity f(x)=0.5*(1-((1-x)^(1/3))+((x)^(1/3)))
Inclination f(x)=1-((1-x)^(1/8 ))*90 degrees
Diameter/Size f(x)=1-((1-x)^(1/49 ))*50km
Semi Major Axis f(x)=x*360 degrees
Ascending Node f(x)=x*360 degrees

Now for the other â€œfunâ€

Post by **mkruer** » Fri Mar 12, 2004 7:56 am

OK hopefully this will more since for everyone. Instead of compiling a list of all the moons and ranking them in order, I have tried to break them down into simpler functions for everyone. Note this may no be 100% accurate but itâ€™s a hell of a lot easier to understand.

There are 3 parts,
First is Grade, i.e. the probability that the orbit will go clockwise.
Second is Retro, i.e. the probability that the orbit will go counter clockwise.
Finally there is Distribution; This is the likely hood of finding an object.

Will all these formulas rolled into one you should get approximately the same curve as seen on my first post about the moons.

Again I need help finding the math aspect for the curves

Eccentricity, and Inclination are a little more complex but they seem to follow the same general curve as Distribution

This seems to be common formula for the Semi Major Axis, and Ascending Node for orbiting moons
Semi Major Axis f(x)=x*360 degrees
Ascending Node f(x)=x*360 degrees

oligo · Post by **oligo** » Fri Mar 12, 2004 6:21 pm

I like playing with curves here's what I found
retro: f(x) = x^2(3-2x)
or f(x) = e^(15(x-.5))/(e^(15(x-.5))+1)
play with the 15 value to change curve inclination, and the .5 value to shift the curve center.

grade seems to be 1-f(x)

For the distribution curve, I don't know if this one can fit : f(x) = 1-e^(-(10(x-.5))^2)
with 0<=x<=1 for all.

oligo

Post by **mkruer** » Fri Mar 12, 2004 9:44 pm

I will check out the curves this weekend.

After this, then there are only 5 more things to do.

1. Comets (very similar to NEOâ€™s)
2. Asteroid belt
3. Trojan Fields (very similar to Asteroid Belt) But limited to LaGrange points L4, L5 around the larger inner planets, i.e. Jupiter, and perhaps some around Saturn.
4. Planetary rings (very similar Asteroid Belt, smaller partials, and
5. And finally the system itself. (this is perhaps the biggest unknown)

The last stage is to standardize all the stuff. If you have noticed, I have been trying to make all the values 0-1, this way we can have the system determine the approximate distance for each object to appear at.

Then we will have all the information to generate a realistic fractal orbits for any system.

Then we will get to the more interesting part of textures, and 3D objects.

Post by **mkruer** » Sat Mar 13, 2004 10:59 am

@oligo

Thanks for the help.

I was playing around with the Curves i.e. formulas, and the first one looks good. its opposite is fine for standard orbit, however maybe you might know this, I want to limit the min and max of the curves at 5%. In theory this say that no matter what, toward the ends there is all ways a 5% chance for the object to go in the opposite direction.

The other formula Needs work essentially what I am looking for a flat line but at some point(s) there needs to be dips where it is a lower probability to find objects.

oligo · Post by **oligo** » Sat Mar 13, 2004 4:21 pm

glad to help

mkruer wrote:I want to limit the min and max of the curves at 5%. In theory this say that no matter what, toward the ends there is all ways a 5% chance for the object to go in the opposite direction.

With f(x)=x^2(3-2x)*.95 the min value is 0% and the max is 95%
or this one f(x)=x^2(3-2x)*.9 + .05 the min is 5% and the max 95%

mkruer wrote:The other formula Needs work essentially what I am looking for a flat line but at some point(s) there needs to be dips where it is a lower probability to find objects.

For the distribution curve ? could a trigonometric function (sin) do the job ?

oligo

Post by **mkruer** » Sun Mar 14, 2004 11:44 am

Orbiting Moons (136)
Eccentricity; See Chart
Inclination; See Chart
Diameter/Size; See Chart
Semi Major Axis; f(x)=x*360 degrees
Ascending Node; f(x)=x*360 degrees
Orbital Distance; See Chart

Post by **mkruer** » Sun Mar 14, 2004 11:46 am

Asteroids (2686)
Eccentricity; f(x)=0.5*(1-((1-x)^(1/3))+((x)^(1/3)))
Inclination; f(x)=(1-((1-x)^(1/8 )))*90 degrees
Diameter/Size; f(x)=(1-((1-x)^(1/49 )))*50km
Semi Major Axis; f(x)=x*360 degrees
Ascending Node; f(x)=x*360 degrees
Orbital Distance; See Chart

Comets (41)
Eccentricity; f(x)=0.5+(x/2)
Inclination; See Chart
Diameter/Size; Unknown/No Information
Semi Major Axis; f(x)=x*360 degrees
Ascending Node; f(x)=x*360 degrees
Orbital Distance; See Chart

Post by **mkruer** » Sun Mar 14, 2004 11:48 am

Asteroid Belt (See Chart; est. total mass 3.6E+22 kg current, > 2E+25, when system formed)
Eccentricity; Enough to keep it with the defined area
Inclination; Enough to keep it with the defined area
Diameter/Size; <1000km
Semi Major Axis; f(x)=x*360 degrees
Ascending Node; f(x)=x*360 degrees
Orbital Distance; 2-4AU

Trojan Fields (very similar to Asteroid Belt) But limited to LaGrange points L4, L5 around the larger inner planets, i.e. Jupiter, and perhaps some around Saturn. Note: The Trojan field combined have more mass then in all of the asteroid belt

Planetary Rings (very similar Asteroid Belt, smaller partials,)
Working on it

Master System Distribution
Working on it

Shark · Post by **Shark** » Sun Mar 14, 2004 9:52 pm

Lots of asteroids!
No elliptical orbit?

Post by **mkruer** » Sun Mar 14, 2004 10:26 pm

Actually, there are a lot fewer asteroids then I original though. This makes it so in real life the density in the belt is perhaps one object every 1000km or more. Not very dramatic is it. As for the orbits, most of them are probably elliptical, is that there are just so many of them, and they are spaced more or less equality that it appears, and is on a hole very circular.

With the updated orbits, the asteroids will not be as static as they are in the current VS. If you were in the field it would be more as if you were a cross fire with massive boulders. Itâ€™s just the way the numbers play out. Where it becomes really fun is in planetary rings. The density of those are much greater.