Can a Random Curve in R^n be C1 and Differentiable?

[Dragonfall]

This came up a while ago in a post. What is a sensible way of defining a "random" curve in R^n? Let's say n=2 in order to keep things simple.

 

[trambolin]

random points + bezier curves on computer?

 

[Dragonfall]

Then I'm guessing we can only have finitely many random points. I don't think all curves can be described this way.

 

[HallsofIvy]

What do you mean by "curve"? Graph of any function? or relation? Or continuous function?

What do you mean by "random"? Obeying some probability distribution? What distribution?

 

[Dragonfall]

Curve = continuous map from a real interval to R^n

Random is what we're trying to define.

 

[Office_Shredder]

Start with a random derivative, and randomly pick the second derivative for each t? Then you have a continuous derivative and probably a well-behaved random curve

 

[trambolin]

Then I'm guessing we can only have finitely many random points. I don't think all curves can be described this way.

I got it wrong. I thought you are trying to generate curves. But why do you need to define "random"? And HallsofIvy is right, with respect to which distribution etc. ?

 

[HallsofIvy]

Curve = continuous map from a real interval to R^n

Random is what we're trying to define.

Who do you mean by "we"? I know how to define "random": according to some probability distribution. That's why I asked what probability distribution you wanted to use.

 

[AUMathTutor]

If you wanted to define a random parabolic curve (I guess this is what you mean by order n=2) then you would just have to pick 3 random numbers.

For each of the random numbers, you would need a distribution. It doesn't make sense to just say "pick a random real number". That cannot be done without further imposed constraints.

 

[Dragonfall]

Who do you mean by "we"? I know how to define "random": according to some probability distribution. That's why I asked what probability distribution you wanted to use.

"We" as in those who have posted in this thread before you. So how exactly do you define a probability distribution over the set of curves in R^2? Say according to N[0,1].

 

[Dragonfall]

I guess this is what you mean by order n=2

n is the dimension of the space we're in.

 

[AUMathTutor]

Oh right, sorry.

I don't think a random curve in R^2 is well defined. But here's an example of one:

Let an alphabet consist of the following symbols:
x, +, -, *, /, (, and ), and the digits 0...9, with a decimal point, and a special symbol END which means the string is over and a special string START which means the string is beginning.

(1) Start with the START symbol.
(2) Use a probability table associated with START for picking the next character.
(3) Use that character's probability table to select the next character.
(4) Repeat step 3 until you get the END character.
(5) The string between START and END can be interpreted as an expression in x... the probability tables can be chosen judiciously so as to make such strings well-formed. Call this expression e_x.
(6) Let f(x) = e_x. There you go.

 

[AUMathTutor]

For instance, let the tables for START, +, -, *, and / be this:
x: 1/11
0: 1/11
...
9: 1/11

Let's exclude parentheses from this one.

Let the table for x be this:
+: 1/5
-: 1/5
*: 1/5
/: 1/5
END: 1/5

Let the table for digits be this (let's exclude decimal points)
+: 1/15
-: 1/15
*: 1/15
/: 1/15
0: 1/15
1: 1/15
...
9: 1/15
END: 1/15

You can imagine rolling dice or writing a program or whatever that would give you a string of random (or pseudorandom) numbers which determined one function in R^2.

I believe this method can be easily extended to R^n. How? Add as many variables as you want to the alphabet.

 

[gel]

This came up a while ago in a post. What is a sensible way of defining a "random" curve in R^n? Let's say n=2 in order to keep things simple.

The question is rather ill-defined. There's lots of ways of generating random curves. The method you choose depends on what properties you want. A standard one is the http://en.wikipedia.org/wiki/Wiener_process" (aka Brownian motion), which is nowhere differentiable.

 

[ice109]

random points + bezier curves on computer?

The question is rather ill-defined. There's lots of ways of generating random curves. The method you choose depends on what properties you want. A standard one is the http://en.wikipedia.org/wiki/Wiener_process" (aka Brownian motion), which is nowhere differentiable.

does that generate a c1 curve? what stochastic process does?

 

[gel]

does that generate a c1 curve? what stochastic process does?

No, its continuous but not differentiable. Could integrate it to get a C1 curve.