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

  • Context: Graduate 
  • Thread starter Thread starter Dragonfall
  • Start date Start date
  • Tags Tags
    Curves Random
Click For Summary

Discussion Overview

The discussion revolves around the definition and characteristics of a "random" curve in R^n, specifically focusing on the case where n=2. Participants explore various methods of generating such curves, the implications of randomness, and the mathematical properties that may arise from different definitions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question how to sensibly define a "random" curve, suggesting that it may depend on the probability distribution used.
  • There are suggestions to generate random curves using random derivatives or by selecting random points, though some express skepticism about the completeness of these methods.
  • One participant proposes a method involving a probabilistic approach to generate mathematical expressions, which could define a curve in R^2.
  • Another participant mentions the Wiener process (Brownian motion) as a standard method for generating random curves, noting that it is nowhere differentiable.
  • There is a discussion about whether certain stochastic processes can yield C1 curves, with some asserting that while processes like Brownian motion are continuous, they are not differentiable.

Areas of Agreement / Disagreement

Participants express differing views on the definition of "random" and the methods for generating random curves. There is no consensus on a singular definition or approach, and the discussion remains unresolved regarding the properties of the curves generated.

Contextual Notes

Participants highlight the ambiguity in defining "random" and the need for specific probability distributions. The discussion also reflects on the limitations of certain methods in producing curves with desired differentiability properties.

Dragonfall
Messages
1,023
Reaction score
5
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.
 
Physics news on Phys.org
random points + bezier curves on computer?
 
Then I'm guessing we can only have finitely many random points. I don't think all curves can be described this way.
 
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?
 
Curve = continuous map from a real interval to R^n

Random is what we're trying to define.
 
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
 
Dragonfall said:
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. ?
 
Dragonfall said:
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.
 
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.
 
  • #10
HallsofIvy said:
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].
 
Last edited:
  • #11
AUMathTutor said:
I guess this is what you mean by order n=2

n is the dimension of the space we're in.
 
  • #12
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.
 
  • #13
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.
 
  • #14
Dragonfall said:
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.
 
Last edited by a moderator:
  • #15
trambolin said:
random points + bezier curves on computer?

gel said:
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?
 
Last edited by a moderator:
  • #16
ice109 said:
does that generate a c1 curve? what stochastic process does?

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

Similar threads

Undergrad Generic Curve in R^n: What We Know
  • · Replies 5 ·
Replies
5
Views
3K
Graduate About computing the tangent space at 1 of certain lie groups
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Finding Minimal Mean Distance Curves on the Unit Sphere
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Induced orientation on boundary of ##\mathbb{H}^n## in ##\mathbb{R}^n##
  • · Replies 6 ·
Replies
6
Views
3K
High School Solving for integral curves- how to account for changing charts?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Parallel Transport Along Geodesic
  • · Replies 6 ·
Replies
6
Views
5K
Undergrad Affine parameter and non-geodesic null curves
  • · Replies 1 ·
Replies
1
Views
1K
Graduate Differential forms on R^n vs. on manifold
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Differential for surface of revolution
  • · Replies 30 ·
2
Replies
30
Views
4K
Graduate Differential movement along a curved surface
  • · Replies 3 ·
Replies
3
Views
3K