# Possibility of a point moving out of a shape

• Asuralm
In summary, the conversation discusses the problem of a point moving out of a shape, such as a square or triangle, when a noise of fixed distance is added to it in an arbitrary direction. The possibility of the point moving out of the shape depends on the shape and can be calculated through integrals and probability distributions. The conversation also mentions the generalization of the problem to higher dimensions and the concept of Brownian motion. Ultimately, the goal is to find an equation to model the problem.
Asuralm
Dear all:

The question is like this:

Given a point in a shape like square or triangle. Let's take square as the simplest one. Say the length of the side of the square is $$l$$. If add a noise to the point which will move the point [tex]r[\tex] distance and the direction is arbitrary, what will be the possibility that the point will move out of the square?

And what will be the possibility if the point is in a triangle? Also what about a cube or tetrahedron?

Does anyone have any idea to solve the problem please?
Ideally if there is a equation to model the problem.

Thanks

YOu are talking about "Brownian Motion". As I recall, one can show that, with a step size of r, the point will move on average, a distance $r\sqrt{n}$ from its initial position in n steps. Whether that will move it out of the figure depends on shape of the position. You could probably do an integral over $d\theta$ with $\theta$ going from 0 to $2\pi$ using the distance from the point to boundary at each $\theta$.

I will simplify the shape to a line. If you can solve the problem in 1-D (a line), you can generalize to 2-D (a plane).

If x is given, then the random part of y is r. Without loss of generality, assume the endpoints of the line are 0 and 1, and $\ell$ = 1. Let y = x + r. Then the probability you have inquired about is given by Pr{y < 0 or y > 1|x} = 1 - Pr{0 < y < 1|x} = 1 - (F(1|x) - F(0|x)) where F is the cumulative distribution (CDF) of y|x, which you need to derive.

For example, one might assume that given x, r is distributed uniformly over [x-0.1, x+0.1]; show that distribution as U(r) = (r - (x-0.1))/0.2. Then F(y*|x) = Pr{y < y*|x} = Pr{x+r < y*|x} = Pr{r < y*-x|x} = U(y*-x) = (y*-x - (x-0.1))/0.2 = (y*+ 0.1)/0.2.

When generalizing to a plane, you need to define the distribution of r over a circle around x, instead of an interval (the interval [x-0.1, x+0.1] in my example).

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The expression (y*+ 0.1)/0.2 in my above post should have been (y*-2x + 0.1)/0.2.

I'm not 100% sure what the OP is asking for. If the "noise" is a single shock of size r then you need to calculate the length of the portion of the circle of radius r centred at the point that lies outside of the given set.
If it is a mathematical brownian motion (http://en.wikipedia.org/wiki/Wiener_process" ) and want to know the probability of it lying outside the set at a given time T, then it becomes an integral over the normal distribution.
If you want the probability of it going outside the set at any time before T, then the problem is much harder, and becomes a statement about the distribution of the http://en.wikipedia.org/wiki/Hitting_time" of the brownian motion.

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I don't think I am talking about Brownian motion as the energy of noise is fixed. Given a length r, the point should only noised by r.

Thanks EnumaElish, It really helps.

## 1. Can a point move outside of a shape?

Yes, it is possible for a point to move outside of a shape. This can happen if the shape is not closed or if the point is moved beyond the boundaries of the shape.

## 2. How does the shape of the object affect the possibility of a point moving out of it?

The shape of the object plays a significant role in determining the possibility of a point moving out of it. Shapes with irregular or open boundaries are more prone to allowing a point to move out, while closed and regular shapes provide more confinement.

## 3. Are there any factors that can prevent a point from moving out of a shape?

Yes, there are several factors that can prevent a point from moving out of a shape. These include the shape of the object, the boundaries of the object, and any external forces or constraints acting on the point.

## 4. How can we calculate the possibility of a point moving out of a shape?

The possibility of a point moving out of a shape can be calculated by considering the shape's boundaries, the point's current position, and any external forces acting on the point. This can be done using mathematical equations and principles of geometry and physics.

## 5. Is it possible for a point to move out of a shape in a controlled manner?

Yes, it is possible for a point to move out of a shape in a controlled manner. This can be achieved through precise manipulation of external forces or by using specific shapes and boundaries that allow for controlled movement of the point.

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