Integrating Along a 2D Line Segment

In summary, the conversation discusses the evaluation of an integral for a squared exponential covariance function, which calculates the covariance between two points. The function is dependent on a lengthscale constant and the Euclidean distance between the points. The question at hand is how to evaluate the function from a point to a line segment, which involves integrating a modified equation that incorporates the equation of the line. The individual has been able to solve the 1-D case, but is having trouble solving the 2-D case without rotating the line segment. They are seeking help in determining if there is a way to solve the integral without resorting to this method.
  • #1
Teg Veece
8
0
I've been trying to evaluate an integral for the last few days now and it really has me stumped.
I was hoping that maybe someone here would be able to help me out.

So the function, cov(x,x'), is fairly basic. It's called a squared exponential covariance function and it evaluates the covariance between two points.

[tex]
cov(x,x') = e^{-(\frac{r^2}{2l^2})}
[/tex]

r is just the Euclidean distance between the points x and x'. l is the lengthscale constant for that dimension.

So for example the 1-D case, with an lengthscale of say 2, the covarance between a x = 5 and x' = 9 is:
[tex]
cov(5,9) = e^{-\frac{1}{2}(\frac{(5-9)^2}{(3)^2})}= 0.4111
[/tex]

In the 2-D case with an x lengthscale of 2 and a y lengthscale of 1 the covariance between x = (5,3) and x'=(9,6) is:

[tex]
cov([5,3],[9,6]) = e^{-\frac{1}{2}(\frac{(5-9)^2}{3^2}+\frac{(3-6)^2}{1^2})}= 0.0046
[/tex]So that's the function evaluated from a point to another point. However, I want to evaluate it from a point to a line segment.
I've solved this for the 1-D case but the 2-D case seems to be a bit trickier.

I've rewritten the covariance function so that it incorporates the equation of the line:

[tex]
cov(([xstart,ystart],[xend,yend]),[xpoint,ypoint])=\int e^{-\frac{1}{2}(\frac{(x-xpoint)^2}{lengthscalex^2}+\frac{((mx+c)-ypoint)^2}{lengthscaley^2})}dx
[/tex]

[xstart,ystart] are the starting coordinates of the line segment and [xend,yend] and the ending coordinates. [xpoint,ypoint] are the coordinates of the point. Everything in the equation is now a constant except for x. (m is the slope of the line segment and c is the y intercept of it)
So the limits of the integral would be xstart and xend.

I'm not sure how to integrate this however.

I have been able to get a solution by rotating the line segment and the point about the origin so that the line segment's slope is zero. Then it's very similar to the 1-D case. However is it possible to solve that integral without resorting to this.

Any help at all would be greatly appreciated. If you need me to clarify anything, just let me know.

I've summarised my problem in the pic below.
f9nms7.jpg
 
Last edited:
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  • #2
Or I guess in other words, what I'm asking is it is possible to determine the mean value of the Gaussian distribution along a line segment like in the diagram below?
2w7n33m.jpg
 

1. What is the purpose of integrating along a 2D line segment?

The purpose of integrating along a 2D line segment is to calculate the total value of a function over a specific line segment in a 2D space. This can be useful in various scientific fields, such as physics, engineering, and computer graphics.

2. How is the integration along a 2D line segment different from regular integration?

The main difference is that integrating along a 2D line segment involves calculating the integral of a function over a specific line segment in a 2D space, rather than over an entire 2D region. This requires using a different set of equations and techniques, such as parametric equations and line integrals, compared to regular integration.

3. What information is needed to perform integration along a 2D line segment?

In order to perform integration along a 2D line segment, you will need to have the function that you want to integrate, the parametric equations that define the line segment, and the limits of integration along the segment. Additionally, you may need to have knowledge of vector calculus and multivariable calculus.

4. What are some real-world applications of integrating along a 2D line segment?

Integrating along a 2D line segment has numerous real-world applications. For example, in physics, it can be used to calculate the work done by a force along a specific path. In engineering, it can be used to calculate the total force or torque acting on a rigid body along a specific line segment. In computer graphics, it can be used to calculate the area or volume of a 2D shape or 3D object, respectively.

5. Are there any limitations to integrating along a 2D line segment?

One limitation of integrating along a 2D line segment is that it can only be applied to functions that are continuous and differentiable along the line segment. Additionally, the line segment must be smooth, meaning it cannot contain sharp turns or corners. If these conditions are not met, then the results of the integration may not be accurate.

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