Average distance between points on a circle

In summary, the conversation discusses finding the average distance between two points in a circle of radius r by exploiting symmetry and using a formula involving a quadruple integral. It is determined that the average distance is equal to 2r/3.
  • #1
phoenixthoth
1,605
2
of radius r or a square of side length a? do you need some kind of quadruple or double integral or is there a trick?

sorry, i meant two points either in the interior of said shape or on the boundary.

for the circle, by symmetry, is that the same as the average distance between a point and the origin? that is, i think, 2r/3. note that the max distance is 2r and the min distance is 0.
 
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  • #2
What's wrong with a quadruple integral? :smile:


There is some symmetry one can exploit for the circle, but not what you suggested.
 
  • #3
would this be the formula for the average distance for two points in a circle of radius r:
[tex]\frac{\int_{-r}^{r}\int_{-\sqrt{r^{2}-y2^{2}}}^{\sqrt{r^{2}-y2^{2}}}\int_{-r}^{r}\int_{-\sqrt{r^{2}-y1^{2}}}^{\sqrt{r^{2}-y1^{2}}}\sqrt{\left( x2-x1\right) ^{2}+\left( y2-y1\right) ^{2}}dx1dy1dx2dy2}{\int_{-r}^{r}\int_{-\sqrt{r^{2}-y2^{2}}}^{\sqrt{r^{2}-y2^{2}}}\int_{-r}^{r}\int_{-\sqrt{r^{2}-y1^{2}}}^{\sqrt{r^{2}-y1^{2}}}1dx1dy1dx2dy2}[/tex]?

btw, i get [tex]\frac{\int_{-r}^{r}\int_{-\sqrt{r^{2}-y1^{2}}}^{\sqrt{r^{2}-y1^{2}}}\sqrt{x1^{2}+y1^{2}}dx1dy1}{\int_{-r}^{r}\int_{-\sqrt{r^{2}-y1^{2}}}^{\sqrt{r^{2}-y1^{2}}}1dx1dy1}=\frac{2r}{3}[/tex]
 
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1. What is the formula for calculating the average distance between points on a circle?

The formula for finding the average distance between points on a circle is D = C/(n-1), where D represents the average distance, C represents the circumference of the circle, and n represents the number of points.

2. How do you determine the distance between two points on a circle?

The distance between two points on a circle is determined by finding the length of the arc connecting the two points and then dividing it by the circumference of the circle. This can be calculated using the formula D = (x/360) * 2πr, where D represents the distance, x represents the central angle between the two points, and r represents the radius of the circle.

3. How does the number of points on a circle affect the average distance between them?

The more points there are on a circle, the smaller the average distance between them will be. This is because as the number of points increases, the circumference of the circle remains the same, but the number being divided by (n-1) in the formula also increases, resulting in a smaller average distance.

4. Can the average distance between points on a circle be larger than the diameter of the circle?

No, the average distance between points on a circle can never be larger than the diameter of the circle. This is because the diameter represents the longest possible distance between any two points on a circle.

5. How is the average distance between points on a circle useful in real-world applications?

The average distance between points on a circle can be useful in various fields such as navigation, geography, and engineering. It can help in determining the most efficient routes, estimating distances between landmarks, and optimizing the placement of objects on circular structures.

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