Calculating and visualizing the mean distance between 3 points in 3D

[Mover]

Hi,


First of all my apologees if I am unclear as english is not my native language.


I am looking for 2 things: 1. How to calculate the average disctance between 3 points (a, b, c) in relation to point d. I have the distances between a, b and c (in metres)


a-b: 224m
b-c: 135m
c-a: 356m


And I have the distances of (a, b, c) and point d


a-d: 6681m
b-d: 6471m
c-d: 6363m


I am also 2. looking for a way to visualize this. I have been looking on the Internet, but as this is far from my expertise and I am unfamiliair with the correct terminology i have not yet found what i seek


Summarized:
1. I am looking for a formula to calculate the average distance between 3 points in a 3d environment when I have the distances between points
2. I am looking for a way to visualize this


Is there someone who could help me?

Edit: My own idea was simply to take the average of the 3 numbers but even if that is correct it might not take into account the 3d environment, and I got some of my old schoolbooks but could not really find a solution.

 

[Mover]

My idea was simply to take the average of the 3 numbers but even if that is correct it might not take into account the 3d environment, and I got some of my old schoolbooks but could not really find a solution.

 

[Ishuda]

Hi,


First of all my apologees if I am unclear as english is not my native language.


I am looking for 2 things: 1. How to calculate the average disctance between 3 points (a, b, c) in relation to point d. I have the distances between a, b and c (in metres)


a-b: 224m
b-c: 135m
c-a: 356m


And I have the distances of (a, b, c) and point d


a-d: 6681m
b-d: 6471m
c-d: 6363m


I am also 2. looking for a way to visualize this. I have been looking on the Internet, but as this is far from my expertise and I am unfamiliair with the correct terminology i have not yet found what i seek


Summarized:
1. I am looking for a formula to calculate the average distance between 3 points in a 3d environment when I have the distances between points
2. I am looking for a way to visualize this


Is there someone who could help me?

Edit: My own idea was simply to take the average of the 3 numbers but even if that is correct it might not take into account the 3d environment, and I got some of my old schoolbooks but could not really find a solution.

What are your thoughts? What have you done so far? Please show us your work even if you feel that it is wrong so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting

HINT: Think of the point d as being at the origin, for example in a 2 dimensional Cartesian coordinate system, just translate point d at (x,y)=(x₁, y₁) to (u,v)=(x-x₁, y-y₁). The translation leaves distances preserved. Thus, in two dimensions, you can think of concentric circles centered at the origin. So,
Point a lies on a circle of radius r_a=6681m
Point b lies on a circle of radius r_b=6471m
Point c lies on a circle of radius r_c=6363m
So, if you took point a on circle r_a and drew a circle of radius 224m, where would it intersect circle r_b?

Similarly, in 3 dimensions, think of spheres.

EDIT: By the way, Given the three distances, just compute the averages if that is all you want.

 

[Mover]

First of all, thanks for your suggestions so far

What are your thoughts? What have you done so far? Please show us your work even if you feel that it is wrong so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting

Allright I realized that I should have put my whole case here, so basically the full thing that I am trying to do is to assert the difference of space between A, B and C while they travel towards D. I want to assert if they in general drift apart or come closer together (the general assumption that I would make is that given that they move towards the same point (D) that they come closer together).

Start
The starting positions of A, B and C relative to each other are as follows:

A-B: 400m
B-C: 680m
C-A: 321m
Average: 467m


and the distance towards D is (unfortunately I cannot get this number more accurate than 100m):

A-D: 224217000m
B-D: 224117100m
C-D: 224217200m

Finish
Then A, B and C move closer towards D and their finished positions are:
A-B: 224m
B-C: 135m
C-A: 356m
Average: 238,3m

And the distance to D
A-D: 6681m
B-D: 6471m
C-D: 6363m



Now if I just take the averages of distance between A, B and C to see if they drift apart or come closer together I could make something like this

Which shows the relative distances between the points and how they change.

However this does not show how they are positioned relative to D. After all if we look at the starting and finishing positions relative to D then we see that
A: started out approx 100m (compared to B) and 200m (compared to C) closer to D, but ended up approx 200m (compared to B) and 300m (compared to C) farther from D
B: started out approx 100m closer to D (compared to C) and ended up approx 100m farther

Acutal distance moved (this number is however somewhat inaccurate because the original distance to D was only accurate up to 100m):
A: 224210319m
B: 224210629m (+ 310)
C: 224210837m (+ 208)

So if we would look again at the image it appears in 2D when just looking at the distances between A, B and C, that A-C remained relatively the same, the reality however is that in terms of positions there may have been a 500m drift between A and C because C caught up and passed A relative to D.

And it was at this point that I lost sight of how I should proceed further ...

I like your idea of circles, but given the relative minor distances between ABC and the huge distance with D at the starting position that does not seem feasible on paper, so i am looking for a program that can do so

HINT: Think of the point d as being at the origin, for example in a 2 dimensional Cartesian coordinate system, just translate point d at (x,y)=(x₁, y₁) to (u,v)=(x-x₁, y-y₁). The translation leaves distances preserved. Thus, in two dimensions, you can think of concentric circles centered at the origin. So,
Point a lies on a circle of radius r_a=6681m
Point b lies on a circle of radius r_b=6471m
Point c lies on a circle of radius r_c=6363m
So, if you took point a on circle r_a and drew a circle of radius 224m, where would it intersect circle r_b?

Similarly, in 3 dimensions, think of spheres.

EDIT: By the way, Given the three distances, just compute the averages if that is all you want.

 

[Mover]

There's no integer solution to that system...

Did you write that down at random?

No what you are seeing is data from moving 3 objects in a simulation, where we do not have control over what happens between starting the movement and finishing. So we figured looking a bit further what is happening, we have been writing down the distances between objects at start and finish and the numbers you are referring to is what we found at one such finish.

 

[Ishuda]

First of all, thanks for your suggestions so far



Allright I realized that I should have put my whole case here, so basically the full thing that I am trying to do is to assert the difference of space between A, B and C while they travel towards D. I want to assert if they in general drift apart or come closer together (the general assumption that I would make is that given that they move towards the same point (D) that they come closer together).

Start
The starting positions of A, B and C relative to each other are as follows:

A-B: 400m
B-C: 680m
C-A: 321m
Average: 467m


and the distance towards D is (unfortunately I cannot get this number more accurate than 100m):

A-D: 224217000m
B-D: 224117100m
C-D: 224217200m

Finish
Then A, B and C move closer towards D and their finished positions are:
A-B: 224m
B-C: 135m
C-A: 356m
Average: 238,3m

And the distance to D
A-D: 6681m
B-D: 6471m
C-D: 6363m



Now if I just take the averages of distance between A, B and C to see if they drift apart or come closer together I could make something like this

Which shows the relative distances between the points and how they change.

However this does not show how they are positioned relative to D. After all if we look at the starting and finishing positions relative to D then we see that
A: started out approx 100m (compared to B) and 200m (compared to C) closer to D, but ended up approx 200m (compared to B) and 300m (compared to C) farther from D
B: started out approx 100m closer to D (compared to C) and ended up approx 100m farther

Actutal distance moved (this number is however somewhat inaccurate because the original distance to D was only accurate up to 100m):
A: 224210319m
B: 224210629m (+ 310)
C: 224210837m (+ 208)

So if we would look again at the image it appears in 2D when just looking at the distances between A, B and C, that A-C remained relatively the same, the reality however is that in terms of positions there may have been a 500m drift between A and C because C caught up and passed A relative to D.

And it was at this point that I lost sight of how I should proceed further ...

I like your idea of circles, but given the relative minor distances between ABC and the huge distance with D at the starting position that does not seem feasible on paper, so i am looking for a program that can do so

Part of the problem is that you cannot locate points, A, B, and C relative to one another. Forgetting, for a moment, the point D, imagine Point A as the origin at the starting positions, that is A=(0,0). Point B is on a circle (in 2 dimensions) with center at A and radius 400 m. Let B lie along the positive x axis, i.e. B=(400,0). Now where is C? Obviously, it lies in either the second or third quadrant but how to get the location. Well, let C=(c1, c2), then
c1² + c2² = 321²; point C lies on a circle with center at A and radius 321.
(c1-400)² + c2² = 680²
Thus, substituting for c2²,
-800 c1 + 400² + 321² = 680²
gives a
c1 ~ -249.2
c2 ~ \(\displaystyle \pm\)202.3
So we have two locations for Point C if A were the origin of the system.

However, in reality, depending on the location of the actual origin [point D?], points A, B, and C can be translated and or rotated to still keep the same orientation among themselves. The same applies to the ending position and so something more needs to be known about the relative positions at the beginning and end. Did, for example point C start at the + position and end at the - position?

Note also that if you are just given distances for the A, B, C, and D you have the same kind of problem with a further complication: The actual location of, say point B relative to point A, has two possibilities and, for each of those, you have two possibilities for point C [some of these may collapse but I have a feeling they don't]. Thus you have 4 possible configurations of your triangle ABC at both the beginning and 4 at the end. How do the configurations change with the drift?

As far as the numbers being large at the beginning relative to the other measurements, that might need to be addressed separately unless execution time is of no importance for the program. The uncertainty of the large numbers could be thought of as uncertainties in the initial location leading to a 'circle' (in 2 dimensions) of uncertainty in your final answers. If you proceed that way note that the uncertainty in the positions of A, B, and C would be an uncertainty of their 'local center' and not of the uncertainty of their relative positions given the additional information you would need].

The closer/farther from D at the beginning and end appears to be within the margin of error of the measurements and thus really have no meaning.