Model the relationship between 3d points given two reference points

[dwelu]

Homework Statement

Trying to figure out how to model the relationship between two 3D points, A and B. Both points have visibility to 2 common reference points, C and D. Bearing and angle readings are available from both A and B to the reference points C and D. The reference points C and D are at the same elevation. The goal is to spatially model these (relatively model in unitless 3d space) so the true distance between the points is not needed. Distances can be assumed in order to determine a 3d line equation that enables this modeling.

Homework Equations

Direction vector: (x-x0, y-y0, z-z0)
Parametric 3d line:
x = x1 + at
y = y1 + bt
z = z1 + ct

The Attempt at a Solution

I have assumed distances to the reference points in order to determine direction vectors, and therefore parametric line equations can be determined from A and B individually to points C and D. After that I’m not sure how to use this to relate the two points A and B.

 

[KataKoniK]

Any help would be appreciated.



Thank you for sharing your problem with us. To model the relationship between two 3D points, A and B, we can use the concept of triangulation. Triangulation is a method used to determine the position of a point by measuring the angles between that point and two known reference points. In this case, our reference points are C and D.

To begin, we can define the position of point A as (x1, y1, z1) and the position of point B as (x2, y2, z2). We can also assume that the distances between points A and C, and points B and D, are known and denoted as d1 and d2, respectively.

Using the direction vector and parametric 3D line equations that you have mentioned, we can create two equations for each point, A and B, as follows:

For point A:
(x1 + a1t) = xC
(y1 + b1t) = yC
(z1 + c1t) = zC

For point B:
(x2 + a2t) = xD
(y2 + b2t) = yD
(z2 + c2t) = zD

From these equations, we can solve for the values of a1, b1, c1, a2, b2, and c2 using the known distances d1 and d2. Once we have these values, we can use them to create a system of equations that relates the positions of points A and B:

(x1 + a1t) = (x2 + a2t)
(y1 + b1t) = (y2 + b2t)
(z1 + c1t) = (z2 + c2t)

Solving this system of equations will give us the values of t and the direction vector components (a, b, c) that represent the relationship between points A and B. This will enable us to model the spatial relationship between the two points.

I hope this helps. Please let me know if you have any further questions or concerns.