Hello there!!
Could you please help me to find the interpolation for such a point with respect of the distance. please check the picture bellow
**broken link removed**
I want find the value of point F respecting the distance between the other points
You can determine the location of F by using any of the other points using a method called triangulation.
1) You know the distance from point A to F, so you can draw a circle around A with a known distance (radius). YOu now can say that F will lie at some point on that circle (all points on the circle are the correct distance away from A).
2) You know the distance from B to F. Draw a circle around B using that distance. The circle for A and B should meet at two points. F will exist at one of those two intersections, since F must lie on both of the circles for A and B.
3) You know the distance from C to F. Draw a circle around C using that distance. The A, B, and C circles should intersect at one point. That is where F is located. If you drew the distance-circle for the last point (D), it should also intersect at point F.
Mathematically, you could write the equation for a circle with the known distance from three points, then solve the equations simultaneously (read: algebra workout).
Amateur radio operators (HAM's) use a similar method in fox-hunt competitions. In that competition, you use a receiver and antenna to locate a hidden transmitter. By determining the signal power and approximate direction from several sample points, the operator can do triangulation and determine where the transmitter is located (power is related to distance from the source, and the antennas will give you a reasonably narrow region (say 10-20 degree arc).
GPS uses a similar technique, expect that it needs four reference points because each reference point creates a 3D sphere w/ constant radius. 2 satellites give you the intersection of two spheres, which is an ellipse. 3 satellites will give you two points, and 4 satellites gives you the single common point. More satellite signals just help improve the accuracy because of propagation delay and other signal-degrading effects caused by the atmosphere.
From what I see, you can make a very simple linear system out of the distances. Here is how:
Let us consider four variables:
a = distance of point F from point A
b = distance of point F from point B
c = distance of point F from point C
d = distance of point F from point D
Now we want to develop a relationship between these distances. I shall assume that you can find the values for a, b, c, and d from the four diagrams given. Simply solve the system as simultaneous equations.
Consider the following general equation:
k1a + k2b + k2c + k4d = 0 ; where k1, k2, k3, and k4 are constants of which the values we want to find out in order to develop a linear relationship between the distances. One by one, enter the values of a, b, c, and d from the four diagrams into these equations and you shall have four equations and four unknown constants which you can solve by method of elimination, matrices, or substitution - whichever you prefer.
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By the "value of point F," I have assumed you simply mean the distance of point F from other points. After developing a linear relationship between a, b, c, and d, you will be able to input values of any 3 distances and compute the fourth one.
Tnx a lot albbg for your huge effort for helping me and what you have done is fully clear but you forget that i need the magnitude of pint F not distance.
please help. thank you a lot
a --> (ax,ay)
b --> (bx,by)
c --> (bx,cy)
d --> (bx,dy)
F --> (Fx,Fy)
ax^2-bx^2-2(ax-bx)Fx+ay^2-by^2-2(ay-by)Fy=da^2-db^2
cx^2-dx^2-2(cx-dx)Fx+cy^2-dy^2-2(cy-dy)Fy=dc^2-dd^2
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