Possible to get Lat/Lon coordinates of triangle vertex point?

Question

I have two known points (A and B) on latitude / longitude coordinates. Certainly I know the distance between A and B in meters.

I have third point C with unknown coordinates, but I know the distance between AC and BC in meters.

Is it possible to get the lat/lon coordinates of point C with math formula?

---

Here is an example:

The all known distances between coordinate points are:

AB = 120.9234 meters AC = 226.7206 meters BC = 213.1376 meters

The known coordinates of A and B point:

A lat = 47.49784; A lon = 19.08939;

B lat = 47.49893; B lon = 19.08941;

and I calculated all angles of triangle in deg: B: 80.341932306607 A: 31.722055823679 C: 67.936011869714

and I measured the result what I looking for in map: Lat: 47.498575374822 Lon: 19.09129858017

but I like to use a math formula to get these two coordinates of point C using the upper input values.

Answer 1

Assuming you know AB, AC, and BC, then the answer is yes. It requires trigonometry. The following assumes a flat plane and that points A anb B are not on the same line of latitude or longitude. To get the lat/long of C you need to figure out the ΔX and ΔY from a known point (let's use A).

You're going to need the value of the angle θ, which is 90-a-λ

Let ΔXB and ΔYB be the lat/long difference from A to B (both of which have known coordinates). From basic trigonometry we know that sin(λ) = ΔYB/AB. Therefore λ = arcsin(ΔYB/AB).

By the Law of Cosines cos(a) = (AB²+AC²-BC²)/(2*AB*AC), which means a = arccos((AB²+AC²-BC²)/(2×AB×AC))

Therefore θ = 90 - arccos((AB²+AC²-BC²)/(2×AB×AC)) - arcsin(ΔYB/AB)

With θ calculated, we're back to basic trigonometry again: sin(θ)= ΔX/AC and cos(θ) = ΔY/AC. Therefore ΔX = sin(θ)×AC and ΔY = cos(θ)×AC. Just add those values to the coordinates of A, and you have your answer.

TLDR version:

• C(x) = A(x) + AC × sin(90 - arccos((AB²+AC²-BC²)/(2×AB×AC)) - arcsin((A(y)-B(y))/AB))
• C(y) = A(y) + AC × cos(90 - arccos((AB²+AC²-BC²)/(2×AB×AC)) - arcsin((A(y)-B(y))/AB))

Answer 2

I trying the formula, and the result is close to the answer, but something wrong.

I have coordinates of A and B points:

        $Ax = 47.497846896196;
        $Ay = 19.089394211769;

        $Bx = 47.498934174013;
        $By = 19.089415669441;

x is latitude, y is longitude.

The distance between A and B is

        120.9047 meters

This is counted by simple formula. If I use MySQL GLength(), I got another values, but I think the trigonometry formula is good too.

Counting:

        $DxAB = ($Ax - $Bx); = -0.0010872778169997
        $DyAB = ($Ay - $By); = -2.1457671998348E-5

        $Lambda = asin($DyAB / $AB); = -1.774759128334E-7

        $aAngle = acos((pow($AB, 2) + pow($AC, 2) - pow($BC, 2)) / (2 * ($AC * $AB))); = 1.0422581825838

        $Theta = deg2rad(90 - rad2deg($aAngle) - rad2deg($Lambda)); = 0.52853832168699

        $DX = asin($Theta); = 0.55687780990758
        $DY = acos($Theta); = 1.0139185168873

        $Cx = $DX + $Ax; = 48.054724706104
        $Cy = $DY + $Ay; = 20.103312728656

but the result we measured is:

        47.498575374822 and 19.09129858017

While I have 2 different list of distances between points (counted by trigonometry and by database GLength()) the output of formulas are not exactly same as we measured:

Distances between input coords by trigonometry:

        $AB = 120.9047;
        $AC = 164.3976;
        $BC = 146.9635;

distances between (same) coords by database handler:

        $AB = 120.9234;
        $AC = 226.7206;
        $BC = 213.1376;

I understand why we can got different distances.

The result of this formula if using database handler's distances as input:

        $Cx = $DX + $Ax; = 47.893151605351
        $Cy = $DY + $Ay; = 20.264885829409

This result is closer than the previous one, but not valid. Maybe something wrong in my side?

Answer 3

I cancelled latitude and longitude coordinates as inputs, and I would like to solve a simple example with any unit.

I have all inputs on the image, but no Cx and Cy. What is the best method to get these data as accurate result? I tried to implement formulas to programming language, but not always returned acceptable result. I have non 90 degree, non equilateral triangles only.

thanks!