# Determining 3D position after time

I'm trying to figure out how to calculate an object's angle to a fixed point after a certain amount of time, given the following information:

1: The origin 0,0,0 is your stationary point of view (the aforementioned fixed point)
2: The distance and bearings to the object from the origin are known
3: The bearings are given in one horizontal angle and one vertical angle
4: The speed and heading of the object are known and constant

I will use the following values in an example:

1: Object speed: 100 m/s
2: Object starting distance from origin: 2000 m
3: Object bearing from origin: 1°00′00″ horizontal, 2°00′00″ vertical
4: Object heading: 130°00′00″ horizontal, 5°00′00″ vertical
5: Position to calculate after t = 60 s

I'm guessing the solution would involve the following method:

1: Determine the starting coordinates of the object, based on the horizontal and vertical bearings, and the distance.
2: Use the object's starting coordinates, its heading, and its speed multiplied by t = 60 to determine its coordinates after the specified time.
3: Determine the angle from the origin to the coordinates at t = 60.

It looks like there is a lot of trigonometry involved in this. Perhaps this is easier to solve than it looks, but to me, it appears to have a lot of steps involved in it. Or perhaps there is a better method to solving this? Some formula I'm unaware of?

• I'm not sure what any of that means. As for what I tried to do, I thought I could use the distance formula and solve for the variables, as I knew the distance and the origin coordinates, but I quickly realised that I'd be solving for three unknowns (the x, y and z of the starting coordinates). – Hiigaran Jun 6 '17 at 3:43
• The bearing from origin to the start? That's the known information listed in point three of the example. Or did you mean what I did to try and solve for the bearing from origin to the t=60 coordinates? If the latter, I didn't make it that far. – Hiigaran Jun 6 '17 at 3:51
• Ok i see the edit. This is trivial then.. Just apply the line equation and you are done. – Brethlosze Jun 6 '17 at 3:54

You list an object bearing from the origin, presumably that is at the start of your time period. That gives you an $(x,y,z)$ based on the distance which we can call $d$. Then add the vector that is the object motion in $60$ seconds. That will give a final location that depends on $d$. You still don't have any information that allows you to measure $d$. It started somewhere on a ray from the origin and is now on a different ray, but you can't find the point along the ray. Once you find another piece of data to determine $d$ you will have a unique solution, which will involve a bunch of trig to change angles into distances. Why are you opposed to the trig? That just generates constants to multiply by.
• Just do everything in Cartesian coordinates. Where do you measure the bearings from? If from the $x$ axis, the starting position is $(2000 \cos 1^\circ \cos 2^\circ, 2000 \sin 1^\circ \cos 2^\circ, 2000 \sin 2^\circ)$ Compute the displacement the same way. Add that to the starting position. You will need some $\arccos$ and $\arcsin$ functions to get the angles at the end. – Ross Millikan Jun 6 '17 at 4:37
• You are probably putting degrees into the spreadsheet but it expects radians. The first set looks right since $1^\circ \approx \frac 1{57} \approx \sin 1^\circ$ – Ross Millikan Jun 6 '17 at 5:08
• You compute the $x,y,z$ displacements from the motion. The total displacement is $60\cdot 100=6000$ and you apply the same trig functions to get the components. Add that to the starting location and you have the ending location. – Ross Millikan Jun 6 '17 at 14:05