Finding out a third point using two points from rotated planes.

[faceguydb]

I am not sure what genre of math this would fall under so if it is in the wrong section I apologize in advance.

The problem I have is I need a formula or algorithm to take two points that represent the exact same two points in reality but are two points on different planes. I will have access to gps data, I am hoping to take 2 gps positions, put in what their coordinates are according to a map that is rotated and then beable to put in coordinates and turn them into gps data. Also I would need to take gps coordinates and turn them into the rotated map's coordinates.

The story is I need to beable to find very small boxes in a very big plant and I am given coordinates to those boxes that is from a plane that is rotated compared to true north because the plant is built on an angle. The hope is I could go to some boxes I already found and know the plant's coordinates for and make a point that links both the gps and the plant to beable to make a relative comparision to find unknown points on one of the planes.

Any help would be awesome, even just a push in the right direction.

 

[DrPhil]

I am not sure what genre of math this would fall under so if it is in the wrong section I apologize in advance.

The problem I have is I need a formula or algorithm to take two points that represent the exact same two points in reality but are two points on different planes. I will have access to gps data, I am hoping to take 2 gps positions, put in what their coordinates are according to a map that is rotated and then beable to put in coordinates and turn them into gps data. Also I would need to take gps coordinates and turn them into the rotated map's coordinates.

The story is I need to beable to find very small boxes in a very big plant and I am given coordinates to those boxes that is from a plane that is rotated compared to true north because the plant is built on an angle. The hope is I could go to some boxes I already found and know the plant's coordinates for and make a point that links both the gps and the plant to beable to make a relative comparision to find unknown points on one of the planes.

Any help would be awesome, even just a push in the right direction.

I hope this is easier than you made it sound!

Start by identifying the point in the plant that you want to use as reference .. your office, the delivery dock, whatever. Get a good solid gps reading at that point. Presumably you can define an x-y coordinate system with that point as the origin, that lines up the the plant's shape. It would be nice if you knew the distance you are from the center of the earth, but you can calibrate for that. Each micro-° of latitude (the true N-S direction) is 0.365 feet, or reading 5 places to the right of the decimal would give units of 3.65 feet. For the longitudinal (true E-W), the linear distance decreases as you go north (or south) from the equator, so a micro-° is (0.365 ft)/cos(latitude). Again, reading to the 5th decimal place gives reasonable units. Now you can take the gps of the box you are looking for, subtract the gps of the origin, and convert to (x,y) in feet.

Let \(\displaystyle \theta\) be the angle the plant makes with true north, measured CCW. Either read that off the plant site drawing, or guess and calibrate. To rotate the "true" coordinates into "plant" coordinates,

\(\displaystyle x_{plant} = \;\;x_{true}\ \cos\theta + y_{true}\ \sin\theta\)

\(\displaystyle y_{plant} = -x_{true}\ \sin\theta + y_{true}\ \cos\theta \)

Two numbers to calibrate .. (1) the conversion of angle measure to feet, and (2) the angle of rotation. Sounds easy enough.