Measurements (LDU) of Lego tracks Set 151 I have an old set of 7851 curved tracks.  I want to use the outer tracks (the ones with the bumpy surface).  If I was to put one end down on a baseboard so that is tangent/parallel with the grid of studs, how many LDU in the X and Y direction would the other end be?

I'd also like to the know the X, Y cartesian coordinates (in LDU) of the middle 2 slots as well.
Well, obviously one way to measure that would be empirically - you have the tracks, you have the baseplates...

As for the maths, let's see... a full circle of these tracks has a diameter of 5 straight tracks as measured from the centre of the sleepers, that translates to 80 studs. The radius is 40 studs, or 800 ldu.

The outer, ridged track, is 2 to 3 studs further depending on which border you consider (and assuming you want borders instead of stud positions). So the radii for the border of that piece of track are 840 and 860.

So, assuming the track circle is centered around (0,0), and using the usual way of considering circles in math (counterclockwise from (1,0)), your first piece of outer track starts at (820,0) and (830,0).

There are 16 segments of track to a full circle, so the angle for one segment is 22,5°, or pi/8. So the end of your first track part is at (840*cos(pi/8),840*sin(pi/8)) to (860*cos(pi/8),860*sin(pi/8)), or (776.06,321.45) to (794.54,329.11)

Expanding further, you get the following segments for the short border of the outer track:


(840.00,0.00)-(860.00,0.00)
(776.06,321.45)-(794.54,329.11)
(593.97,593.97)-(608.11,608.11)
(321.45,776.06)-(329.11,794.54)
(0.00,840.00)-(0.00,860.00)
(-321.45,776.06)-(-329.11,794.54)
(-593.97,593.97)-(-608.11,608.11)
(-776.06,321.45)-(-794.54,329.11)
(-840.00,0.00)-(-860.00,0.00)
(-776.06,-321.45)-(-794.54,-329.11)
(-593.97,-593.97)-(-608.11,-608.11)
(-321.45,-776.06)-(-329.11,-794.54)
(0.00,-840.00)-(0.00,-860.00)
(321.45,-776.06)-(329.11,-794.54)
(593.97,-593.97)-(608.11,-608.11)
(776.06,-321.45)-(794.54,-329.11)


Hopefully that makes sense. Watch out, these are the coordinates of the borders; for the coordinates of the studs (that is, the center of the studs, which I think is more interesting, but I don't know what your goal is) we'll have to use a radius of 850, and move 10 ldu along the tangent to find the stud. So to find the stud after the segment, we'll do (850*cos(pi/8)-10*sin(pi/8),850*sin(pi/8)+10*cos(pi/8)). For each segment N starting from 1, the studs are as follow:


start stud: (850*cos((N-1)*pi/8)-10*sin((N-1)*pi/8) , 850*sin((N-1)*pi/8)+10*cos((N-1)*pi/8))
end stud: (850*cos(N*pi/8)+10*sin(N*pi/8) , 850*sin(N*pi/8)-10*cos(N*pi/8))


End the two studs in the center of the track are basically and end stud if the angle was pi/16, and the start stud of the next track, or:


first middle stud: (850*cos((N-1)*pi/8+pi/16)+10*sin((N-1)*pi/8+pi/16) , 850*sin((N-1)*pi/8+pi/16)-10*cos((N-1)*pi/8+pi/16))
second middle stud: (850*cos((N-1)*pi/8+pi/16)-10*sin((N-1)*pi/8+pi/16) , 850*sin((N-1)*pi/8+pi/16)+10*cos((N-1)*pi/8+pi/16))


The four studs of the first part of track are thus at coordinates:


start stud: (850.00,10.00)
first middle stud: (831.72,175.63)
second middle stud: (835.62,156.02)
end stud: (789.12,316.04)