Some basic understanding of the track geometry is essential if you want to build large layouts that join together properly. It's pretty frustating if you're putting together a masterpiece only to find that it doesn't match up when you get to fit the last piece.
The fundamental track piece is the straight, which is 8 studs wide by 16 studs long.
The curved piece of track is approximately the same length as the straight but its real relationship is the fact the the diameter of a circle made with the curves is exactly the same as 5 straights.
This relationship is more important than it may seem at first because it allows you to build layouts where there are curves on one side and straights and the other. For example the Bell Shape Loop relies on this fact. You can think of this layout as an oval with a bulge, where the bulge consists of two quarter circles and one semicircle. Therefore 10 straights are required on the opposite side to match the bulge (2 x 2.5 + 5 = 10).
Since curves are 1/16th of a circle they cover an arc of 22.5°, and any combination of LEGO® track you can think of will always result in angles that are multiples of 22.5°, e.g. 45°, 67.5°, 90° etc.
The following table shows the exact dimensions of curved and straight tracks at the various angles. The units are in studs and the dimensions are the x,y offsets relative to the centreline of the track.
| Angle | 0° | 22.5° | 45° | 67.5° |
|---|---|---|---|---|
| Straight | 16.000, 0.000 | 14.782, 6.123 | 11.314, 11.314 | 6.123, 14.782 |
| Curve | 15.307, 3.045 | 12.977, 8.671 | 8.671, 12.977 | 3.045, 15.307 |
The points are designed so that branch track when followed by a curve is offset by exactly one straight, as shown above. This fact is demonstrated in Oval with Passing Loop, each point offsets the outer track by one straight so there are two straights inserted in the semicircular siding.
The gap between the two tracks is exactly 8 studs.
The straight part of points is exactly equivalent to 2 straights, i.e. 32 studs. The branch is offset by 32.693, 12.955 studs (x,y relative to the centrelines) and at an angle of 22.5°.
By crossover, I mean a track that joins two parallel tracks and allows a train to cross from one to the other. It would be nice if you could take two like-handed points and just connect them branch to branch, and even though you can do it, you are left with a virtually useless setup. The geometry just doesn't work out.
To make the geometry work you need to use the curves supplied with points to bring the middle of the crossover parallel to the main tracks. See Euro Two Track for an example of this setup.
For a slightly more compact solution there is an alternative setup that works fairly well. This is not mathematically exact but it is so close that nobody would ever know!
Using the figures from above, the dimensions are 80.168, 32.033, and we are trying to approximate 80.000, 32.000 so the error is only a tiny fraction of a stud. Twinslip is an example of this geometry.