Mike asked an excellent question in the last post regarding how radii are laid out in the field. In the model world, we have the advantage of slipping some sectional track together, and wherever it lays, it lays, or for those using flex or hand laying, we draw a couple of straight lines representing where we want the track to go and we swing an arc using a pencil on a string secured by a nail.
A route preliminary survey doesn’t start much different than the latter. A couple of disclaimers first; I do not work for a railroad nor have I surveyed a route. The knowledge conveyed next is one of a few ways of inserting a radius in the field per my 1907 ICS Reference Library.

The Survey team will carry with them, among other things, an engineer’s transit, a 100 foot tape or chain, two transit poles, lots of stakes and a drafting board, scales, triangles, paper, pencils, radius protractors in the “office equipment.”

Let’s start with an example straight route, arriving from the left, as shown. The numbers represent staked stations, by 100, such that Sta. 246 is 24,600 feet from the origin of the route. Due to some obstacle, the route needs to diverge from its path at Sta. 248 + 25 (24,825 ft). Moving he transit to the last station, the Surveyor swings his transit about and notes an acceptable divergent route, Δ = 30° to the right of the previous path. Staking continues.

Diverging Routes with Stations

At the end of the day, the Surveyor conveys his notes to the draftsman, who translates the measurements onto a topographical map. At the divergence, the draftsman uses one of the radius protractors to select a suitable radius to be inserted.

Example Radius Protractor

In this example, I have chosen to include a 6° curve, which equates to a radius of 955 feet, therefore the distance T is 256 feet. As mentioned in a

previous post, the radius R can be found through 5730 / D, where D is the degree of curvature.

Using mathematical tables or geometry, the Point of Curvature (PC) and Point of Tangency (PT) are determined with respect to Sta 248 + 25, also known as the Point of Intersection (PI). For our reference, the equation to determine the distance T from PI to PC or PT for a given divergent angle (Δ) is R tan (Δ / 2), were the curve radius R is in feet and the angle Δ is in degrees.

Points of Intersection, Curvature and Tangency

When the Survey team returns to lay the final right of way, they will “backup” from the PI 256 feet, or advance from the prior station stake if they feel like doing a little more math that day (these distances shown as reference). The transit will be placed on the PC and swung in the direction of curvature ½ the selected degree of curvature, 3° in our example. A stake will be placed 100 feet out, representing the first chord.

Establishing the Radius Chords

With a pole at the PC, the transit is moved to the new stake placed 100 feet and 3° out. Using the preceding pole and the new stake, the transit is used to sight placement of the next stake, now a full degree of curvature (6° here) from the last chord. The remained of the chords will all be placed 100 feet and 6° divergent from the prior chord until it aligns with the PT on the divergent route.

Finally, with all stakes in place, the roadbed and rails can be centered on the stakes, thereby laying tracks along the desired routes with selected curves.

Final Route, Highlighted

Thanks for the request Mike, and I hope this answers your question.