# Geometry problem - calculating curve coordinates from versines

1. Oct 22, 2013

### banksie

Hi,

I was wondering if anyone can help me. I don’t have a homework problem, but a problem I have encountered at work. I am a mechanical engineer working in the railway industry and I am struggling with a problem of reconstructing the vertical geometry of a rail in terms of height and distance along the rail.

The rail is measured using a chord measuring system, with versines taken at two points along the chord:- a point halfway along the chord and a point between the mid-point and the end. The measuring chord is moved along the rail at equally spaced intervals and the versine pairs measured at each point. The chord length is an integer multiple of the measuring interval.

Currently the process used to obtain the original rail position from the versine information uses the centre versine only. However this means that wavelengths equal to half the chord length cannot be reconstructed, hence the requirement to somehow incorporate the information encoded in the second versine measurement. The current method uses a simple geometric relationship to calculate the height of the rail at one end of the chord, given initial starting values of zero (i.e. the other end of the chord is at zero). I don't see how I can use this method and incorporate the second versine though, and i am struggling on how to approach this problem. Can anyone help point me in the right direction?!

I hope that I have defined the problem clearly enough, if not please do ask if there is any more information I can provide.

Many thanks

2. Oct 22, 2013

### Staff: Mentor

Welcome to PF!

Can you draw a picture of what you're describing and what you want to compute?

http://en.wikipedia.org/wiki/Versine

3. Oct 22, 2013

### banksie

OK,

So the first picture hopefully describes what's going on. We have a profile P on which we lay our chord AB at two points, we then measure the offset from the chord to the profile at two points C and D. These offsets are termed the versines, which I've labelled u and v. This process is repeated along the profile - the chord is advanced along the profile by some distance and the versines are measured again. The second picture tries to illustrate this for the first three points. Note that this process would continue along the profile beyond the point where A is at B

So, we know the chord length AB, the sub chord lengths, AC, CD and DB, and the versines u and v. Given a number of these versine measurements collected along the length of the profile, how can we reconstruct the profile P from this information? In particular we need to utilize both u and v, as I mentioned in my first post...

Many thanks

Banksie

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4. Oct 22, 2013

### Staff: Mentor

I dont have an answer as it seems that there's not enough info yet.

You mentioned that the profile is advanced some distance each time. Do you know this distance?

Also shouldn't the versines be perpendicular to the chord?

Lastly, do you know the angle of the profile at each measurement?

Last edited: Oct 22, 2013
5. Oct 22, 2013

### banksie

yes you're right the versines are measured perpendicular to the chord. I sketched that up in Word and just couldn't get the lines in the right place!

The chord is advanced along the profile at regular intervals. Lets say that the interval is 1m and the chord length is 20m, the distance AC is 7m and distance AD is 10m. The angle is unknown.

You would assume that the profile starts at zero and the remainder of the profile would be relative to that.

6. Oct 22, 2013

### banksie

Also, as we are talking about railway track geometry, the versine measurements are small compared to the radius of the curve so I think we can neglect the angle of the versine measurements relative to the chord.

7. Oct 23, 2013