If you go directly from a straight piece of track to a circular curve you get quite an abrupt change of direction. This doesn't look very good and, in real life, would be uncomfortable for the passengers.

For the technical details about this, and also "cant" (the angle of the track to the horizontal) I suggest you read the article on Model Railways Online.

One thing I will say at this point though is that whether transitions are used and how they are designed is entirely up to the individual modeller. If you don't want them, don't have them! If you want to draw something on the back of the proverbial fag packet that looks OK then do so. For me the cubic parabola works well as a solution.

Having dredged back in my memory from my maths courses at University and recollections from watching the Open University programmes late at night I recalled some information on these. In the early days a smooth curve was created by putting pegs for the known points into a wallboard and placing thin wooden strips (or splines) onto them. These were then weighted until they just touched all the pegs, giving a smooth line which could be then copied onto the drawing.

Later calculations and computers took over this function and various methods were then developed to replicate these smooth curves for both lines and indeed for solid objects (I remember installing a 3D modelling program called SWANS - Surfaces With A Nice Shape onto a MicroVax when I worked at a local college in the 1980's).

One of the main advantages of the cubic spline is that not only is the speed of the object going round the curve smooth but so is its acceleration. This is a desirable trait for moving vehicles. This doesn't quite work as applied to the railway curve transition as there is an asymptotic change in acceleration from the straight to the curve so other methods are used for high speed lines, but it's certainly good enough for my model!

Based on the information on MROL for the cubic parabola I developed a simple spreadsheet to calculate the X & Y coordinates of the points along the transition curve. I then plotted these on screen in CorelDraw and linked them with a smoothed curve. I then drew in the circular curve and joined the two together.

The same method would work with any line-drawing package, with a pencil and a sheet of graph paper, or indeed you could draw the points directly onto a baseboard and use them as a guide for the track centres (the track itself would then smooth the curve).

Based on the available information I set a transition length for a scale 20m, and calculated a suitable radius for the curve using a bit of trial and error. I'm aiming to keep the track at least 25mm from the edges of the baseboard, and need room for the halt at the top, so this defined my maximum overall radius.

The spreadsheet for the calculations is here...

This is set up for N gauge and a 20m prototype transition length, but you could change the values as required for other scales or lengths.