Here is a crude hand-drawn image showing what Paul is saying. Note that at the apex of the curve, the radius is matched between the perfectly circular curve and the eased curve. In fact, in this example, it is quite extensive...the range over which they share the same radius at the top of the curve. However, as the eased curve begins to diverge slightly, showing how its radius is changing to a wider one, you can see that it has still not reached the 90 degree point of tangency until past the point, on each side, where the fixed radius curve has turned precisely 90 degrees.
Not all curves are 90 degree curves, but you get the idea how an eased curve must be slightly/somewhat longer than a curve of fixed radius when they otherwise are tangent in parallel lines. The two red lines make that clear...as crude as this depiction is.
Crandell
one more drawing:
Paul
I never tried to calculate easement on curve,I find it out of purpose for or small trains, but I must admit, I didn't like mathemetics.
I use a very simple and so easy method, the yard stick design track.
See more info's about it and pictures on www.aorailroad.com, the layout of mister David Stewart.
He use this very easy and effective method.
The yarstick is a fine piece of wood, whith a lot of flexibility.
A few points traced whith a pencil, as check point, put in alignement and keep in place whith a few nuts and his flexibility allow gentle curves and the easement comes naturaly.
The flowing of the track is amazing; I use it in Nscale since the expansion of my layout whith great results.
Marc from Belgium
Marc, we know it as "the bent stick method". A meter length of flex track, a length of hardboard or Masonite, or doorskin, or even a long stainless steel ruler on edge, will accomplish the same sort of curve with minor variations.