Figure-1 illustrates turnouts having the same frog number with different lead-lengths. The non-diverging straight rails are shown in gray, the frog in black, the closure rail in blue and a straight rail between the closure rail and frog in red. The closure rail radius, lead-length and frog number are indicated from left to right.
The lead-length decreases along with the closure rail radius. Correspondingly, the straight rail connecting the closure rail and frog increases in length.
Turnouts with shorter lead-lengths are better suited for modeling in limited spaces. But the benefit of a larger turnout is compromised by having a smaller closure rail radius.
The primary restriction on the closure rail is that it ends at the same angle as the diverging rail of the frog, the frog-angle. This angle is the same as the angle between the green and orange lines in Figure-2 when the orange line is perpendicular to the closure rail and is the radius of the closure rail.
The blue line also intersects the end point of the closure rail and therefore indicates the vertical distance of the closure rail, the distance between it and the leftmost point of the closure rail. This is further illustrated by the length of the green line, the difference between it and the closure rail radius.
The cosine of the frog-angle is the ratio of the lengths of the green and orange lines. The sine of the frog-angle is the ratio of the lengths of the blue and orange lines.
Figure 3 - Prototypical Turnout
Given only the frog number and frog angle and for the case where the curved section of the closure rail ends at the frog
radius = track-gauge / (1 - cos(frog-angle))
For the same case, the horizontal length of the closure rail will also be the lead-length,
lead-length = sin(frog-angle) * radius
These equations can be modified to handle the case when the closure rail ends before the frog, as shown in Figure-1. The lead length limits the maximum radius of a constant radius closure rail.
Figure-3 illustrates HO scale turnout dimensions using the above equations. The lead-length increases proportionally with the frog number. The radii listed are for the closure rail. The track radii (both rails) are these values less half the track gauge (0.33").
The Catskill Archive - Frogs and Switches web page, describes full-scale turnout dimensions and equations for determining them. The curved section of the closure rail in the full-scale turnouts are constant radius that end at the frog. Both the equations above and the turnout dimensions in Figure-3 are consistent with those listed for full-scale turnouts.
Figure 4 - NMRA RP Dimensions
Turnouts built with the RP lead-lengths with constant radius closure rails result in smaller radii than described above and straight sections of various lenghts as illustrated in Figure-1.
Lead-length (in.) Difference Proto RP 4 5.28 5.06 0.22 4.2% 5 6.60 5.69 0.91 13.9% 6 7.93 6.25 1.68 21.1% 7 9.25 8.44 0.81 8.7% 8 10.57 9.00 1.57 14.8% 9 11.89 9.56 2.33 19.6% 10 13.21 10.06 3.15 23.8%
Commercial turnout lead-lengths are similarly compared in the following table. Having a slightly shortened lead length using a constant radius closure rail will result is a short length of straight rail between the end of the curved section of the closure rail and frog which will align the wheels before crossing the gap at the frog, reducing the chance of a derailment.
Lead-length (in.) Difference Description Proto Measured 4 5.28 4.88 0.41 7.7% Shinohara #4 6 7.93 7.31 0.62 7.8% Atlas Custom Line #6