Model railroad vertical transition calculator.

How the transition is calculated

Where one grade meets another there is a corner, and a long car or locomotive can lift a coupler or bridge the dip. A vertical transition removes the corner by changing the grade gradually. This calculator uses the standard full-size definition of a vertical curve: the grade changes at a constant rate along the curve. Caltrans' Bridge Design Details 2.3, Rate of Change of Vertical Curves (June 2025) writes it as R/C = (G2 − G1) / L, where G1 and G2 are the grades at the start and end, in percent, and L is the length of the curve. Caltrans measures L in stations, a full-size surveying unit; this page uses inches or millimetres.

A constant rate of change makes the track a parabola. The height at distance x from the start is G1/100 × x + (G2 − G1)/100 × x² / (2L). That follows directly from the constant rate, so the rate of change at every point is the one the first formula gives. The middle of the curve sits |G2 − G1|/100 × L / 8 away from the corner where the two grade lines would otherwise meet. The page also runs the formula backward: given the rate you want, the length is |G2 − G1| divided by the rate.

What the page does not do is tell you which rate is gentle enough. The NMRA's index lists no standard or recommended practice on vertical curves or grades, and the Caltrans sheet defines the rate without giving a model railroad limit. Pick your length by testing the longest locomotive or car you run, and use the table to cut the roadbed.

Worked examples with real numbers

• Level into 3% over 20 inches. The rate of change is 3 / 20 = 0.15% per inch. Heights at 5, 10, 15 and 20 inches are 0.019, 0.075, 0.169 and 0.3 inch. At 10 inches the curve is 0.075 inch below the sharp corner.
• The length for a rate you choose. To change from level to 3% at no more than 0.10% per inch, the curve must be at least 3 / 0.10 = 30 inches long.
• A crest. From +3% to level over 20 inches the rate is −0.15% per inch, and the table shows the same offsets mirrored.

What this calculator leaves out

It does not know your equipment or its couplers. It assumes the transition is a single parabola between two straight grades. It ignores the thickness of the roadbed and the stiffness of the track, so check by running real cars over a trial section.