Model Railroad Curve Easement Calculator
Enter your centerline radius, easement length, and total turn to plot a cubic-spiral transition from tangent track into a circular curve. The results give dimensions you can transfer to roadbed, plywood, or a full-size paper template.
Total centerline track length (in)49.7
- Curve arrangement
- Two easements with a constant-radius circular arc
- Easement end angle (degrees)
- 14.32
- Circular arc angle (degrees)
- 61.35
- Curve shift (in)
- 0.25
- Tangent distance to curve intersection (in)
- 30.24
- Circular arc length (in)
- 25.7
- Easement endpoint coordinates (in)
- x = 11.9252, y = 0.9955
Entry-easement stakeout coordinates (in)
| Distance s (in) | Along tangent x (in) | Offset y (in) | Tangent angle (degrees) | Instantaneous radius (in) |
|---|---|---|---|---|
| 0.0000 | 0.0000 | 0.0000 | 0.0000 | Infinity |
| 2.0000 | 2.0000 | 0.0046 | 0.3979 | 144.0000 |
| 4.0000 | 3.9997 | 0.0370 | 1.5915 | 72.0000 |
| 6.0000 | 5.9977 | 0.1250 | 3.5810 | 48.0000 |
| 8.0000 | 7.9901 | 0.2960 | 6.3662 | 36.0000 |
| 10.0000 | 9.9699 | 0.5775 | 9.9472 | 28.8000 |
| 12.0000 | 11.9252 | 0.9955 | 14.3239 | 24.0000 |
The exit easement is the corresponding reversed geometry from the other tangent.
Coordinates are track-centerline offsets measured from the tangent-to-spiral point; the first row has an infinite instantaneous radius because it is straight track.
How to use this calculator
- Choose inches or millimeters and use that unit for every length you enter and mark out.
- Enter the centerline radius of the constant-radius curve and the length wanted for each transition.
- Enter the total direction change between the two straight tangent tracks.
- Choose the number of equal stations wanted in the entry-easement stakeout table.
- Mark the table coordinates from the tangent-to-spiral point, then mirror the geometry at the exit.
How the curve easement is calculated
An easement is transition track whose curvature changes gradually from zero on the tangent to the selected radius at the circular curve. This calculator uses the cubic spiral described in the NMRA data sheet. It reports centerline geometry, so the same calculation works in inches, millimeters, or scale dimensions as long as every length uses one unit.
Let R be the circular radius, L one easement length, and Delta the total turn in radians. The spiral parameter is A² = R x L. At a distance s from the tangent end, the tangent angle and instantaneous radius are phi(s) = s² / (2A²) and R(s) = A² / s. The coordinate table evaluates the clothoid series for x(s) and y(s); at s = 0, the radius is infinite because the track is straight.
At the spiral-to-circle point, the easement has turned theta = L / (2R). The remaining circular angle is Delta_c = Delta - 2theta. Its length is R x Delta_c, so total centerline length is 2L + R x Delta_c. The calculator also finds the curve shift p = yL - R(1 - cos(theta)) and tangent distance T = (R + p)tan(Delta / 2) + k, where k = xL - R sin(theta).
Using the stakeout table
Start at the tangent-to-spiral point and establish the straight tangent as the x-axis. For each station, measure the listed x distance along that tangent and the y offset inward toward the curve, then fair flex track through the marks. The endpoint coordinates locate the join to the constant-radius portion. The exit transition uses the same coordinates in reverse, measured from the outgoing tangent.
Easement length and radius move the layout most. A longer easement turns more before it reaches the selected radius, so it reduces the circular-arc angle and needs more room. If the two easements consume all of the turn, they meet with no constant-radius section. If they consume more than the entered turn, the plan overlaps and needs a shorter transition, a larger radius, or a larger total angle.
What this calculation leaves out
This is horizontal track-centerline geometry, not a clearance or operating guarantee. It does not calculate parallel-track spacing, car overhang, superelevation, grades, vertical easements, turnout geometry, or a locomotive's ability to handle the chosen radius. For grade and clearance inside a multi-level route, use the model railroad helix design calculator separately. Flex track can spring away from plotted marks, so test the finished alignment with the longest and most restrictive equipment before fastening it permanently.
Worked example
For a 24 in radius, two 12 in easements, and a 90 degree turn, the end angle of each easement is 12 / (2 x 24) = 0.25 radians, or 14.32 degrees. The remaining circular arc is 61.35 degrees and its length is 25.70 in. Total centerline length is therefore 24 + 25.70 = 49.70 in.
The spiral endpoint is 11.9252 in along the starting tangent and 0.9955 in inward. Its curve shift is 0.2494 in, and the tangent-to-intersection distance is 30.2370 in. Mark the six equal stations in the table, fair the first easement through them, lay the 25.70 in circular section, and use the reflected station marks for the exit.
Common questions
What is a spiral easement on a model railroad?
A spiral easement is transition track between tangent track and a constant-radius curve. Its curvature starts at zero and increases gradually until it matches the circular curve, avoiding a sudden kink at the joint. The cubic spiral is a common railroad transition shape because its curvature changes smoothly with distance.
How long should a model railroad easement be?
The answer depends on the available space, the radius, the equipment, and the amount of turn available. A longer easement is gentler but consumes more of the total angle and benchwork length. Use the table to compare alternatives, then test the plan with the longest cars and body-mounted couplers you expect to run.
Does adding easements change the effective curve radius?
The entered radius remains the radius of the constant-radius middle section. The transition itself has a changing radius, from infinite at the tangent to the entered radius at the join. Adding easements changes the total path, curve shift, and the amount of constant-radius arc left in the same overall turn.
How do I mark easement coordinates on plywood or foam?
Draw or snap the incoming tangent, then measure each x coordinate along it from the start point. Square inward from that line by the matching y offset and mark the station. A flexible batten or the flex track itself can be faired through the marks before the roadbed is cut or the track is fastened.
Can I use this calculator with sectional track?
Sectional track cannot normally follow an exact cubic spiral because it is made in fixed-radius pieces. Flex track is the direct way to use the coordinates. With sectional track, approximate the transition with progressively broader radii between the tangent and the final curve, then check for a smooth, kink-free alignment.
What should I do when two easements leave no circular arc?
Two meeting easements are valid when their end angles exactly use the total turn, but there is no constant-radius middle section. If the tool reports overlap instead, shorten each easement, increase the radius, or allow a larger turn. Avoid forcing the track through an overlapping layout because it cannot join tangent-to-tangent as entered.