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Degrees of Curve

Modelers use radius when describing curves, but the real world uses Degrees of Curve. What is a degree of curve, it is the degree of offset per 100 feet. It would not be practical to draw a 300 or 400 foot radius curve in the real world, we can do this in the modeling.

 

There are 360 degrees in a circle, if you offset by 10 degrees per segment, you will form a circle in 18 segments, 180/10=18. It then follows that 18 degrees of curve will take 10 segments and 45 degrees of curve will take 8 segments. See the following PDF files for a diagram of this.
Degrees of Curve 10.pdf, Degrees of Curve 18.pdf, Degrees of Curve 45.pdf
Translating from Real World Degrees of Curve to HO radius is easy if you know just a little Trigonometry. If it takes X segments, at X degrees of offset to form a circle, then each segment is X degrees of a circle. Trigonometry tells use that in a right triangle, sin(Alpha)= Opposite divided by the hypotenuse. If we draw a line from the center of the circle and perpendicular to the segment we get a right triangle. So,

  1. sin (1/2 A) = 50/R - Multiple each side by R

  2. R * sin(1/2 A) = 50 - Divide each side by sin(1/2 A)

  3. R = 50/sin(1/2 A) - Replace 12/ A with A/2 (makes it easier to read)

  4. R = 50/sin(A/2) - This is our first equation. How to find the Radius from the degrees of Curvature.

  5. R * sin(A/2) = 50 - Starting with the equation from step 2, Divide each side by R.

  6. sin(A/2) = 50/R - Take the Arc Sin of each side.

  7. A/2 = arcsin(50/2) - Multiple each side by 2.

  8. A = 2 * arcsin(50/2) - This is our second equation. How to find the degrees of curvature from the radius.
     

 

Using these equations we get two tables.

Degrees of Curve to Radius
Radius to Degrees of Curve

Interesting Number From the above Tables

  Degree of Curve
(Degrees)		 Radius (Feet) Scale Feet Radius (Inches) for HO
Minimum Free-Mo curve for a branch line 22.063 261.300 36.000
Minimum Free-Mo curve 18.880 304.850 42.000
100 foot radius curve 60.000 100.000 13.777
  1.000 5729.651 789.389 (65.78 feet)
  2.000 2864.934 394.710 (32.89 feet)
  3.000 1910.078 263.156 (21.93 feet)

In real life the railroads use 1, 2 and 3 degree curves. Looking at the last three line, to make an HO scale 1, 2 or 3 degree curve takes a 65, 32 or 21 foot radius curve.

Page Last Updated 12/19/2004
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