Railroad Degree of Curve, Radius, and Middle Ordinate
Railroads describe a curve by degree, highways by radius, and the conversion trips up everyone who works across both.
Example
You enter
- Degree of curve (deg) 4
- Radius to convert back to a degree (ft) 1432.4
- Chord length for the middle ordinate (ft) 62
- Total central angle (deg) 20
- Measured ordinate on a 62 ft chord (in) 4
You get
- Radius, arc definition 1,432.4 ft
- Radius, chord definition 1,432.7 ft
- Difference between the definitions 0.29 ft
- Middle ordinate for the entered chord 4.03 in
- Degree implied by the measured ordinate 3.98 deg
- Curve length for the central angle 500.0 ft
Checked against published worked examples from: US Army Transportation School.
Details, formula, and sources
ScopeA one degree curve turns one degree over a hundred feet, and the useful field consequence is the 62 ft chord rule: the middle ordinate of a 62 ft chord, measured in inches, is very nearly the degree of curve. That is a measurement a track inspector can make with a string and a rule, standing on the track, with no instrument. The chord-versus-arc distinction is the trap. A 4 degree curve is 1,432.4 ft of radius by the arc definition and 1,432.7 ft by the chord definition, only about a third of a foot apart; at 12 degrees the same comparison gives 477.5 ft against 478.3 ft, and the gap keeps growing as the curve sharpens. A radius handed between a railroad and a highway designer without stating which definition it carries can be wrong by enough to matter at a grade crossing or a clearance check. On that 4 degree curve the middle ordinate of a 62 ft chord comes out just over 4 in, which is the string-and-rule rule of thumb doing exactly what it claims; an inspector reading 6 in on a 62 ft chord is standing in about a 6 degree curve and can pair that with the superelevation check to get the speed on the spot. Geometric conversion between the degree and radius descriptions of a circular curve, plus the chord middle-ordinate relation. It does not evaluate whether a curve is correctly aligned, which is what string-lining is actually for and which needs a series of ordinates along the curve rather than one, and it does not compute the throws needed to correct alignment. It does not address spirals, compound or reverse curves, or the alignment tolerances in the FRA track safety standards, which are separate limits by class of track, and it does not handle vertical curves. Highway curve layout and stationing are a different calculation. The FRA Track Safety Standards at 49 CFR 213, the railroad's engineering instructions, and the track owner govern.
Formula or tablechord definition D = 2 arcsin(50 / R), R = 50 / sin(D / 2); arc definition R = 18,000 / pi / D; middle ordinate M = R (1 - cos(theta / 2)) with theta = 2 arcsin(chord / (2 R)); curve length = 100 x central angle / D.
Source and editionThe chord and arc definitions of degree of curve and the middle-ordinate relation, by name, with the 62 ft chord field rule; 49 CFR 213 named for the alignment limits this does not evaluate. First-principles curve trigonometry. The railroad's engineering instructions and the track owner govern.
Free accessPlane trigonometry on the user's own curve; no proprietary table is reproduced.
What governsEstimate. AHJ and licensed professional govern.
API field namesdegree_of_curve, radius_ft, chord_length_ft, central_angle_deg, measured_ordinate_in, radius_arc_ft, radius_chord_ft, radius_difference_ft, middle_ordinate_in, degree_from_ordinate, curve_length_ft
- Chord and arc are different definitions the gap grows with the sharpness of the curve and must be stated when a radius changes handsrailroad and highway curve practice
- The 62 ft chord rule middle ordinate in inches is very nearly the degree of curve; a string-and-rule measurementtrack inspection practice
- One ordinate is not an alignment check string-lining takes a series of ordinates along the curverailroad engineering instructions