Roads are typically staked ("mapped out in real life") along the centerline or at a designated offset from centerline.

So I think there's a discrepancy between what you're seeing in a text and the rigorous definition of the stationing convention. Stationing starts at a point of beginning (POB), then follows the alignment through the series of curves etc increasing along the distance of the line. So for a given curve, the minimum value would be at the point of curve (PC), maximum at point of tangency (PT), with the Point of Intersection at some intermediate value (that isn't half) of the station at the PT.

So hypothetically, for a curve with a start station of 1+00, and length of 100 feet, the PT would be at Sta 2+00. However, the tangent distance (T) is proportional to the radius and Delta angle (T=R*tan(Delta/2))

Seeing this applied to a hypothetical problem.
Radius = 1909.86'
Degree of curve: 3^00'00"
Tangent length: 422.24
Delta = 24^56'
Assume PI at Sta 568+24.33
Find PC and PT stations

L_c = Delta/Degree of curve * 100 = 831.11 feet
PC = PI- Tangent = (568+24.33) -(4+22.24) = 564+02.09
PT = PC + L_c = (564+02.09) + (8+31.11) = 572+33.20

(From the CDOT Survey Manual Chapter 8 at https://www.codot.gov/business/manuals/survey/chap...)

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Chris Enright
PE, Colorado
Roads and Trains