Spiral-curve-spiral transition
The elements of a clothoid spiral-curve-spiral (spiral angle, X, Y, p, k, long and short tangents, total tangent), its , , , and stations, and deflections along the spiral.
Experimental: not yet fully verified. How results are checked
The spiral runs TS 45+35.46 to SC 47+35.46, the arc to CS 52+33.59, and the spiral out to ST 54+33.59.
| Station | Deflection from the TS |
|---|---|
| 45+35.46 | 0°00'00.0" |
| 45+50.00 | 0°00'36.4" |
| 46+00.00 | 0°11'56.1" |
| 46+50.00 | 0°37'35.2" |
| 47+00.00 | 1°17'33.6" |
| 47+35.46 | 1°54'34.9" |
- X
- Y
- p (throw)
- k
- Long tangent
- Short tangent
Provenance
- Computed by
- survey.curves.spiral 1.0.0, core 0.1.0
- Model
- Clothoid: θs = Ls/(2R); X = Ls(1 − θ²/10 + θ⁴/216 − θ⁶/9360), Y = Ls(θ/3 − θ³/42 + θ⁵/1320 − θ⁷/75600), four terms each; p = Y − R(1 − cos θs), k = X − R sin θs, Ts = (R + p) tan(Δ/2) + k, arc = R(Δ − 2θs) (Ghilani & Wolf 2021, ch. 24)
- Accuracy
- The four-term series is good to far better than 0.001 of the unit for spiral angles up to about 30°
- Notes
- 1 shown with the answer
- Cites
- Ghilani, C. D., and Wolf, P. R., Pearson, Elementary Surveying: An Introduction to Geomatics
Something look off?
How we got thisFormula, worked example, sources, and proof
Model: Clothoid: θs = Ls/(2R); X = Ls(1 − θ²/10 + θ⁴/216 − θ⁶/9360), Y = Ls(θ/3 − θ³/42 + θ⁵/1320 − θ⁷/75600), four terms each; p = Y − R(1 − cos θs), k = X − R sin θs, Ts = (R + p) tan(Δ/2) + k, arc = R(Δ − 2θs) (Ghilani & Wolf 2021, ch. 24)
Accuracy: The four-term series is good to far better than 0.001 of the unit for spiral angles up to about 30°
Worked example: Ls = 200 ft, R = 1,000 ft, Δ = 40°, PI at 50+00. Source: add-survey-suite spiral-curve-spiral scenario.
You enter
- Total deflection Δ
- 40°00'00"
- Deflection interval
- 50 ft
- PI station
- 50+00
- Circular curve radius R
- 1000 ft
- Spiral length Ls
- 200 ft
You get
- Spiral angle θs
- 5°43'46.5"
- X
- 199.8 ft
- Y
- 6.662 ft
- p (throw)
- 1.666 ft
- k
- 99.967 ft
- Long tangent
- 133.403 ft
- Short tangent
- 66.73 ft
- Total tangent Ts
- 464.543 ft
- Circular arc length
- 498.132 ft
- TS station
- 45+35.46
- SC station
- 47+35.46
- CS station
- 52+33.59
- ST station
- 54+33.59
Review: Not yet independently reviewed by a licensed surveyor.
Last verified: 2026-09-23, when a maintainer last confirmed this tool's sources at the issuer. See the sources ledger.
Status: version 1.0.0, core 0.1.0. See this tool in the verification report.
Changes
- 2026-09-23, fixed: Survey tools cite Ghilani and Wolf's Elementary Surveying, 16th edition (2021), but their method notes said "Ghilani & Wolf 2018". The notes now give 2021, the edition cited. No results change. Changelog
Checked against: 6 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.
Sources
- Elementary Surveying: An Introduction to Geomatics, Ghilani, C. D., and Wolf, P. R., Pearson, 16th edition. Chapter 24 (spiral curves: elements, stationing, and deflections).
Terms
- CS: curve to spiral
- The station where the circular curve ends and the exit spiral begins. Source: Elementary Surveying: An Introduction to Geomatics
- PI: point of intersection
- On a horizontal curve, where the two straight tangents would meet. Source: Elementary Surveying: An Introduction to Geomatics
- SC: spiral to curve
- The station where the entry spiral ends and the circular curve begins, at the curve's full radius. Source: Elementary Surveying: An Introduction to Geomatics
- ST: spiral to tangent
- The station where the exit spiral ends and the alignment is straight again. Source: Elementary Surveying: An Introduction to Geomatics
- TS: tangent to spiral
- The station where the straight tangent ends and a spiral transition begins, curving gradually from straight toward the circular curve. Source: Elementary Surveying: An Introduction to Geomatics
Experimental means this tool has not yet met the stable bar: at least 20 golden vectors, differential tests, and an independent worked example.
CS — curve to spiral
The station where the circular curve ends and the exit spiral begins.
Source: Elementary Surveying: An Introduction to Geomatics
SC — spiral to curve
The station where the entry spiral ends and the circular curve begins, at the curve's full radius.
Source: Elementary Surveying: An Introduction to Geomatics
ST — spiral to tangent
The station where the exit spiral ends and the alignment is straight again.
Source: Elementary Surveying: An Introduction to Geomatics
TS — tangent to spiral
The station where the straight tangent ends and a spiral transition begins, curving gradually from straight toward the circular curve.
Source: Elementary Surveying: An Introduction to Geomatics
Learn the concept: Horizontal curves explained