Degree of curvature#Formula from arc length

The degree of curvature is defined as the central angle to the ends of an agreed length of either an arc or a chord;{{citation|last1=Wolf |last2=Ghilani| title=Elementary Surveying|year=2006|edition=11th|isbn=9780131481893}} various lengths are commonly used in different areas of practice. This angle is also the change in forward direction as that portion of the curve is traveled. In an n-degree curve, the forward bearing changes by n degrees over the standard length of arc or chord.

Curvature is usually measured in radius of curvature. A small circle can be easily laid out by just using radius of curvature, but degree of curvature is more convenient for calculating and laying out the curve if the radius is as large as a kilometer or mile, as is needed for large scale works like roads and railroads. By using degrees of curvature, curve setting can be easily done with the help of a transit or theodolite and a chain, tape, or rope of a prescribed length.

The usual distance used to compute degree of curvature in North American road work is {{convert|100|ft|1}} of arc.{{Cite book|last1=Davis|first1=Raymond Earl|title=Surveying Theory and Practice|last2=Foote|first2=Francis Seeley|last3=Kelly|first3=Joe Wallace|date=1966|publisher=McGraw-Hill|isbn=978-0-07-015812-2|language=en}}{{Page needed|date=June 2021}} Conversely, North American railroad work traditionally used 100 feet of chord, which is used in other places{{Where|date=June 2021}} for road work. Other lengths may be used—such as {{Convert|100|m}} where SI is favoured or a shorter length for sharper curves. Where degree of curvature is based on 100 units of arc length, the conversion between degree of curvature and radius is {{math|1=Dr = 18000/π ≈ 5729.57795}}, where {{math|1=D}} is degree and {{math|1=r}} is radius.

Since rail routes have very large radii, they are laid out in chords, as the difference to the arc is inconsequential; this made work easier before electronic calculators became available.

The {{convert|100|ft|2}} is called a station, used to define length along a road or other alignment, annotated as stations plus feet 1+00, 2+00, etc. Metric work may use similar notation, such as kilometers plus meters 1+000.

File:Degree of Curvature Formula Explanation.svg

Degree of curvature can be converted to radius of curvature by the following formulae:

$r\; =\; \backslash frac\{180^\backslash circ\; A\}\{\backslash pi\; D\_\backslash text\{C\}\}$

where $A$ is arc length, $r$ is radius of curvature, and $D\_\backslash text\{C\}$ is degree of curvature, arc definition

Substitute deflection angle for degree of curvature or make arc length equal to 100 feet.

$r\; =\; \backslash frac\{C\}\{2\; \backslash sin\; \backslash left(\; \backslash frac\{D\_\backslash text\{C\}\}\{2\}\; \backslash right)\; \}$

where $C$ is chord length, $r$ is radius of curvature and $D\_\backslash text\{C\}$ is degree of curvature, chord definition

$D\_\backslash text\{C\}\; =\; 5729.58/r$

As an example, a curve with an arc length of 600 units that has an overall sweep of 6 degrees is a 1-degree curve: For every 100 feet of arc, the bearing changes by 1 degree. The radius of such a curve is 5729.57795. If the chord definition is used, each 100-unit chord length will sweep 1 degree with a radius of 5729.651 units, and the chord of the whole curve will be slightly shorter than 600 units.

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• Geometric design of roads
• Highway engineering
• Lateral motion device
• Minimum railway curve radius
• Radius of curvature (applications)
• Railway systems engineering
• Track geometry
• Track transition curve
• Transition curve
• Turning radius

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{{Reflist}}

{{More footnotes needed|date=June 2021}}

• {{Cite web|date=2005-02-12|title=Degree of Curvature|url=http://du.edu/~jcalvert/railway/degcurv.htm|access-date=2021-06-24|archive-url=https://web.archive.org/web/20050212195231/http://du.edu/~jcalvert/railway/degcurv.htm|archive-date=2005-02-12}}
• http://www.tpub.com/content/engineering/14071/css/14071_242.htm {{Webarchive|url=https://web.archive.org/web/20050127082523/http://www.tpub.com/content/engineering/14071/css/14071_242.htm |date=2005-01-27 }}
• {{Cite web|date=2005-02-23|title=Just how sharp is that curve?|url=http://www.steamlocomotive.com/model/curve.html|access-date=2021-06-24|archive-url=https://web.archive.org/web/20050223171348/http://www.steamlocomotive.com/model/curve.html|archive-date=2005-02-23}}
• {{Cite web|title=Interactive Highway Design|url=http://www.cee.mtu.edu/~balkire/ce3401tc/ce3401Lec20.doc}}
• {{Cite web|title=Degrees of Curve|url=http://www.trainweb.org/freemoslo/Modules/Tips-and-Techniques/degrees_of_curve.htm|access-date=2021-06-24|website=www.trainweb.org}}
• {{Cite web|date=2004-12-13|title=CIRCULAR CURVE|url=http://www.fairview-industries.com/standardmodule/circurve.htm|access-date=2021-06-24|archive-url=https://web.archive.org/web/20041213192255/http://www.fairview-industries.com/standardmodule/circurve.htm|archive-date=2004-12-13}}
• {{Cite web|date=2005-03-04|title=Horizontal circular curves are used to transition the change in alignment at angle points in the tangent (straight) portions of alignments|url=http://ceprofs.tamu.edu/rbruner/curves/circularcrvs.htm|access-date=2021-06-24|archive-url=https://web.archive.org/web/20050304064535/http://ceprofs.tamu.edu/rbruner/curves/circularcrvs.htm|archive-date=2005-03-04}}
• [https://web.archive.org/web/20010529135039/http://memun.org/SchoolsProject/html/Resources/Roads/Fundamentals.htm]
• {{Cite web|date=2004-09-17|title=Sec. 5. Final subdivision plat.|url=http://www.cityoffrederick.com/departments/Planning/APPENDIX_C_SUBDIVISION_REGULATIONS/Sec__5__Final_subdivision_plat.html|access-date=2021-06-24|archive-url=https://web.archive.org/web/20040917090858/http://www.cityoffrederick.com/departments/Planning/APPENDIX_C_SUBDIVISION_REGULATIONS/Sec__5__Final_subdivision_plat.html|archive-date=2004-09-17}}

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{{Railway track layouts}}

Category:Surveying

Category:Transportation engineering

Category:Track geometry