View source for Degree of curvature

{{Short description|Measure of a bend's roundness}}
{{about|the measure of curvature||degree (angle)}}
{{inline citations|date=September 2023}}

'''Degree of curve''' or '''degree of curvature''' is a measure of [[curvature]] of a circular arc used in [[civil engineering]] for its easy use in layout [[surveying]].

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

==Usage==
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 (surveying)|transit]] or [[theodolite]] and a chain, tape, or rope of a prescribed length.

=== Length selection ===
The usual distance used to compute degree of curvature in North American [[Roadworks|road work]] is {{convert|100|ft|1}} of [[Arc (geometry)|arc]].<ref>{{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}}</ref>{{Page needed|date=June 2021}} Conversely, North American [[railroad]] work traditionally used 100 feet of [[Chord (geometry)|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.

==Formulas for radius of curvature==
[[File:Degree of Curvature Formula Explanation.svg|alt=Degree of Curvature Formula Explanation|thumb|Diagram showing different parts of the curve used in the formula]]
Degree of curvature can be converted to radius of curvature by the following formulae:

===Formula from arc length===
<math>r = \frac{180^\circ A}{\pi D_\text{C}}</math>

where <math>A</math> is [[arc length]], <math>r</math> is radius of curvature, and <math>D_\text{C}</math> is degree of curvature, arc definition

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

===Formula from chord length===
<!-- See also [[Talk:Degree of curvature#Another formula for chord length]] -->
<!-- See [[talk:Degree of curvature#This does not work]] -->

<math>r = \frac{C}{2 \sin \left( \frac{D_\text{C}}{2} \right) }</math>

where <math>C</math> is chord length, <math>r</math> is radius of curvature and <math>D_\text{C}</math> is degree of curvature, chord definition

===Formula from radius===

<math>D_\text{C} = 5729.58/r</math>

=== Example ===
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 (angle)|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.

== See also ==
{{Div col|colwidth=22em}}
* [[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]]
{{div col end}}

== References ==
{{Reflist}}

== External links ==
{{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}}

{{Rail tracks}}
{{Railway track layouts}}

[[Category:Surveying]]
[[Category:Transportation engineering]]
[[Category:Track geometry]]