2. Arcs
Geometric curved lines are generally sections of circular arcs since those are the most prevalent in horizontal alignments.
a. Geometric elements; Nomenclature


Figure G2 
The primary geometric elements are related through Equations G1, G2, and G3.
Equation G1 
Equation G2 
Equation G3 
When dealing with highway or railroad alignments, an additional geometric element is often specified: degree of curvature, D. There are two different degree of curvature definitions:
 D_{A}, arc definition; the angle subtended by a 100.00' arc, Figure G3(a)
 D_{C}, chord definition; the angle subtended by a 100.00' chord, Figure G3(b)
(a) Arc definition 
(b) Chord definition 
Figure G3 
Arc definition is used in highway and street design while the Chord definition is used in railroad design. Because each definition differs, there is a slightly different radius relationship for each:
Equation G4 
Equation G5 
R in feet, D in degminsec 
Degree of curvature is an exclusively English units system attribute as there is no 100.00' equivalent in the metric system. Because of this, we won't use it here in the COGO section.
b. Sectors; Segments
Sectors and segments are parts of a circle and are used to compute areas enclosed by curvilinear boundaries.
 A sector is a wedge of a circle bound by the arc and radial lines spaced Δ apart, Figure G4.
 A segment is the area between the arc and its chord, Figure G5.
Figure G4 
Figure G5 
Equation G6 
Equation G7 