For a unit speed curve on a surface, the length of the surface-tangential component of acceleration is the geodesic curvature 
. Curves with 
are called geodesics.
For a curve parameterized as 
, the geodesic curvature is given by
![kappa_g=sqrt(EG-F^2)[-Gamma^2_(11)u^('3)+Gamma^1_(22)v^('3)-(2Gamma^2_(12)-Gamma^1_(11))u^('2)v^'
+(2Gamma^1_(12)-Gamma^2_(22))u^'v^('2)+u^('')v^'-v^('')u^']
×(Eu^('2)+2Fu^'v^'+Gv^('2))^(-3/2),](/images/equations/GeodesicCurvature/NumberedEquation1.svg) |
where 
,
,
and 
are coefficients of the first fundamental form
and 
are Christoffel symbols of the second
kind.
