0
$\begingroup$

I am trying to compute the geodesic (or tangent) radius of curvature of the geodesic circle by using the below formula.

$\frac{1}{\rho_c}=\frac{\partial G/\partial S}{2\sqrt{E} G}$

where $s$ is the arc length parameter and $E$, $G$ are the coefficents of the first fundamental form.

Can you please tell me how to perfrom the $\partial G/\partial S$? Since $G=r_v\cdot r_v$ I am not sure how to derivate it with respect to arc length

Thanks!

$\endgroup$
0
$\begingroup$

Geodesic (or tangent) curvature of geodesic "parallel" circles in geodesic polar coordinates is given ( for $ s = u $= constant lines, not $v$= constant ) by:

$ 1/ρ_c= \kappa_c=∂G/∂s/(2G). $

where s is the arc length parameter and G is the coefficient of the first fundamental form. You must choose your particular case parametrization next. For example in case of axis-symmetric cylindrical coordinates $G = r^2$ , where $d $ can be written for $ ∂$, constant Gauss curvature differential equation

$ ∂^2\sqrt G/∂s^2 + K \sqrt G=0 $ becomes $ d^2r/ds^2 + K r=0 $

etc. can be solved in terms of elliptic integrals.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.