The paper Computation of Shortest Paths on Free-Form Parametric Surfaces shows how to compute a geodesic curve on a surface. In this paper they start with two equations which are true for a geodesic curve on a surface (eqn 14):
$\frac{dt}{ds} \cdot r_u = 0, \frac{dt}{ds} \cdot r_v = 0$
From here the paper goes on to transform these equations into two second order differential equations (eqn 17-18):
$\frac{d^2u}{ds^2} + \Gamma^1_{11}(\frac{du}{ds})^2 + 2 \Gamma^1_{12}\frac{du}{ds}\frac{dv}{ds} + \Gamma^1_{22}(\frac{dv}{ds})^2 = 0$
$\frac{d^2v}{ds^2} + \Gamma^2_{11}(\frac{du}{ds})^2 + 2 \Gamma^2_{12}\frac{du}{ds}\frac{dv}{ds} + \Gamma^2_{22}(\frac{dv}{ds})^2 = 0$
Finally they rewrite as a system of first order differential equations (eqn 19-22) which can be solved using the finite difference method.
$\frac{du}{ds} = p$
$\frac{dv}{ds} = q$
$\frac{dp}{ds} = -\Gamma^1_{11}p^2 - 2\Gamma^1_12pq - \Gamma^1_{22}q^2$
$\frac{dq}{ds} = -\Gamma^2_{11}p^2 - 2\Gamma^2_12pq - \Gamma^2_{22}q^2$
I have implemented this paper in code and it is working well. What I want to do is extend the method to compute a curve of constant geodesic curvature, or even a curvature which depends on the magnitude of the normal curvature.
Is there a way to adapt the initial equations to achieve this?
Update 11 Jan (Ted Shifrin)
The paper has equation (6) which I believe is what you wrote:
$\kappa_g = \frac{dt}{ds} \cdot (N \times t)$
It goes on to write this equation as (11):
$\kappa_g = [\Gamma^2_{11}(\frac{du}{ds})^3 + (2\Gamma^2_{12} - \Gamma^1_{11})(\frac{du}{ds})^2\frac{dv}{ds} + (\Gamma^2_{22} - \Gamma^1_{12})\frac{du}{ds}(\frac{dv}{ds})^2 - \Gamma^1_{22}(\frac{dv}{ds})^3 + \frac{du}{ds}\frac{d^2v}{ds^2} - \frac{d^2u}{ds^2}\frac{dv}{ds}]\sqrt{EG - F^2}$
When I try to write this as a system of first order ODEs I end up with:
$\frac{du}{ds} = p$
$\frac{dv}{ds} = q$
$\frac{dp}{ds} = -\Gamma^2_{11} \frac{p^3}{q} - (2\Gamma^2_{12} - \Gamma^1_{11}) p^2 - (2\Gamma^2_{22} - \Gamma^1_{12}) p q + \Gamma^1_{22} q^2 + \frac{p}{q}\frac{dq}{ds} - \frac{\kappa_g}{q \sqrt{EG - F^2}}$
$\frac{dq}{ds} = \Gamma^2_{11} p^2 + (2\Gamma^2_{12} - \Gamma^1_{11}) p q + (2\Gamma^2_{22} - \Gamma^1_{12}) q^2 - \Gamma^1_{22} \frac{q^3}{p} + \frac{q}{p} \frac{dp}{ds} + \frac{\kappa_g}{p \sqrt{EG - F^2}}$
I am not sure how to go forward. $\frac{dp}{ds}$ depends on $\frac{dq}{ds}$. This means I cannot calculate my $F$ function in the finite difference method.
In the paper they seem to suggest I can use the first fundamental form to fix this?