I always struggle to understand how to visualise a geodesic. When given a metric I can easily deduce the geodesic equations but I do not know how to visualise them. For instance consider a torus parametrised as follows: $$ x = (A + B \cos θ) \cos \phi, \space y = (A + B \cos θ) \sin \phi , \space z = B \sin θ, $$ where $A$ and $B$ are two constant parameters.
We can define the metric as follows:
$$ds^2 = B^2 dθ^2 + (A + B \cos θ)^2 dφ^2 $$
I know the geodesic equations are as follows:
$$\frac{d}{d\lambda}((A + B \cos θ)^2 \dot \phi) = 0$$
$$\frac{d}{d\lambda}(2B^2 \dotθ) = −2(A + B \cos θ)B \sin θ \dot \phi^2$$
My question is how could one look at those geodesic equations (or better still any geodesic equations ) and understand the path of the geodesic.