Maths abuse alert: All the following is done intuitively, and I've completely failed to formalise it, which is a bit embarassing given I've read a book on differential geometry.
It occurred to me the other day while wheeling the children to Greenwich Park, over the somewhat unever ground, that pushchairs try to follow geodesics. That is, if one of the wheels is taking a longer path, the pushchair will tend to turn away from that side. So, locally, it seeks a shorter path, and in the limit it'll run along geodesics.
Another way of looking it is that the pushchair will run along a path such that each side wheel runs along the same distance. So, along the path the derivative of distance stuff perpendicular to the path is zero, so that said distance stuff is locally maximised or minimised (and I guess we're looking at a minimum).
This in turn helped me understand a Feynmanism. Feynman said that light takes the fastest route between points, but I never really got why this should fit with the other formulations of the behaviour of light. However, I now see this is rather like the case with pushchairs. In refraction, light changes direction as it goes through media with different speeds of light, so that the wavefronts match up across the interfaces (GCSE description of refraction). In other words, it makes the light turn towards the faster medium, and away from the slower medium. And as long as it dos this in the appropriate way (which it does), once again it'll follow a geodesic. Lightbulb appears above my head.