My comments were not intended to be taken seriously
But
If we take the river-crossing scenario, if the food is some distance downstream and the current is sufficiently slow, the dog might actually be moving at right angles to the flow, or even in a somewhat downstream direction when it reaches the food.
Or in the case of the Hare running directly away from the dog, the entire path of the dog will be a straight line. Given the range of possible scenarios, we cannot say that the path will always be a spiral or even a curve.
I have not bothered to go through the maths in the link, but if it is correct, then presumably it will cater for the cases examples which I have described above.