I'm trying to understand the geometrical meaning of acceleration in circular motion. When I consider a particle moving in a circular path, I can clearly see that the acceleration vector can be composed of a tangential and radial component. But since the change in velocity happens over a period of time, the particle has to move a small distance along the circle (arc), so my question is this: when we say that the acceleration has a radial and tangential component, at what point on the path are we talking about? Because we could say tangential and radial to the initial point, or the final point. Now another thing I don't understand is that if the speed (magnitude of velocity vector) is uniform, the particle's velocity vector consequently only changes in direction. This has to mean that the initial and final velocity vectors have to be of equal length, but drawing them with tails in the same origin still shows that the acceleration has a tangential component not equal to zero. I know I'm very off track somewhere in this thinking, because it clearly makes sense in an intuitive way, but not geometrically.