What I mean when I refer to of the arc of the running stride is that, once your driving foot leaves the ground, your body is either going up, or it is going down. There is no such thing in physics as a thrown or pushed object traveling horizontally within a gravitational system. Obviously, you don’t want to drive your body forward and down, so you send it forward and up — but just enough up so that you have time for your recovering leg to reach the ground in front of you before you begin the downward phase of your running stride arc. Your objective is to get the foot that will receive the body’s weight on the ground before the body begins to fall, so that the descending component of the arc of the body’s forward movement is decelerated by the leg, without loss of forward momentum. Further, the muscles that take the body’s weight should be the same muscles which will drive it forward and upward on the next extension. Thus, as the body comes over the foot that has been placed on the ground in front of the body, the bi-articular muscles are quickly loaded to the point at which they maintain tone to stimulate mono-articular muscles (see Biarticular muscles in running) to contract to extend the leg.
As stated elsewhere, speed is directly linked to the amplitude of extension. This is obvious in a vertical jump – to jump high, one begins with a deeper flexion than one would assume for a small jump, and then one extends the joints fully to accelerate through the extension. While one could do small jumps without fully straightening the legs, one can also do small jumps that extend the legs fully, beginning from a condition of less flexion. What I am trying to get at here is the notion that you have some choice, when running at moderate speeds, about where you place the amplitude of your extension – you can run slowly up on your toes, or you can run slowly in deeper flexion without full extension. There two ranges are useful in running hills. When one runs uphill, it is easier to use the driving power of full extension and to be more up on the toes. The leg can extend fully below you, and it will not recover very far uphill. Driving off of a very flexed leg while running uphill would expend too much energy.
On the other hand, running downhill at a moderate pace, one will probably not fully extend the driving leg. And when the leg recovers, it should reach well forward and down to find the ground before the body falls too much. In fact, the foot should already be moving back toward the body before it touches the ground. Running downhill, one will tend to use deeper flexion of the hip joints to allow the transmission of shock to the glutes. So the legs may be more bent when under the body than they are when running uphill.