About pi quan for those who don't know:
The appearance of pi quan:
It's basically a downward hammer fist, but it goes through a circle, the hand coming down by the waist and circling up past the opposite shoulder to drop down about sternum high ...
as a movement of ki, or I directed my ki through the inner equivalent of the outer pi quan. The circle became more important than the snap down at the end. With the circle, combined with stepping and the intermingling of your own ki with the attacker's, this could put his lower body forward while bending his upper body back over, as if you were stringing a bow, bending it back into the opposite curve. ... a no-form application of ki/body response to an attack, and it was basically the same as described above, but with a different use of the hands. I thought, you could bend him back and slam him down by making the circle with the ki, blending his movement and leading him into this backward-bent place, then dropping the ki.
And then I thought, "Hey, maybe that's what they mean in daito ryu, dropping him at your feet."
.... it doesn't have to be downward, but can also go upward or both back and forward, which is one reason it's called "splitting".
FWIW -- you are describing buckling. Buckling is caused by the internal stress of compressing a column with a slight off-center load. Shear stresses and bending caused by the small lateral displacement of the earth's resistance and the applied load, concentrates shear and creates bending moments in the body. A stably
buckled column takes the shape of part of a sine curve, A sine curve is the path of a point on a rolling circle, e.g. -- the shape of pi quan
you describe. If the load is dynamic, it is a moving sine wave of shear stress and bending moments (or rotations).
If the circle is done correctly, the two sine waves happen but dynamically at different points in the strike. One begins at the first displacement of the strike and a sine wave of buckling goes through the body, A second one begins at the maximum extension of the circle in the strike. If done in this critical way-- it results in combining
the two waves, the first reflecting against the earth and returning, to double effect that causes the top and bottom parts of the body to move in opposite directions - or shear as we would call it, "splitting" in the traditional vernacular you mentioned, and the tenchi
principle in terms of aiki.
If done at a critical rate ~10 Hz, (the same as tekubi furi and furitama), you get resonance in the human body which disproportionately disrupts structure relative to the applied load.