SACH-SAWCH

When an actual math relationship as an equation is beastly nasty to apply in real time keeping up with the process, engineers ask: Do we really need THAT particularly complex or hard to produce equation? We are not interested in going from negative infinity to positive infinity, just from minus a small amount to plus that same small amount in a potentially useful range. Does some other calculation or mechanism nearly coincide within that very narrow range?

There is a field of math (& engineering) which is focused on exactly that. This simplification is common especially with gravity being useful to have around and the need for having hyperbolic cosines at the ready. If you did that once, can you use something simpler if you are only working within a much narrower bounds? See right? The red curve is complicated. The green is simple, Staying between the two blue pointers, the simple green math will get you there.

Yes. So you see the use of a stiff leg acting like a metronome – an arc pivoting from below & the apex below the floor can be approximated by bringing the arc apex up to the floor surface by standing tip-toe. That also moves the pivot in line with the center of the travel arc (with a touch of knee flexion to help center the pivot point). What??? Does that mean a flexed knee with a plantar flexed foot (equinus) is FUNCTIONAL??? It works quite well for horses and in a pinch, when the Inman determinants are thwarted, in humans as well. That common area in the graph above could also be sine wave & wheel. So, knee flexion and plantar flexion of the foot are NOT necessrily abnormal? Hmm, this is how academics can be rotten clinicians.