Hullo!
I've been doing some disrelated research to energy... so this may seem to be coming from entirely out of left field.
Let me start with some links, and then go on to explain what they are...
https://www.geogebra.org/3d/spxzup4s (arccos)
https://www.geogebra.org/u/d3x0r (various graphs I've done with this neat widget, mostly disrelated to the eventual discussion)
https://www.geogebra.org/m/a5dauj6s (this was early attempt, it is very slow; it's essentially the same as the first, but 2 dimensional instead of a slider)
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I did figure out, however, extended arcsin and arccos that are valid; but arccos(cos(a)+cos(b)) is a number that only that operation can produce. (or approximate with a bunch of other things). Makes picking a random secure key kinda nice - suppose that's where elliptic curve stuff is based.
https://d3x0r.github.io/STFRPhysics/math/ This is arccos(x) - (x*2/pi) ... +/- 1 around the origin... The difference in the change for adding a little bit at the ends are a very steep slope. (sine, is similar, but harder to graph, because it loops back on itself and doesn't really go anywhere).
The slope of the arcsin(x) curve is similar if reverse biased with a counter instead... the existing arc-sin can sort of work as-is, but store the number of wraps in a counter floor(angle/2pi). in parallel with the sin result, and then the original angle can be recovered also; otherwise sin() can only return the principle angle. cos(), however, can return the number of wraps ...
This is hard to explain exactly what this graph is showing... but it does remind me of the electron orbital shapes. This also is why from the middle (0) to negative.. until a maximum of absolute 0, the next step is the hottest thing. (If the thing itself is very hot then it will give energy to the system and prevent the system as a whole from going below 0. There's a good chance the same behavior happens at the top end, so you have to put in more and more energy exponenetially (leaving the origin) until a maximum, which inverts very which quickly to reverse the state to the other directions.
The feedbacks only occur where arccos( cos(x) + n ) > 1 .
while resolving what the graph of arccos looks like, it actually simply extends, because the 'forward' is +/-1 and balances out in a 'real' way. It is basically the shape of a closed surface; although vertically up and down it is also the same as the sine graph. The sine graph in its 'natural' state is the shape of open space...
I spent some time developing some meshing algorithms a while ago...
https://d3x0r.github.io/IsoSurface-MultiTexture/ In this demo, there is a 'sine waves' Input data, which is a 3d plot of sin(x)+sin(y)+sin(z), and the surface at 0.... The surface itself is entirely 1 surface, but has inifinite holes in it(?) ... the arc-sin function at >1 wraps backwards on the x-axis ... the space immediately 'in front' of sine is real and imaginary (open)... so the arcsin function is the graph of the open space side... Anyway; it's all a matter of perspective, if you just are the thing rotating, it's all just silly to think about.
Okay so then, A while ago I was playing with `mobially` wound coils (ABHA), and my royer/scope had huge feedback especially under load... The graph of the normalized arccos( cos(x)*2 ) looks a lot like that... such that what was coming back was greater than 1x cos wave (?). Probably a coincidence. However, this other graph, but to go back to the numbers, the projection of power outside the curve reflects differently immediately in the view of deltaX... at deltaX+1 it's very hard to get some places... The function of arccos( >1) is also negative, but odes not reach the full range of the negativity... this would be a very good solution for rubber banding/exponentiated limiting...
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Hopf Fibrations -
https://www.youtube.com/watch?v=AKotMPGFJYk Fasciating
These are those neat interlink toroidal links, which are a subset of quaternions (all that wonderful math I've been doing). They resemble the shape of magentic fields... (although I suspect moving iron filings is a projection of the field, and not really the field itself.... )
So; I built this fancy new mesher that can generate arbitrary shapes, and dynamically reshape them (dig out some of the closed space); but then I need to know where it 'is' of the physics of the thing, so I started on this physics engine, but Matricii (Matrixes) are limited in their ability to carry rotations except for very small ticks; and quaternions are really no better... there is, however, a layer under the quaternions called 'log quatnerion' or apply the natural log function to a quaternion, which just makes it a 0 ( exp(0) = 1 ). and a bunch of imaginary rotations for each axis in the total angles each... these then get 'truncated' to 2pi, or their principle values, and can be applied as quaternions or matricii, and get some rotations... but there's been NO math on these.
It's like all this time there's this huge issue that noone's figured out a mapping from the rotation space ( arcsin( sin(a)+sin(b) ) and arccos(cos(a)+cos(b)) ) being the axises... The arcsin and arccos functions are 'continuous' in rotation space.... but somehow when sine rotates it affects the cos part.
(wow I'm really sorry, I'm realizing this is probably all about 99% useless, which makes it effectively a nil thing)
J
