Cosine, my friend


Feb 12, 2016
That was always going to be the tile of this for quite some time now.
Originally meaning cosine, as my friend; but then here now to friends in dual meaning as well.
I never get the sine cosine bits. Seems to be referenced without explanation.
The Art and Science of Practical Rigging vs. TCIA Best Practices for Rigging in Arboriculture thread
i look at cosine as directness and sine as deflection >>any potential connected to scenario is 1 of the 2
A column with weight on top capitalizes on cosine in this model as the alignment of load to support
>>while at 90degrees from this line capitalize on sine across column like Samson or wrench across a linear column
>>column @pure vertical uses all length for reach, but can't use full length nor strength vertically @ lean
to this model a stone drop is cosine benchmark axis all rest of sine rings are measured from that benchmark etc.

Cosine is the simpler to start with; and has been my friend once got to know(and quit kicking my azz) as a decoder to events.
>>often taught as visible lengths/distance usually, but applies to force (as a distance reciprocal so one always implies the other)
Several ways of using the measuring stick tools of cos/sin to decode the changes all to same end, just how much work between..
Many find cosine as horizontal axis only from school, and can only see that way.
While builders etc. may only see it as gravity load bearing vertical aspect, as a Greek column of support.
Quite a chase trying to self teach.. Is best to be able to use cosine in any direction more fluently i think.
i walk it around a scenario like a measuring tape/stick, and claim one primary axis of scenario as the benchmark cosine.
>>to judge rest of scenario from this benchmark line of simplest 1D, for a scenario MUST have dimension(or no place to change to).
Cosine is a linear 1D so i align it to a similar 1D in scenario(force line or support against usually) for least calculations

Cosine can be seen as a remaining virtual column of support in a tree lean as support work achieved against weight and space(height)
>>the side force byproduct as the leverage making a triangle actual lean, and the vertical and horizontal EXPRRESSIONS of length and force in the actual lean as the total potential that can be expressed.
The main work i still seen as support CoG but can't do as well in strength nor height with present fixed assets:
We can capitalize on the leveraged byproduct of lean to fall tree in this chess game tho.
Cosine is a graduated scale, but of unequal graduations per degree for a 1Dimensional axis benchmark line.
Easiest to L-earn as gravity load for our work i think, i look at it as a Greek column, co(lumn)sine i say(sine as the sin against laterally/crosswise) as memory aids
A slanted pillar can't hold as much weight as pure inline, and can read effect as cosine dropping as the precise scale of change to loss of efficient usage of strength and length potentials.
Cosine ranges as a multiplier from 0 to 1; i look at as 0-100% multiplier of efficiency of benchmark alignment.
Pure vertical is 100% strength and length potentials used, 30degrees deflection for support and is only now efficiently:
able to use 86.6% of full length (as a potential) for rise AND able to use 86.6% full strength(as a potential)for support.

A rigid device can internally resist on at least 1 cross axis, but rope/flexible devices cannot resist against on any cross axis.
>>so need an external imposer of side force or will self adjsut to none
>>the green rope below @250# is that, or another opposing support leg side force in opposite direction , pulley in center

BUT>>incurs a byproduct side load of 50%, a big jump from Zer0 vs. 13.4% drop in cosine each in 1/3 the travel
>>cos/sin give us these numbers,
>>but (in my imagery) the primal FFF caveman preprocessor brain assUmes different numbers to hand to brain mainframe(prev.discussion)..
>>does not expect the 250# side force below nor the slight loss in vertical drop at 1/3rd travel, must retrain/intercept that process and force new look.
This is as a system of balances, the above configurations MUST give these numbers; cos/sin is the key decoder to define change like it was a balance scale !
i used to look at cosine from the column of support perspective, as pictured Greeks did to hold up roof
>>but have 'evolved' to generally setting the initial force as cosine as flows to that column support,
>>to same numbers but think is more correct , at any angle, like used in electric/wind/water/light etc. waveforms as concept is consistent even to those depths
(from same numbers found decoded the stars movements) child's coin slam force transference:


These are all the same cos/sin to decode, the same rules just expressed in water, tree, rope, pulse wave examples of same things etc.
As are all displacements against physical space and against physical force, so must be able to define most pivotally as cos/sin.
>>as even the stars are commanded to do and so all below the Ancients found, we could never prove them wrong on that.
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Waaaay too deep for my walnut-sized brain, but I dig "In Rope We Trust" :dude:
I have no doubt that this is very interesting and informative but when I look at any of Kenny's posts, all I see is,

"Βλέπω το συνημίτονο ως ευθύτητα και το ημίτονο ως εκτροπή
Vlépo to synimítono os efthýtita kai to imítono os ektropí..."
A background in advanced pathophysiology and Boolean algebra would certainly be helpful.
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  • #11
If lean 100" pipe @ 30degrees deflected from straight against a tall building, how high will the pipe reach upward?
How efficiently (cosine) does it use it's potential (length) to displace against space(height)?
How well does it line up on this single target(up) with all that it has(potential)?
>>Is the same math that is used to decode all of this in ALL physical distance( and thus force) instances.
(Because only force can overtake distance as they trade for each other so are entwined as reciprocals)
My clock calculation in head would say 86" at a glance constantly read in everyday life w/o looking anything up, just a decoder.
>>everything daily self confirming everything else around me, once have decoder ring(clock) to talk to cosine, my friend.
And tough as it may seem, the decoded numbers make it all make sense logically just like the needle on a balance that they in fact are.
This is how they put man on moon and bring'em back whole, with just how much fuel starting when in what cycle to not crash into anything and sling shot around on moon's pull to throw back to where, when by calcs on a slide rule!
This stuff kinda works..they could only even think to do that because even the heavenly bodies had to obey the same laws.
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  • #12
This decoding gives a full open world view of the happening, rather than usual peek thru crack under the door to read world thru..
Otherwise can be as a language in a foreign country, guessing what is said from a few hand motions hope reading right; unsure of next move in these strange rules of this 'society' that all rest of world moves so Naturally with around us as a flowing logic.
Most all things fail from the side force overload from sin(e) much more than aligned loss of cosine/efficiency inline part of support against load.
These fails show across and express as a buckling, shearing etc. from overrun of across/not inline forces
>>not expressed as inline crush w/o side blow out etc. type fail etc.
This is how stuff is commanded to work, up until it doesn't, and how to read, tune and manage it; w/o getting killed by it.
>>also how to even use byproduct as a power utility to assist rather than just tax efforts; turn things around in the chess game.
Against a target distance we can show these things more visually as examples to all else;
but as work we can tune to feel them as do and burnt into memories and even scars of past to sort.
The numbers truly, truly faithfully decode and explain what saw and felt in present or excavated past well how to plot and command in future.
"Nature, to be commanded, must be obeyed" -Sir Francis Bacon
is all that it is; done thru /read thru this language to command, or even see what are inadvertently commanding, or sift out why something really works so can use it else where etc. Once catch enough glimpses in shituations etc. ; more easily can sift the same silhouette from several instances commanding stuff in each expression as same; for 1 lesson not 5 to command same amount, and expands to more as in fact commands all.
Once understand the language, decode it to your own internal Native; 'easily' understand the 'expressions' of the rest of the society 'hustling and bustling' around you in physical distance and physical force enough to miss getting hit by a bus.
The biggest G'D bus ya ever seen; leading w/drooling grin to make another juicy dark reddish grease spot pancake outta someone
....even has scorecard on the back as it goes by; still hungry as a shark while you now just watch scene(not be part of scene) w/locals.
(brotherly rant over)

This pic starts with counter-intuitive warning of borderline limits of no support @90degrees/180 spread of Zer0 cosine .
Shows at end standard safety thumbrule of no wider than 60degrees deflection/120spread as now in red zone of last 3rd of of deflections from load
Then support singly efficiently cos100%, and shared support cos100% efficiency thru pulley with straight support legs aligned against work;
and then shared supports used less efficiently, needing more tension for same work at angle not aligned fully to work.
Work done is as any other expenditures;
>>cost$ more if assets lent inefficiently, as now carry byproducts of tax etc. for same raw 'purchase' of work done in trade.

120 degree spread (thumb to baby finger spread angle as checkmark)thru a pulley weight in center,
where each leg of rope support deflects 60degrees from the center load line;
is usually safety limit on spread apart, as at that point the rope tension = load as a benchmark w/o math
The efficiency of both support legs is 50% each, so together the 2 x 50% = capacity of single leg of line.
DANGER: the cosine falls very rapidly in the final 3rd range of deflection after efficiency of .5cos @60degrees ea.leg/120 total spread before blows out....
>>just 15degrees later @75 degrees on each of shared support legs/150 degrees total spread the cosine drops to .2588
>>look for rope tension to be ~2xLoad and doing nothing but getting frightfully worse fast if continue wider deflections/spread of supports past 120degrees...
>>and rapidly expands to theoretically infinite 15degrees later @90degrees ea.leg deflection of trying to carry load w/0cos as no support
>>upwards towards this is how stuff breaks, recoils, maims, kills etc. like rope center break and recoil back like breech end of rocket over if on the power axis.
I use a few of the words mentioned but can’t follow what Kenny is putting down. Obviously I’ve forgotten a lot of advanced math courses I took over two decades ago
Sounds like a fancy way of sayin a 90 degree bend in arope has mare of it!s rated tensile strength than beyond 90 degrees of deflection.

The exact reason Inutilze Double Cloves locked off with a TimberHitch, on serious big wood when catching it.

Cosine can be used Fourier representing a lot of things.

edit - no one got the EE joke? Fourier transform? summation of series of increasing frequency cosine waves, helps understand harmonic content, nice square waves in digital data transmission etc. Now I feel a bit let down.
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  • #16
Very true, another function extruded from this organic/Natural pattern in all things physical, and then some.
i looked hard at so many things, cross comparing what might be something to then see in all else; sifting, sorting etc.;
to come around back to a kinda: wow, this does just seem to fall into place easily from this view, that i try to lend.
Only after running at cosine term/not from as once did.
Also in wonder on how long ago we were told about it, and still stands...
There is an if/then logic here when can read the potentials, and in fact logic spawned from such math/geometry.
Two of these things(there were many) are a constant surviving companion model for decisions, as Substance & Sail
A boat sail gives strong power broadfaced across the wind, leveraging the amount of capture to max, capitalizing on the maximum wind tunnel effect/position. While the boat itself is inline with the wind, to cut thru water with minimal resistance, capitalizing on the minimum wind tunnel profile, 90 degrees from sail. Maximizing each, by setting the function of each at opposing ends of the 90degree paradigm.
i view wind as the initiating linear force, start with it as cosine cos:100%, sine:0%. The boat sits same, both capitalizing on extreme cosine.
The sail is opposite, to the other 90 degree extreme, capitalizing on the sine100%, cosine:0%
Substance massive vs. minimal profiles: Model of granite / marble slabs: but also going down to glass panes etc. as 'feel' for these things
The way these skinny slabs are stored, shipped and then placed into use; and simply the math that decodes from those functions, that rules all so commonly around us hard to sift out.
Once crunched enough math to catch sight of the silhouette of the constantly lurking principles, can sift w/o (so much) math the same pattern in other things to just an expected logic. Numbers show limits of how far might can go then, where at now etc. is all.
Once recognized how same principles in all, gave more wider breadth parallel examples/experiences to sort against any present puzzle, even if once some of those seemed unrelated; now can lend direct light on subject.

As anything will find most strength amassed across the greater/DENSER STACKED PROFILE axis of most densley packed real substance,
'exposing' only thinner, minimal axis (minimized wind tunnel resistance position) to flow of broad facing against any force, gravity etc. as smallest potential/blunt cross-axis horizontal facing to any side leverage against self, w/own weight at 90 degrees from gravity force flow.
>>vs. as so find the weakest strength across the minimal profile axis, 'exposing' then the greater axis to leverage against self as mostly a poor trade(a greatest sail position in wind tunnel).
In 3D space these all (can)have RIGID profiles of a long, very densely stacked maximum axis, side axis(may be same) but then also a very, very minimal profile axis.
>>as their Achille's Heel by comparison to the denser stacked axis profiles of thus more substance to commit against.
>>the minimal weakness model focuses on for earliest signature of change: massive fail of it's frailty vs the other 2axises that can hold constitution longer under greater load.
Set a column of support on a long axis, as also baby a frail baby/minimal/less dense materially axis from weight bearing when can.
vs. the same scenario
>>the same amount of load is now supported by a column of less of same substance, while also using longer axis for leverage against self compounding to further complication.
note how all supports are such substance RIGIDITIES AGAINST load as a capacity to firmly stand own ground; even against own weight, w/o displacement even to deform, to be firm.
Real examination here is in the vertical storage/shipping or even installation of common very thin profile glass.
And how aligns purposefully to gravity to bear own weight on a longer axis NOT across the minimal axis.
Store, ship, install thin profile glass totally vertically aligned with gravity's pull of own weight against it's own self supported by a long, not minimal axis/profile.
pure vertical alignment to gravity :0 deflection from force is cosine:100%, sine:0% when punched into fone calc
>>all force exerted against a long stacked bulk of own material's long column axis of most amassed together in unison against own self as load.
>>NO, Zer0, nada force exerted against frailest profile axis of minimal dimension at other 90degree extreme.
In traffic seeing/sitting behind beautiful slabs of marble, granite;
some of which that might not be able to buy anymore than lift directly with a finger.. warehouse storage.jpg
(Slab Distributor | Granite & Marble Slab - note specifically designed and engineered racks always seen for this)
Sneaking up on them in parking lots, find tilts of 10degrees, no more on fone app.
fone calculator 10, cosine returns : .9848 of total strength and length maintained at this angle against vertical of gravity and vertical space.
fone calculator 10, sine returns : .1736 of total strength and length maintained at this angle against vertical of gravity and horizontal space.
on watch if had one, would think 2 mins/12degrees and honestly guestimate : .98 and .2 at a glance walking by truck in parking lot w/retrained eye,
maybe put hand blade up and start from high noon and click 1min, 2, then 3 while comparing to slabs angle *as guide.
Reading, re-affirming, cross comparing every step of the way daily.
The minimal axis takes enough in lean to lock against flip/flop forces might have as a balanced vertically seems target.
Thicker than glass, presents more strength on this frailest axis.
Longest axis generally on travel direction as a force, and a long axis for gravity bearing
>>leaving then the minimal axis at minimal force position.
Note how any balanced position doesn't have the power that going to either side would;
>>but has the most immediate power tho to go either way immediately, more than any other
>>invoking that high impacting amplitude of change as goes from 0 to 1 degree deflection (here with so much weight multiplier)
>>the small lean gives a determinate locking position against that i think
These ALIGNMENT considerations are how things are COMMANDED to work, or fail.
The number scales give the maths of how.
These same patterns are in all things, even trees, hinges and ropes.
Not how vertical weight bearing of gravity force minimizes sail catching/leveraging more force against own form
but horizontal bearing gravity weight gives more sail against gravity on that profile.
The cleanest profile for that downward direction in wind tunnel too etc., all the same

Perhaps now drilled to more innate, but these 2 models are a constant reference model for cos/sin functions to me that i share hear.
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  • #17
In the range of use extremes, we have support column theory as application as shown in this thread.
>>And it's bastardization of support theory, turned upside down as mechanical inverse of function (not direction to same axis, but to the cross axis) of wrenching.
To this model as cosine as columnsine , wrenching then must be capitalizing on sine.
As extreme ends of the same power paradigm to choose from and the hybrid range of the mixed extremes between.
What attributes are chosen, are as if a volume dial to strengthen one extreme at the cost of the other extreme in the hybrid positions between extremes. column-vs-wrench-as-opposing-extremes-in-90-degree-range-force-paradigm.png
The full cycle would continue another 90degrees clockwise to same axis as column/opposite direction(1 stroke)
another 90 to same axis/opposite direction of wrench, another 90 back to start for total 360 degrees (1 cycle).
The internal part of wrench can even be hollowed, as is mostly distance between the real workers of tension/compression.
Wow base60!
But, w/o calculator or sliderule find more even divisors excluding identity divisors of self and 1:2,3,4,5,6,10,12,15,20,30
than our base10(2,5), for more division w/o remainder in base60; they made it work, especially as encompasses the first 3 prime numbers as multipliers..
>>and factors can add up to many more perhaps..
The original Babylon clock was also a calendar and both built on the circle as defined by the stars.
Part of my reason and logic of the clock minute ticks (6degrees each) for angles and values both from the same clock.
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  • #19
That is probably a good thang.
Note how this can be cinched up tighter on the host pulling Standing Part out harder and using Half Hitch as like a 2/1 just to crank/wrench architecture tighter on host:
BUT, totally at the cost of strength/efficiency, for then it must use that HH to shear across the single column of support to the load harder.
Just as the previous drawing shows, have a choice of strong cosine for support or strong wrenching as 'antagonistic reciprocals' of increase 1 decreases other within the space of the sum whole.
i look for only 3 rope elements: linear, 90, 180, in reaching for binary decisions like computer end up at binary +non for trinary/ternary as correct/organic/Natural. So i say they are all arcs, linear as the non in array_position:0 of the list:
arc0(linear), arc90, arc180 to cover all rope parts. Note how the arc90 wrenching tighter above is traded more for an arc0 linear of more support/less wrenching tighter w/Round Turn on Standing Part(SPart) to pull along SPart more than shear across, this evolves to Bull and Cat.

Similar in Basket vs. Choke sling positioning, even if basket tensile is half strength of Choke to even out the 4x legs of Basket vs 2x legs of Choke.
Still the Choke shears across columns of support to the load, wrenching tighter on the host;
>>while Basket presents no wrenching grip tighter so much, but does maintain more strength/efficiency in the unbent legs of support to load.

The Choker 'bottom brace' also affords more side pull stability than Basket, Choke on side pull more likely to grip where Basket may slide.
Below see how Ring/Dbl.Ring Hitches shear more across support columns to load, as wrench tighter on host.
While the Bull and Cat's Paw pull more decidedly along, than across the support legs against load as then MUST grab host less in trade.

Legendary in Natural ropes on loading docks/decks is story of Cat's Paw 'loose splice' having enough frictions to maintain a shearing or even separated rope part while loaded, soon as load comes off, effect gone.
cosine vs sine choices extruding column vs wrench results to choose from.
Must be getting lazy, didn't have to draw nothing this time!
I have no doubt that this is very interesting and informative but when I look at any of Kenny's posts, all I see is,

"Βλέπω το συνημίτονο ως ευθύτητα και το ημίτονο ως εκτροπή
Vlépo to synimítono os efthýtita kai to imítono os ektropí..."
If Spock were a tree man....
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  • #21
One of my chases as i worked with cosine was how does it define where the 2x in a pulley pull comes from?
>>this is also the best nip position logically as peak seating force to the host in either function (pulley or nip).
Fair chase around and this is my model for it that several models spawned from once could see that far, after this point.

2-1_effect_in_pulley_as_also best_nip_from_cos-sin_model_b.png
For me this starts to describe many things, note the sine hugs inward too.
>>i define as need rope force on opposing sides and directions as grip
>>note how the grip here is only nominal 1xInput as the most elusive rope control to chase.
While the frictions and nip top out at 2xInput, a sum greater than the initiator /nominal
>>with another arc180 opposing, in lossless model, THEN could then show 2xInput for grip by this model.
note:arc180 in ANY material is very unique in that it is the only form that the end, center, whole form work in the same direction
>>thus usages in bridges etc.
Here show Ashley's short list of Half Hitch nip positions from chapter_21:HITCHES TO SPAR AND RAIL (RIGHT-ANGLE PULL) pg.289+
The common nip near Standing Part for Half Hitch is not in model i showed but is WORST nip position, much less than nominal 1xInput..
Note how part of the rope pulls away from spar at this point, and is 'roped' back in for partial seating.
>>the arc nips by contrast TOTALLY seat to host w/all force and only nominal 1xInputt at the extremes of input and output to the arc itself.
>>i model nip forces on this side of the arc to be sine force only
>>vs. the arc itself i model as cosine + sine forces as a sum to seat to the host
That includes Timber Hitch, i plot a nip in the 2x area
>>that makes the preceding twirls of the 'loose splice' as more spacers + helper nips to the grand daddy nip in 2x area

When i could see how to explain stuff like this that followed the patterns shown;
the lights started to really turn on..
cosine, my friend(s)
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  • #22
Please note how the mid-point between 0-100% is NOT an easy 50%, but rather 70.7% !!
And in the same amount of time it takes cos to drop 29%, sine JUMPS 71% in trade!!!
These fine points are counter-intuitives and must be taught to see, the Ancients found; and so handed down.
These things are key to how things really respond and work.
Best nip is at strongest 2/1 pull point in pulley, as both the same thing, from same maths.
>>i show this as the back to back 90's giving double cosine downward.
Also, thru a range of deformity who's input/outputs endpoints are not in opposing directions (as a linear)
>>between the endpoints the external forces against host are leveraged to greater against the host than the initiating internal forces
As another counter-intuitive to be familiar with, or at least not surprised by..
This is how chit really works day to day....decoded by numbers to now a viewable pattern, to then now be commanded..
this is how work is commanded to getter'done, whether it plots against you/target of efforts or for same; these numbers faithfully decode the happening to be really read, and then so commanded.
cosine, my friend(s)
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  • #23
i in no way was born seeing, nor came to see these things over night;
much getting back in the ring, taking hits and beatings until the spaghetti thrown at self stuck well..

As a doctor listens to the most reactive points on graph, or of breathing or even pulse etc. for deep initial reading; we can just begin to read the knot works from the most reactive point as well.
This is the apex of the Primary Arc in rigging, Hitches and Bends all where focused linear force is converted over/to be handled by dispersing to radial controls.
The 1st ropePart to look for in hierarchy then is the arc180, the most reactive point is the above post's apex 2x position for pulley and best nip etc.
if not see if arc90(peak 1.414 xTension) if not then arc0(peak 1 xTension)to describe next ropePart/function in chain.
(starting all from if Primary Arc conversion is arc180 or arc90)
>>the math unveils this in segments/degrees, to then be read as most logical in a flow comprehensively
>>reveals as more like steps of computer program logic that must flow to a certain point by setup/commands given
>>only to a mechanical logic of staging steps; but still to the outcome commanded by those stages all the same, not by chance..
>>Find the Standing Part as ending where Primary Arc begins
>>then rest of ropeParts from output of primary arc
Tearing down to those 3 root elements(arc0,arc90,arc180) describes anything i have found in rope work, rigging, Hitches and Bends.
The importance of focused linear input to radial conversion (at loss) control(and any radial vs linear sample):
Classically, a Constrictor, Strangle Bag or Groundline loads internal input force for Radial Binding, differently than same lacing then used as a Hitch of external to the controlling arcs LINEAR input that degrades around, where in Round Binding same exact arcs are all equal force around...
(pic as example of knots in this knot family rooted in 3x arc180 Crossed Turns on host , but not of this Binding vs Hitching concept so much)

The input force type (focused linear direction vs. unfocused radial directionless) greatly determine how forces are handled thru the same rope structure.
ANY time going from linear to radial is dispersing focused to unfocused force to then control(fixed bollard).
>>BUT going from radial dispersed input of no specific direction/all equal to fold all that force to funnel to a focused linear of direction is opposite; so is concentrating, not dispersing, force (bollard cranking rope tight as capstan/used as winch)
It is all related to distance and direction of force volume of dispersed radial(lesser per) or focused linear(laser beamed focused from source volume) . The linear maintains then the power axis linearity until converted to other power axis, even around radial host at cost of conversion of friction tax against tension passed thru.
Leaving then less rigidly tensioned section of line(not seen in Round Binding until nip constricting force flow).
As side note: see how Strangle is more of an outie to Constrictor innie !
>>and if lower leg was loaded would fall back neatly into 3x arc180 Crossed Turns on host type Hitch.
>>older pic, theories more evolved now. Originally Strangle was not even on here, is add in years later
(my fave is a Slipped Bag especially on small host, and can show mathematically why that feels right/fits..)
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  • #24
In preceding post#23 described linear force to radial conversion.
Radial vs. Linear matters in most things, perhaps mostly in the host face(s) that lend then radial or linear to rope support architecture at mating surfaces rope to host.
So this also extends next preceding post#22 where math pulls back curtain to show radial frictions even more fully here to contrast then how weak nonRadial host frictions are(to show how really great round host is).
All to move forward with linear vs. arc rope elements in all working rigs and knots.
Model: arc0(linear), arc90, arc180 as the only 3 elements making up all working/loaded ropes.
Round host as Natural, organic, premium solution of continuous, harmonious FLOW.
>>square, pentagon, stop sign etc. linear faced hosts as not smooth, non-harmonious flow but stop/reset/flow again.
>> inOrganic, un-natural , abortive, separate/segmented individual flows rather than single contiguous flow.
Kinda another version of it is not the destination but the journey ...!
arc180 as1 smooth continuous, don't stop , Natural flowing , organic arc180 on motorcycle/bike
VS. getting off bike and rotate 90degrees , restart. At each corner/joint between linear faces, herky jerky, unNatural, in-organic half-azz substitute

Models i show usually assume round host (and assume round rope as another radial vs. linear aspect)as premium target thus key outer potential benchmark.
>> but here show how to 'discount' for linear host face conditions.
>>and then the value of round arc as go forward
Even tho it seems the rules change, math shows the actual consistencies:
>>only deformities from basic linear get compound frictions
>>apexes of arc and corner deformities get cos+cos xTension
>>other deformity points (found in arcs/radial faces only) get cos+sine xTension
This is how stuff works. Arc puts the arch in (nonSimplest/non1D)support architecture!(literally)
cosine, my friend(s)
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  • #25
Above pic shows cos/sine bisection as stethoscope to read/listen to what is going on to give greater usable support and frictions for radial not linear forms.
ALL displacements against space and/or force can be decoded to cos/sine; to then read most clearly, like on right frequency w/o garbled loss.
Same gem, from a different facet of view, to same central sum; can look like this:
Simpler, Linear expressions of friction (and support) can't use cos+sine as 1 sum to a single purpose like friction, support etc.
>>here cos is used for direction of movement maths, and 90 degrees later sine to surface/friction contact maths SEPARATELY.
>>1 element runs across linear path(direction), the other into the path(friction)
>>if travel is into linear form, that is collision, not friction, that would be at the 90degrees to the sides as collision went thru
Very simple, as expected scalar/evenly graduated maths of progressive changes/increases
CoF(CoEfficient of Friction) for the mated materials X distance X weight = Frictions, Efforts needed.

More complex, Radial expressions of friction (and support) can use cos+sine as 1 sum to a single purpose like friction, support etc.
>>here cos is used for direction of movement maths, and 90 degrees later sine to surface/friction contact maths COMBINED!!
>>Both elements direction and byproduct run into/around radial !!
>>if travel is into linear form, that is collision, not friction, but in radial that flows around and becomes part of the support and frictions!!
This is then COMPOUNDED EXPONENTIALLY and by degrees not distance (180 degree units as a 'stroke' ) !!
>>Example shows when give 3x arc180s can return 12.5x more friction than a linear/scalar math of 3xDistance giving 3xEffort needed.

This is how stuff works..

Link list:
Engineers Toolbox: Standards Table of Friction Coefficients for some Common Materials and Materials Combinations
The Mechanics of Friction in Rope Rescue Stephen W. Attaway, Ph.D. (International Technical Rescue Symposium 1999)
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