Naming ropeParts as components in a working support structure/architecture

[KC]

Concentrating and diluting/dispersing force to 1D focal:

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i picture Bends and Hitches as linear fed force to controlling arcs checked by ballast, such as this function:

Capstan Cranking w/Tailer man
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Note how to suddenly load tailer man would want to be preset, braced, cheating pull some, to get jump on /against load rather than that load have same advantage over him.  Similarly i seek to brace, pre-cheat advantage to the ballast side first in knots, then groom out in dressing

[KC]

cosine vs. sine: setting cosine as a benchmark initially in scenario, not outside scenario.
ALL Physical Displacements Against physical space and/or physical force;
can be pivotally expressed every time as cosine of directness;
or it's non of sine(indirectness, not in same simple , single dimension of cosine benchmark) ; to cover all aspects of connected scenario.
Cosine/Sine are keys to the kingdom, of decoding what is going on in scenario.
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Classically schooled to statically set cosine to horizontal as benchmark to any scenario framed, but any single dimensional length aspect can be used, so i use a linear force or support against that force for fewer calcs of comparisons.  By dynamically setting cosine to scenario; then even cosine so Native to scenario it is one of the comparative elements.  Such as using this linear force line as the cosine benchmark, anything not in that single unique dimension flows into the sine of the connected reference.  Only the cosine is focused and direct as sine runs out diluting in all directions. 

It is this sine of segmented dilution that we use in ropes as controls across rope for the forces running thru rope length to Load.
For rope mechanics are simply no different; same completely ruling maths Ancients witnessed forward to us as even commanding the stars who's revolution took 1yr increment cycle on their clock of 12mos in 4x90 degree quarterly seasons, as a microscopic view of the workings(at least certainly for 800bc) of what they they were dealing with.
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Rope elements work generally in 2 different ways, powered by 2 different aspects of the imposed/input force.
The force down the length or the rope around the host bears the loading at the end of the rope as cosine in this model.
While the forces across the rope length control the load with frictions, nips and in opposing multiples grips.
This gives the major direct force down the length of the line to holding the load/cosine of rope length
>>while the lesser, deflected force of sine is used for the controlling frictions, nips and grips(if opposing multiples).
EXCEPT: at points of change/conversion; then in addition to lesser sine, we then have the major cosine force along with sine controlling the load.  This can be on a corner between linear faces or much larger point/range of change thru a 180arc in radial face host.  The host lends the structural form of the rope, but if radial host can still have some linear parts.  Where the rope shows 180 arc structure, it opens the door to those specifically uniquely powerful arc(h) maths.
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As in all structures, the 180 arc is a king in that it can use both the cosine and sine to same target, not to the usual, lesser separate utilities!
In this model, cosine exhausts out of existence to Equal & Opposite, while sine dilutes out of existence.

[KC]

Slice by Slice;
as i drive car immediate pure inline, flat forward direction is my cosine of 1 dimensional focus(no side nothing);
i can feel the sine/numbers of a curve or hill that is off the MINIMAL distance/base/nominal (for work) pure flat linear UNIQUE 1 dimensional path.
If know the associated numbers, can feel the numbers to 'see' even more sense to them more innately i think.
See this in everyday things becomes so common is not exercise, easier to see in work things.
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Clock Allegory, have shown, is my always handy 'decoder ring' of angles and associated cos/sin/tan that tell story eye to simple to see.
Surprise sideward force on small curve, is the amazing, maximum sine changes in that end of the powerband just off pure alignment/straight that eye only sees thru decoding chart.
While same soft angles of deflection not giving much softer RR track crossing (until greater angle to be effective at dropping cosine faster) is how conversely the cosine barely moves at same range as sine so intense, as are antagonistic reciprocals, that eye doesn't catch.
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For 6degrees deflection from pure inline (1 minute on clock) returns 10% side force on the turn a surprising amplitude of force that is immediate so carries impact wallop too,
but going across  speed bump or RR tracks, same 6 degrees only reduces the confrontation to the RR tracks 1% as drop in cosine/unnoticeable.
This makes the counter-intuitives seem even farther away, like as inconsistent; until can decode and sort to proper piles.
Increase angle hit tracks to 30degrees deflection, drops  cosine by 14% as also giving longer speed reduction time to compound in on reduction multipliers chain, were can appreciate some of the effort, less confronting, more along tracks direction.
(i do prefer aiming at angle, but actual crossing wheels square with car frame).
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On non radial rope elements; can model force path(as all above) slice as cosine down center of rope of simplest needed (anti)Load force, reducing influences as sine across that cosine as benchmark line in scenario.
Radial elements more to overall scenario force path model i think(than segmented as above) for benchmark cosine, from the input Load direction(until over-ruled to new cosine reference from there from shear across cross axis).
Showing dynamically setting benchmark line of cosine to the scenario, rather than statically a preset frame guide of cosine, that set scenario inside of to weigh and measure scenario as classically taught i guess.
For cosine/sine are just graduated rulers with associated values; that like a measuring tape can take and walk around problem points if want more so to me.
Each way of use, just measuring the same facets/relationships from a different benchmark; sometimes to show different insights , or same insights better at that 'angle' of view.  So each paradigm very illustrative, but not exhaustive model alone to me.
Model: "Wages of Sin(e)" as a side tax shearing across your work as Samson to pillar:

In ropes lessor sine across causing seating to host is automatic, implicit in the Load x Angle as Load cause/cos, sine is incurred as host interrupts that direct flow. 
Radial host interrupt/redirect of cosine flow to Load, is most organic/gradual/Natural interruption.
>>That Linear face host can't offer, for can only give harsher impact of change/interruption.
Seating to host gives utilities of : potential frictions, nips and grips controlling the Load held by cosine thru core of rope(slice by slice model).
Co$t of Conversion at change of direction is when the greater cosine(to lessor sine)joins in as a controller of the Load.
>>on a Linear faced host this would be at a spot on a corner
>>but on a rope arc, especially 180, the conversion co$t is over a more usable RANGE not a point of the deformity causing conversion/direction change.,
>>therefore also the returns traded(for co$ts) of controls from seating to host(friction, nip, grip potentials).
>>arc also increases the sine value and uses both cosine and sine together for control
>>where previously only used sine at a lessor value, to power controls over Load (thru seating to host )of :friction, nip and even  grip in self opposing multiples
>>in 180 arc from Linear source/initial input force type, find seating to host as most compound for strongest compound controls of compound friction and compound nip possible w/some nominal, sine(that is increased in arc) powered only grip potentials.
>>compound grip achieved as go again to opposing multiples; this time of 180's for 1D grip w/2opposing 180s potential and if more 180s 2D framework of grip potential.
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Another place can feel the sin(e) shearing across efforts is vertical Prusiking or camming etc. leg compression lifts.
Benchmark cosine as inline target, aligned to gravity line, every single drop of your center of gravity not inline with cosine is less efficiency;
AND carrying force of sine thru bod as a device too, to a new and crossing direction.
If input pull tension from cam handle overhead, sine more cradles and grooms you to more efficient inline as tension pulls to centerline.
But if input is from more powerful leg, the sine is fought more from pushing you more out of line to less efficient as part of the work.
>>for compression pushes E&Os apart more
The more you give in to that side/sine push across to more deflection from centerline
>>the more effort it cost$, in 2 different ways, each to a different direction(cos and sin across) as it tries to beat you in the open weather.
The harder you piston hit with leg(s)upward; the more upward return and also this crosswise, trying to break you as a force vessel (d)effect in return also in this dbl.edged sword.
The thinness of own powerband at that point/range between twilight of can and can't;
is to the good side; the most sensitive to change range; so yielding most body sense.
Sometimes put self(or scenario) to point between can and cant /almost upside down of what can do; to sift by this measure purposefully in many things, for the magnified/more sensitive view.
Other times, only way to find there is a line, or where it drops off like a cliff, is to step over it/or close to feign to.
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Driving, rope, water hose etc. can be shown as slices of focused direction as core linear of focus; directly to cause/cos(minimal path) and the co$t of sin(e) shearing across; to contain all connected parts.
Different insights to each can be found from another, if know what parts of the separate knowledge banks link as one.
Including feeling with body the forces in driving, for a more body sense/knowing, gut level familiarity once experience realized and sorted to correct piles.
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[agent_smith]

Hello 'KC',
I note that you have posted almost identical information over at the 'treebuzz' forum.

I'm going to give you some feedback - and its all done in good faith - which means you should not feel harmed or insulted in any way!

In the first instance, your writing style is very difficult to follow.
This is not an insult - its simply that your sentence structure tends to be incoherent and hard to follow - and I speak and read English as a first language (and I struggle reading your narrative).

I know that you are very passionate and want to share your passion - and that's a very noble cause
I am sure that readers are trying to decipher your narrative, to try to gain insight into your concepts.

Here's an example of awkward sentence structure:

Clock Allegory, have shown, is my always handy 'decoder ring' of angles and associated cos/sin/tan that tell story eye to simple to see.

For the average layperson, I think this may be hard to follow.

and this:

Sometimes put self(or scenario) to point between can and cant /almost upside down of what can do; to sift by this measure purposefully in many things, for the magnified/more sensitive view.

...

With regard to your use of trigonometric functions:

I note that you often try to use trig functions to describe forces.
I personally disagree with this approach - as it assigns a different meaning to trig functions and what they're used for.

A force has both a magnitude and a direction - which makes it a 'vector' quantity (as opposed to a 'scalar').
Sine, cosine, tan, etc.. are functions which relate to right angle triangles and angles (they are not 'forces' per se).
Trig functions are derived from the 'unit circle'.
The SI unit of force is the Newton (N) - it is not 'sine' or 'cosine'.
For example, the tension force in a rope is measured in Newtons - not in 'Sines' or 'Cosines'.
And pressure is measured in Pascals.

Furthermore, the core of a knot is 3 dimensional (not 2D / planar) - and so spherical trigonometry would also apply.
In a spherical triangle, the angles don't add up to 180 degrees!
However, I do accept that on a small/local scale, plane trig functions can be used to determine vector quantities and angles (where locally, a spherical triangle is very close to 180 degrees).

What needs to be understood is that force is at its maximum when entering the core of a knot (via the 'SPart') but eventually reaches zero at the tail end.
The reduction in force from initial entry into the core to eventual zero at the tail end is not linear.
I think that in most cases, the force reduces exponentially.

[KC]

However, whatever names other than rose to a rose; Direction vs. Drag, Cause vs Co$t
what i think have decoded from these views i try to offer; simply 2 separate reciprocal force parts of the whole sum force, as foretold millenniums ago, that sit at 90degrees offset to each other....
No matter how i chase it; i come back to there as the math continues to back it up;
in Linear as well as Radial examples; in all things that rope is just simply a part of that parent set.
>>And, am most powerful and clear when start analysis from there that always end up at; after i begin to catch on..
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1 force is focused against the Load, with primary workforce as feeds AROUND host as Load CONNECTION in rope.
The other a deflected byproduct incurred , peak expression calculated at any offset 90 from primary workforce, here seating INTO host w/secondary, deflected force, that gives CONTROL of the Load, everything else is just basic CONNECTION to Load(thru force AROUND not INTO host).  Can also name these direction and drag/exploited here as CONTROL.  But still part of the focused and deflected/dispersed contexts foretold.
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Thus, the primary invoked forces mostly race by next to host/NOT seating into, as Load CONNECTION, at little/nominal or no taxation against tensions, or to be too inefficient for many uses. 
Until, deformity of either:
>>'brashly' immediate, unNatural, forced hard,  corner (mating between linear faces) vs.
>> gentler span/less immediate, more 'organic' , Natural , softly/ deftly changing, less disruptive 180 arc as Nature teaches in Her own usages.
These deformities then employ the primary Load CONNECTION force, as also now adds to controls by seating into host, only at prescribed points.  But get this compounding effect as a conversion loss, that linear doesn't have (much) any; until deformation.
These are inefficiencies might cry over in other things of life, but here capitalize on friction inefficiency as control of Load, the other controls of nips and grips powered by same byproduct side force also to complete tool package.
Because 180 segments have this most powerful use of deformation over such a span, we have this compounding effect of ALL rope tensions , not just the lesser deflected part of force, used for frictions etc.  This is why capstan theory shows compounding frictions BY DEGREE, as using ALL TENSIONS.
vs. standard Linear friction that is still in 2 force parts direction/drag etc. model, of only using PART of the total forces found for drag and part for direction/cause, so compound by DISTANCE, not degree.
>>births 2 models of usages then.
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The 3 ropePart elements expressed as all arcs : 0(non-arc/Linear), 90 and 180; each are different direction and axis definitions.
0degree arc/Linear endpoints pull in OPPOSING directions along same DIRECTIONAL axis, only in pure Linear axis in straight line/most efficient version
 90degree  arc endpoints pull on cross-axises and so perpendicular directions as well.
180degree arc VERY uniquely has both endpoints pulling in SAME direction around host and even a greater 3rd force reference at the apex INTO host
>>so that the
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This is simply the direction of focused travel and the side drag at 90degrees as in may things, like pushing a cement block across flat surface as shown 2x in Dr. Attaway paper, then in expanding ways as walks us thru to radial theory. 
>>the other seating controls of nip and grip follow the same pattern of usually nominal, but in arc nip can be compounded but only have nominal grip(from only deflected, secondary force seating to host) across arc, until an opposing 180arc.  For that is only time can get COMPOUND grip; that contains the primary Load force with an opposer of same on opposite side.
(on linear faced host 4x4 etc., see these effects 'momentarily' in corners).
The paper shows to count arcs to calc frictions in 180 increments /radian Pi's in capstan theory just the same, as rest of frictions are more nominal/if any.
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Half Hitch gets skull warning in ABoK, but same form with Round Turn instead of simple/single Turn is shown as 'Anchor' dependable icon.  RT + HH much more secure than HH, as RT reduces propensity for Bitter End(BE) to pull out. 
We see this RT preceding in Anchor Hitch too, only the BE is now nipped by the HH's 90 degree Nip arc converted to MUCH more powerful 180 arc nip.  To now trap the lesser propensity to pull out after RT BE; with MUCH more nip, by changing 90arc element to 180arc element.  This might be the greater change alone in the security, and the reduced tension/propensity for BE to pull out; just a pre-fixxing frosting on already sweet cake!

[KC]

Thru all this digging, re-assembling one of largest focuses is how rope/knots should be taught to scouts etc.
Reverse engineering from target back, more assures aligned to target when working forward type strategies.
So, at that level none of this math...
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BUT, should start off; so if go the distance, aligns to the math, real view, so consistently, rather than scattered from the start then changing view.
Thus i think should focus on SPart as just input handle, arcs as leveraged controllers, final nip as tailer man after capstan.
>>show how boy could hold man pull if turns on tree in between
>>show how braced boy, rope tight can take a shock pull from man MUCH better than 2 boys not set, rope slack (dressing lesson).
No math, no waves, but consistent to.
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We treat rope many times as a complication of same rules use for many other rigid things (if even get that far).
>>i think because idea support must be rigid against Load is hard to grasp when holding flexible rope in hands that can be rigid against Load is more something might say(if get that far) than understand.
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Rope is just a construction material; with self linking part functions, that can be formed at room temp unloaded, then forged rigid just enough to stand against Load instantly on loading.
It is only rigid against Load on inline axis and then only in tension direction on that axis, very 1D
>>instead of looking at this as more complicated, it can be seen as less so actually, as is more minimal
>>so can L-earn many lessons of world workings in simple rope, then expand to more rigid examples later of more complications of direction and dimensions.
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Simply best if all lessons are contiguous, aligned to each other in architecture of simple fact.
An intro to how 'man' COMMANDS force extensions of will outside of self, thru simplest tools;
that give pivotal lessons that DON'T change , to throw out to kidz, see what sticks to whom etc. as possible evolves to better math, engineering, working FEEL as goes on from these 'exposures' to these things that trace our evolution of thought etc. 
Carrying with them unspoken lessons thru the ages, not found on the pages!
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Truly one of the best focuses of al this.

[KC]

Following the forces, in simplest scenarios; of 1 active/imposing/'bully' onto other wise peaceful/calm/passive, but responding (rope) support 'village' of operations:
The raw input, imposed force, is therefore the greatest force w/o other active/only responding elements.
The whole 'village' are subject to input's domain of reigning force then, until it is exhausted at extremes and/or nipped from traveling/flowing forward.
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On round host with linear input i look at the focused, linear input as the benchmark cosine of alignment, all day everyday; that defines the rest of the domain or reign, until hits 90degree switch to cross axis, new reign of new domain.
For a child, squarely aligned for maximum, anything less is less; is good enough.
>>and is directly to the truth, even if grow to see it as cosine etc. or not, is contiguous to target, just depends on were you stop with it in growth.
>>in scouting, outside of raw skills, this raw 'exposure' to the raw works to COMMAND, can lead to math, engineering, geometry, rescue, seamanship etc. or not, each to their own liking of taste received.  And lessons consistent as does grow with these things, w/o waste, fully confident in steps taken, so more confident in where they consistently lead from seeds placed.
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In workplace ability to see any linear initiating force as cosine, and responding rest all in relation from that benchmark inside it's domain of broadcast/reign is very dynamic decoding on the spot, w/o lab drafting board, calculator etc. for me.  Where to stand and not, greatest forces, what to expect where etc.

Shows penny hits of aligned and lesser deflected, dropping to Zer0 at 90 degrees force deflections; as a minimal, very true, functional utility; but also very expandable as already in line with those truths would travel to.
Also can key in at young age in bare bones simple example, trying to visualize unseen force thru the transfer.
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Very minimally for points to watch:
Key forces in forming round rope on round host: square alignment(many life lessons just hear) and friction
rope elements/parts:
arcs of power (half circle/180arc) primarily;
can also extend to arc of change (quarter circle/90arc);
other nonArc/0arc are simply connectors until crossing can add more utility friction/nip.
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Rope simply auto connects these simple working elements in groups,
like each element had magnets that could easily connect separate utility functions in chain.
Working Rope is a chain, of non generic force utility parts, that is formable material at room temp, if unloaded.
Working Rope is a glimpse/intro into several aspects of mechanix.
Working Rope only works inline to it's long axis length, and then only in the tension direction on that axis
>>if temporarily set aside what think know about simple mechanix
>>rope can be seen as a starter simplification of 1 direction on 1 axis
>>and then more commoner rigid supports as a complication adding 2 cross axises and more direction variety as well.
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Rope Work is special because gives an immediate,  formable support structure at room temp, that hardens just enough when loaded to be rigid enough against Load to give support. 
We can see it still as this flexible device, but the held static Load's perspective is that rope is as like forged iron jailer that is captive of!

[KC]

So many things want to move forward with, trying to set in order to show etc.
and these things are the basis of.

With regard to your use of trigonometric functions:

I note that you often try to use trig functions to describe forces.
I personally disagree with this approach - as it assigns a different meaning to trig functions and what they're used for.

A force has both a magnitude and a direction - which makes it a 'vector' quantity (as opposed to a 'scalar').
Sine, cosine, tan, etc.. are functions which relate to right angle triangles and angles (they are not 'forces' per se).
Trig functions are derived from the 'unit circle'.
The SI unit of force is the Newton (N) - it is not 'sine' or 'cosine'.
For example, the tension force in a rope is measured in Newtons - not in 'Sines' or 'Cosines'.
And pressure is measured in Pascals.

May i please say that instantly seeing cosine line in work can be life saving as well as focus for efficiency etc.
Am all self taught vs formal, but again and again, inescapably what i see AND feel:
Work Achieved= Displacement Gained x Effort Input x Cosine (as efficiency of effort to target application)
Work Needed  = Work Needed - friction drag inefficiency
If all other parts in scenario are passive/but responding and introduce a single linear force to rope etc.
>>That force will be the reigning domain of full efficiency that can set cosine benchmark to until Zer0 force
>>or Zer0 resistance of cross axis 90 degrees to main axis of active force, now changed axis (round rope on round host)
In very generic, simpler flat terms:

For cosine as target flow of the Linear input force, that defines the rawest force in the scenario.
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Furthermore, the core of a knot is 3 dimensional (not 2D / planar) - and so spherical trigonometry would also apply.
In a spherical triangle, the angles don't add up to 180 degrees!
However, I do accept that on a small/local scale, plane trig functions can be used to determine vector quantities and angles (where locally, a spherical triangle is very close to 180 degrees).

Am not sure if by core mean knot or rope internals please.
But, i think in working with the cosine/sine from aspect of 2, into 3 dimensions, that the rope needs rigidity and perhaps even a linear part that round rope on round host don't present?
Basically i see, just as ABoK presents, rope right angle pulls as a knot basic, then may also have a lengthwise pull. 
But i believe these work in 2D, and also in drawing can show axis thru host minimal axis and long axis on page
>>but for a 3rd Dimension would have to draw out towards the reader.
i just don't think the rope that only works on long linear axis and then only in the tension direction, can quite extrude 3D from such a fluid substance with just 2part cosine/sine then?
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What needs to be understood is that force is at its maximum when entering the core of a knot (via the 'SPart') but eventually reaches zero at the tail end.
The reduction in force from initial entry into the core to eventual zero at the tail end is not linear.
I think that in most cases, the force reduces exponentially.

This can see way to most, but still think that the linear essence of an input linear force type is key;
as is different thru the same rope part members if input radial force of binding against swell even with SAME lacing, not untied, just force input altered to same framework.  Specifically speaking of using family of Constrictor/Bag/Ground Line force patterns change if used as a termination/Hitch utility of Linear input vs. use for Round Binding against swell as a Radial input force as the ONLY scenario change.

[agent_smith]

Hello KC,

You appear to be directing questions to me?

I admit to having significant difficulty trying to understand your narrative - and English is my first language (ie, I am a native English speaker - Australian).

I will repeat what I had already pointed out several posts ago...

I disagree with your use of trigonometric functions to describe knot geometry and response to load.
Trig functions are not forces.
In a 2D planar coordinate space, trig functions relate to the 'unit circle' with x and y axis coordinates.
Planar triangles add up to 180 degrees.
Spherical triangles do not add up to 180 degrees!

A knot is a 3D object - not a 2D planar object.
If you really wanted to try to determine vector quantities in a 3D knot - you would have to use spherical trigonometry. However, on a small local scale, a planar triangle will be 'close enough' to the real larger 3D environment.

You appear to make many posts in this forum and other forums where you assign trig functions to 3D knots.
Again, you would need to provide a coordinate reference frame - showing the x/y/z axis. Without a reference frame with a defined coordinate system, your trig functions are arbitrary and make no sense.

You have also referred to a paper authored  by Stephen Attaway ('mechanics of rope friction in rope rescue' ?) - and it seems that this is the source/inspiration for your use of trig functions?
Attaway was using trig functions to determine the contact angle (in radians) of rope against a metal surface (eg a belay device) - and this relates to the 'capstan equation'. He wasn't specifically examining a knot.

In other words, Attaway wasn't specifically stating that certain parts of a knot are 'sine' or 'cosine'. He never did this because to do so would be arbitrary.

...

In my personal view, I think you should invest in a thermal imaging camera (eg a FLIR camera)
Link: https://www.flir.com.au/browse/professional-tools/thermography-cameras/
Apply load to a knot and observe its thermal response as load increases.
You will be able to pinpoint localised stress concentrations which causes heat buildup.
A significant proportion of tension force in a knot is converted to heat (energy is transferred/transformed as heat).
We always see evidence of melting when synthetic ropes are loaded to MBS yield point.

EDIT NOTE:
Tried to find youtube videos of thermal imaging of knot tests...no luck at this stage.
I think Richard Delaney had produced some videos using a thermal imaging camera?

[SS369]

Here is a link to one of Mr. Delaney?s FLIR tests.
[url][https://www.youtube.com/watch?v=P5qDr3qYZ9o/url]

SS

[KC]

Wow, that is really great TY!
Also have this from before:

i hope in our time, will see computer sensors even built into rope for thermals, tensions, stretches, nips, AND SEATING PRESSURES TO HOST monitored in pulls as deeper reveals.  Always wanted to get a nip sensor for squaring presses across surface also, but outta my humbled economic league for a real McCoy/ the Binford 9000.
.

I admit to having significant difficulty trying to understand your narrative - and English is my first language (ie, I am a native English speaker - Australian).

I will repeat what I had already pointed out several posts ago...
...
I disagree with your use of trigonometric functions to describe knot geometry and response to load.
Trig functions are not forces.
In a 2D planar coordinate space, trig functions relate to the 'unit circle' with x and y axis coordinates.
Planar triangles add up to 180 degrees.
Spherical triangles do not add up to 180 degrees!
....
You appear to make many posts in this forum and other forums where you assign trig functions to 3D knots.
Again, you would need to provide a coordinate reference frame - showing the x/y/z axis. Without a reference frame with a defined coordinate system, your trig functions are arbitrary and make no sense.

.
NP, have always had a different perspective on many things; but have L-earned to trust my instincts, that i triple verify so much anyway to true up.
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After many searches have found more normal verbage i guess for what i see(unlike the many fruitless searches for 'clock' mnemonic etc.) .  i think the formula etc. shown reference cosine for force more freely w/o overhead of fixed, formal coordinate system.  They do show more to a Linear 2D scope, that i then seek to take to radial as next tier complication, and would think another tier of complication to scenario of 3D to follow suit.  But again, feel that because rope only resists against loading on the long inline/not cross axis doesn't have the rigidity complication to take this from 2D to 3D force wise from a raw 'Alpha' force initiator with only diploid of cosine/sine especially round rope on round host.  Would need some unique axis 90 to the other 2 force AGAINST, but no rigidity to force this level complication (or linear unique axis device etc.)
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http://courses.lumenlearning.com/physics/chapter/7-1-work-the-scientific-definition
http://sciencetrends.com/the-formula-for-work-physics-equation-with-examples
http://drijal.com.np/2021/05/why-cos-theta-is-used-in-formula-of-work-done.html
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i think hauntingly give same logic i present, even to virtually the same formula logically deduced...

You have also referred to a paper authored  by Stephen Attaway ('mechanics of rope friction in rope rescue' ?) - and it seems that this is the source/inspiration for your use of trig functions?
Attaway was using trig functions to determine the contact angle (in radians) of rope against a metal surface (eg a belay device) - and this relates to the 'capstan equation'. He wasn't specifically examining a knot.

In other words, Attaway wasn't specifically stating that certain parts of a knot are 'sine' or 'cosine'. He never did this because to do so would be arbitrary.

This paper was very confirming and expanding for me, and a reputable doc to present, but don't really think he created the capstan formula.  It does show the same block push as i do, with the same tangent drag frictions at 90 degrees to target, in his words as builds from the linear model to the next tier of complication of radial.
The EXPONENT of the capstan formula who's base is the logarithm of 1(Euler's Number).
The exponent of compounding contains engineering coefficient of friction of mated materials that can vary, for nylon on steel, aluminum, wood even nylon on nylon etc.  then multiplied by Pi (to convert form linear coefficient to radial coefficient in my imagery) then multiplied by the number of 180 arc count sum (of even partial arcs added up to totaling degrees he shows).  Laws of friction would logically persist blindly not knowing if was in knot or not.
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There simply would be no reason for him to differentiate cosine and sine for the arc study for him tho please.
They are used together x tension in the deformity of the arc for seating to host is the key.
In deformity, as arc, the seating to host is fed with force= (cosine x tension) + (sine x tension)
BUT in linear rope parts only (sine x tension) powered rope seating to host;
>>also with lesser sine when on a linear faced host like 4x4.
So i break apart cosine/sine, to study the arc and the nonArcs in a HH for total package
>>where paper treats them as 1 , but narrower scope to the arc aspect alone. w/o need to differentiate
This difference to me is why arc frictions compound by degree, where linear frictions by distance.
EVERY physical displacement of space (or force as reciprocal displacer of space) can be defined as cosine/sine i think the Ancient stargazers tried to lend as an overwhelming principle that engulfs the planet, just as engulfs the heavens that in turn engulf the planet, there simply is in that model, no choice in the matter!

Capstan Theory covering the green 180arcs mostly above.  Real full host frictions only at top of host spar. The 'legs' from the arc are more linear controlled.  So frictions mostly by side force only, not the complete package of force carries.
>>Green arcs give full force carried thru them to feed frictions, don't need to sift cos/sin apart.
Thus nip would be better under arc of full seating force, but is in the worst nip position instead, only fed by side force.
The pic shows the 3 basic rope structure elements i try to show : arc_0,90,180.
The key to all is directions of pull from a linear force input type to the framework.
The magic 180 is most unique, as it is the only time that both ends, thus the whole 180 structure pull uniformly the same direction as a single rope part, including the created 2xPotential at arc apex interior side.

[agent_smith]

Hello KC,

I am taking a risk in engaging with you in this topic thread.
The risk is that you may perceive my replies as calculated to cause offense or to be insulting.
People are easily offended these days - so one had to be careful.

Also, there is a risk that this interaction between you and I will degenerate into a knowledge contest.

I also have great difficulty reading and understanding your use of the English language -  your sentence structure is awkward and your references to the 'ancients', 'heavens', 'planets', and 'electrical circuits' are all arbitrary and at times - nonsensical.

Be that as it may, I do recognize that you are passionate about what you write...


With the politics out of the way - I will restate some basic concepts in good faith (in point form, for clarity):

1. Trig functions are not forces.
2. Your application of trig functions (in the context of a 3D knot) is incorrect.
3. Trig functions require a coordinate system (eg 'x' and 'y' axis) in a 2D planar environment when calculating vector quantities.
4. You can also use trig functions to determine angles, sides of a triangle, and the length of arcs.
5. Trig functions are derived from the unit circle (Link: https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:trig/x2ec2f6f830c9fb89:unit-circle/a/trig-unit-circle-review )
6. Stephen Attaway (in his referenced paper) did not apply trig functions to a standalone 3 dimensional knot. He was looking at belay devices (a metal object) and rope friction created by rope contact with metal surfaces. He used trig functions to calculate the various contact angles. He was then able to apply the capstan equation (after knowing the contact angle in radians). You appear to be interpreting his paper out-of-context.

Note that a knot isn't 2 dimensional (2D). It is a 3 dimensional (3D) object.
To be accurate, you would need to apply spherical trigonometry to calculate angles, arcs and lengths on a sphere. In spherical trigonometry, the angles of a triangle do not add up to 180 degrees.
However, it is possible to slice a knot in cross-sections and then try to treat each section as 2 dimensional (planar) - however, this would ignore surrounding forces which co-exist and play a significant role in the way a knot responds to load (eg riding turns, crossing points, collars, etc).

...

Where do we go from here?

I think you will just continue to reply with further use of trig functions - and a long narrative.
I think you should review your understanding of what trig functions are and how they are used.

[KC]

i am sorry the 3 links did not show to you anything that i have been pointing at; even w/o co-ordinates for i believe there is a lot here.
>>including the ability to see into and dissect in real time outside of a lab.
i think it is inescapable that all physical displacements of space (or force) can be decoded into cosine/sine as handed down to become structural geometry seen here.
>>because cosine and sine are the direct and diffused expressions measurement of the total displacement of space and/or force.
>>degree, radian, gradian etc. are just even increments of change, but the more organic value is cosine/sine per increment of change
>>the 'weight' difference given to the change between increments as not static as the increments divisions.
The arc frictions science can not tell if inside a knot, on capstan or not, so would not know to alter it's science.
The radial/capstan theory can not be applied to the linear applications i point to in a knot, only the arc parts.
>>the linear rope parts are more like pushing the brick the paper shows near start, as grows to the radial complication of same.
180arc is unique in all materials, as the whole structure serves into the same direction; rope simply no different.
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Are you saying the humble HH has a 3D force structure?
Even tho not set up to take a 2D force pull structurally to lengthwise while also gripping the host?
Or other knots, or other standard simplicity to take a 3D load please??

[KC]

Here, the rope mostly works to hold the Load thru linear strength and it's efficiency of inline connection, but control other than the final inline anchor termination comes from the nonLinear deformities of the corners as proven with the dead eye re-directs.  Sliding a matching size and shaped host underneath the rope travel does not alter this except nominally with perhaps added trace frictions by this measure.
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A fully efficient linear means no force byproduct of secondary side force, thus only connection, no control over that connection.  So control dependent on utility of such further downstream.  As efficiency reduces, it makes room for byproduct of side force, to host seating, giving utilities of control.  Efficiency and byproduct force feeding controlling frictions are reciprocals that displace each other; and together make up the total force in scenario of single input force.
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As an overall pattern, chance linear frictions can give negligible control of the connected chain value.
But the real , reliable, major frictions to control are still at the deformities (corners in this case), not the chance side frictions byproduct to the linear work of the rope run.  The corner/deformities from simple/minimal/benchmark linear shape between the external facing input/outputs are the controlling features, the rest is just connection chain.
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Still persisting we have the same model of tracking work force by it's
A>Unique benchmark axis of nominal, minimal, inline, target directness lower floor limit that can never be less and maintain scenario
B>and deflected byproducts(to target work) off of the unique benchmark axis(of everything else that is connected in scenario)
Then how they are used separately and in combination of their sum total of the force(or force reciprocal of distance) volume.
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To this model, linear faced corners are not recognized, almost a separate system rudely, unNaturally forced in collection to utility.  Thus tighter scope to round hosts giving some linear, 90 and 180 elemental parts.  Linears are nonArc force connection of various efficiencies to same directional axis, 180arc also maintains same axis, but both ends pull in same/not opposing directions.  90arc changes from this power axis to a cross axis.

[KC]

Here we see the same as before, the corners start a new force line system, the arcs are continuing extensions.
The FULL rope tensions are used: (tension X cosine) + (tension X sine), note how if the cosine is NOT equal to 0 or 1 in an arc90, the sum is larger than the tension!
In a 90degree arc (or harsh corner)the sum will be greatest at 45 degrees of 1.414 x Tension potential, but at apex of arc180 has a potential of 2.00 xTension.

Rules even hold true for an almost round stop sign vs real round; segments between corners are new forcelines, but 180 is a continuous, organic flow vs the unNatural corner reDirect.  An arc180 can give 2xTension potential nip at apex, an arc90 only 1.414x Tension at apex that (shows the greatest potential).  These produced forces are against, not around host.  And so are a byproduct of holding against the Load.  These byproducts of seating to host then provide the Load controls of friction, nip and grip potentials, in addition to the more generic task of holding the Load.

The finished structure is dependent on the geometry assumed.  Usually in working with plastic or metal might soften(w/heat), pour the material into the mold and let harden.  If the shape is rectangular is much different than radial etc. in any material, rope simply no different.  We 'pour' the softened(unloaded) rope around the host form.  We can have potential for arcs if round host.  Then we forge to rope solid(by loading it); and the material X geometry provides support architecture. Only with rope we can't remove the rigid form, as is now host.


Degrees more of a visual increment, but cosine and sine the actual affect, expressed forces of that increment of change.  The latter controlling the knot, not the former/visual increment of change.  The eye can lie about such things, so use cos/sin as a decoder key for visual to actual.  Follow the forces has been the battle cry, and also watch the direction of those forces.
Defining w/o corners leaves 3 elements, express in terms of arc for concise, comparison: arc0, arc90 and arc180 all per direction.  The arc180 most unique for the endpoints and WHOLE STRUCTURE comprehensibly, orchestrate to move the SAME direction as a single unit/element.
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Even in the most minimal Half Knot, these things can be witnessed, with very few other things going, and minimal needs of this knot family/class revealed.

The disjointed/cornered host form strategies just don't work, and then this is inherited along with these lessons and many more by Squ?REef and the typical extensions of that fam, make a 'linear' RT/gauntlet of 3 arcs like rappel rack for surgeon's fam branch and at least 1 leg crossing for locks of SheetBend etc. to mitigate weaknesses of Half and later Squares etc. and bring from Bind to Hitch and Bend types etc.