Theory of cos/sine decoding host seating forces for nip positions etc.

[KC]

Radial nips, grips, frictions to control a load expect as 1+ xTension fed; so greater than linear(usually linears  MUCH less as <1 xTension host seating force)
>>also see that position in radial arc per input direction determines how the xTension of the rope is used, as radial force is by degrees/not distance :
nip force values translated for radial and linear
arc180 nip force = (2cos + 1sine) xTension (radials) theory
arc000 nip force = (0cos + 1sine) xTension (linears) theory
Cosine being organic force axis, vertical in gravity force load.  Cosine is single, unique sole/soul benchmark for rest of range to measure from/relations to.
Ancient's found cos as percentage of pure alignment(to single dimension) vs. sine as percentage of pure crossing(to other dimension totally outside of benchmark)

Seating force of rope to host powers nip force for a single position, same force also powers inside a range of frictions and/or grips, nip force is the single start position of rope seating usage.
A practical example of linear force into arcs and how changes nip is ABoK Lesson#0277 pg.049: Groundline also Groundline ABoK Lessons 1243 pg.224
>> Groundline used as Hitch showing how diameter of rope to host ratio can affect nip force in the arc
>> per the dismal-best nip regions given above


Sibling Bag Knot is similar but gets better start of Fig8 upgrade to Half-Hitch ABoK Lesson#1666 or Lesson#1668 fig8 upgrade to Timber Hitch both on page.290
>>in all 3(Groundline, Bag, Timber) get extra frictions, pressure, and spaced more towards arc apex with the strategy
>>Bag/Groundline ABoK Lessons #1242,1243 pg.224 can also use helpful correct slipped form to push nip towards TDC nip/best region
Especially like Double Slipped Bag as spacer when small rope/cord to host diameters.
>>similarly i view the final twirl in Timber/Killick as most positive nip trying to use the preceding twirls (each also a reducer) as spacers for the final twirl/nip to be in the best nip zone. i still give at least 3 tucks tho , not dropping a stitch cuz say can for #1668; and really plot for nip in the best nip zone as go, and generally can use the space of the 3rd tuck for that. i consider keeping the 3rd tuck minimum as a safety factor of using slippier ropes now, than the Naturals rope written for.

[agent_smith]

Thank you for the nice images.
Your use of 'cosine' and 'sine' makes no sense to me (sorry).

Sine and cosine are derived from the 'unit circle' and are mathematical functions.
In 2D space, all 3 angles of a triangle add up to 180 degrees.
But not in 3D curved space, because spherical triangles (angles do not add up to 180 degrees).
Also, trig functions are applied differently to 'right triangles' and non right triangles.
Note that Sine and Cosine have the exact same value at 45 degrees.
Note also the concept of 'complementary angles' - the cosine of one is the sine of the other (eg 30 degrees and 60 degrees).

I note that you are attempting to infer some type of relationship of a hitch with sine and cosine.
In fact, for all hitches, it is the capstan equation that plays the most significant role.
However, we need to first find the coefficient of friction between the rope and the host - which can be difficult to determine precisely.
For nylon rope in contact with alumina - 0.25 has often been used.

Note also that within a hitch structure, there may be 'riding turns' - where the rope overlaps itself (an example of which is the Clove hitch).
There is also compression.
So in a hitch, we have:
[ ] capstan effect caused by turns formed around its 'host' (where we also need to determine the coefficient of friction)
[ ] riding turns
[ ] compression
[ ] direction of force (alignment of a hitch with respect to its host - classically can be longitudinal or perpendicular)
[ ] the diameter of the rope relative to its host
[ ] the shape of the host (eg round bar, square, ellipsoid, rectangular, triangular, etc)

The combined effect of these physical factors defines the performance characteristics of a hitch.

...

A knot has a different mechanism to a hitch.

A hitch requires a 'host'.
A knot requires no 'host' - it is a self-supporting structure.

Applying a mathematical model to a knot is a difficult proposition.
In the first instance, we need to examine the knot structure - and identify all of the axes of force being injected to the core.
For example, in an eye knot (aka 'loop knot'), there are 3 axes of force.
In a 'bend' (an end-to-end joining knot) - there are 2 axes.

In my view, a thermal imaging camera can be a useful tool to examine eye knots and bends - to see in real time the heat signature of these knots in response to an injection of force.
You've probably seen this paper: https://bioinspiredoptics.mit.edu/wp-content/uploads/2020/09/Patil-et-al.-2020-Topological-mechanics-of-knots-and-tangles.pdf
Where color changing fibres are sued.

Here is a link to some research papers that you may find very interesting:
Link: https://www.epfl.ch/labs/flexlab/research/thin-rods/
From the above link - there are some nice research papers as follows:
[ ] Clove hitch research paper: https://www.epfl.ch/labs/flexlab/wp-content/uploads/2022/05/102_Sano_EML_CloveHitch.pdf
[ ] Frictional response of knots: https://www.epfl.ch/labs/flexlab/wp-content/uploads/2017/12/53_2015_Jawed_PRL_MechanicsTopologyFrictional-ResponseLongOverhandElasticKnots.pdf

There is another research paper that I am currently reading here: https://www.researchgate.net/publication/360186425_A_Discrete_Element_Method_model_for_frictional_fibers

EDIT
And I just found another interesting paper here: https://journals.flvc.org/UFJUR/article/download/128717/131755

...

I think I have given you enough material to digest and perhaps rethink your understanding of sine and cosine.

[KC]

TY for your response, own views and studies. i always really like the thermal/stress color knot photography/vids as hopefully the coming science; as like a view of internal specialty knot forces, blurring past the usual skin/generic view of rope as all the same/not the internal workings!
i guess for me cos is universal pattern of organic change, circle just being a form of this rule taken, still the same.
i think these knots must certainly follow the cos/sine math faithfully, for could get nowhere w/o; as would be then the only things that do not follow the universal pattern.
Here is how am stating this universality of these rules statically, with rope w/o knots, just principles knots must adhere to i think:

.
Next pic shows extension of theory to motion where can decode with cos/sine decode in both horizontal and vertical motions of change, even waveforms.  The described breathless stall is one of the things have not seen described before, and my own plea to read own senses some; for what have experienced from earliest memories; reaching for a more innate sense of these things.

.
  In machine works view of loaded knots/ropes, arc180 is the true leader, workhorse, force converter.  arc90 mostly converts to cross axis but at some costs of tension.  arc0/linears are simply connectors, that if not true/pure linear have nominal tension tax during operations.
i choose nip positions to start here, as a single position force at Bitter End from range of nip choices, in the most powerful arc180 for most magnified view.
>>then that same range of nip force choices itself collectively power frictions exponentially in capstan formulae, and with their own opposer give grip on host sandwiched with fullest force. Even w/arc180, if it has no arc180 opposer, can at most use sine for grip, to get full cos and sine usage w/arc180( like do for frictions and nips) need an opposing arc180.

The arc0/linears are therefore simpler, they have a simple stance of endpoints in opposing directions to define.
Linears connectors (to real more powerful functions arc180/90)are so simple also that they can NOT use both cos and sine together to a single function.  In rigids we see this as support column (cos) alignment vs. perpendicular lever (sine).  In rope/flexibles cos holds th load like down center of rope, and 90 to the side it the exterior of rope that can have host contact for nips, grips, frictions.  Cos can be used for nips, grips, frictions in arcs, not linears!!  This pic is not about the arc180 consistently shown but purely about the changing leg connectors to the arc:

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Many times in the blurring flood of rope forms/knots tinyurl.com/abok-online urges to verify "nip adjusted to bear at the top of the spar"(lesson#1663, fave chapter so also have registered tinyurl.com/abok-chap21 /right angle pulls).  And i think the normal, to mid to best nip of the HH series that open chapter are from the radial math i point to, at least that is what pointed me to it...  We also can see the radial force for nip math from squared linear faces even on deck in the previous 'hold fast' chapter  (tinyurl.com/abok-chap20). i believe these points are directly to this radial logic.  "longitudinal or perpendicular" i can go with as tinyurl.com/abok-chap21 Right Angle Hitches and Lengthwise tinyurl.com/abok-chap22 as one real basis of my studying.
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i too was taught complimentary angles swapping cos/sine.
But see it more helpful to view 45degrees unique as only matching cos=sine, center of 90 span, and greatest sum of cos+sine as median(only time cos+sine=1 is at the 0 and 90 degree points, all other positions cos+sine>1, and that complimentary angles are simply equidistant from the 45 median, and therefor mirror/flip the cos/sine vals.  The Fully aligned vs fully crossing give the right angle needed for the trigon-ometery, then the hypotenuse is the actual occurrence range, the cos/sine legs showing the percentage of full potential expressed at given spot on hypotenuse.  i also view the extremes as purebred potential expressed as 100% cos or sine, child position hybrids of % inheritance influence; between the purebred/parental points/potentials of full influence expressed.
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Been a long road, i wasn't looking for all this really; just simply what found.
Some of these drawings are different facets to view the same central truths thru to me.
Like forensic lighting of different colors, to catch different aspects to then focus on.

[agent_smith]

Hello 'KC'.
Thank you very much for your reply to my post - your work is appreciated.

I'm sorry - but your use of 'sine and 'cosine' with respect to a hitch still makes no sense to me.
You appear to be attempting to assign trig functions to segments of a hitch, without defining a coordinate system.
To ensure that we are both defining trigonometric functions in the same way, here is a link that explains what Sine and Cosine is: https://www.youtube.com/watch?v=vuoNyvMvDtA
Here is a link for inverse sines, cosines, and tangents: https://www.youtube.com/watch?v=9jnxmoxqu5E
I am proceeding on the basis of the definitions published in these videos.

I would point out that knots, hitches, and bends are not 2D planar objects, they a 3D objects that exist in 3D space.
A basic overview of 3D coordinate reference frames is here: https://www.youtube.com/watch?v=5sJdfciNM20
Also here: https://www.youtube.com/watch?v=6D-MEb-599A
I base my understanding of 3D coordinate reference frame on this video.

I am also confused about your use of trig functions to explain the properties of a 'hitch'.
That is, I am unclear as to what exactly are you attempting to define?
Are you modelling the propagation of tension force from the S.Part into the core of the 'hitch', and then mapping its dispersal and transformation into something else?

I also note that you only appear to be modelling hitches - which by definition require a 'host'.
I have not seen your use of trig functions with respect to 'knots'.
Here I am defining a 'knot' to be a self-supporting structure (ie requires no host).
And a subset of a knot... an eye knot - has three axes of force injection into the core.
Whereas a 'bend', only has 2 axes of force injection into the core.

Given that knots are 3D objects existing in 3D space, if you wish to map segments/components, it seems that you need to use a 3D coordinate system and spherical trig functions.
Here is a link to spherical trigonometry: https://www.youtube.com/watch?v=hcXbLRPq5vc

Are you able to apply your sine and cosine trig functions to 3D knots using a 3D coordinate system (ie eye knots and bends)?
Can you explain how tension force propagates through knots using your Sine and Cosine definitions (I'll include 'hitches' in this request - since it is a subset of 'knots')?

[KC]

Agent Smith, et al,
For me, Hitch/termination of force flow thru rope to another device as a function (not necessarily a knot name) is logical start to ID separate items rope and host as focus, yes.  Clean, perfect loaded line, then node swell at termination.
Then expand to Bend/continuation of force (usually thru a jointed node of deformity from pure linear to either side) i see as kinda dual sided Hitch, only each is the other's host (usually).  To this imagery i find we have 2 versions of HH : 1 SPart to end termination and 2 competing (as if ) SParts to shared internal termination/0point between pulls model.  But are not each other's host, but rather each other's stopping point shared Zer0 point where Termination HH ends too on it's own.
Knot as a standalone, i see takes it's own self as host; but once again am trying to stay to loaded rope reference as trace force; so Bowline eye/SPart would be loaded. The 1st 2 youtube vids are definitely part of my self educated background.  2nd 2 less so, ty.
Some 'electric' symbols made in the past of imagery of Hitch vs. Bend etc.

.
Focused linear/dispersed from center evenly to all axises(only 1) in 1D (    differentiated model) vs.
diffused radial/dispersed from center evenly to all axises(multi )  in 2D (undifferentiated model)
Can make a big difference in all these things.
i find an arc to be an organic continuous flow on radial face on a spar, but a 4x4 host presents segmented flow of non-organic, sudden, harsh reset/not flow of force at corners.  The force runs parallel to 4x4 host on the linears, so no host seating pressure pre-exist for nips, frictions and grips until deformity from pure rope line at corners. Drop in cos of pure linear, no inwards to host force direction, to drop in cos>>raise sine to give frictions, nips, grips.   Linear rope part is just extension then between the more active machine conversion points of corners here, again.  Length of face doesn't matter force change wise, is just an extender.  You can keep the corners and remove faces to same math force wise.  To this imagery, radial is more a gradual flow of deformity so gets some host seating forces at all points, but not a reset of the linear force flow thru in big picture.

Previously you had mentioned diameter ratios of rope to host; i hope 4,5 in pic above speak to that some along with the previous post Groundline pic.
i do recognize also aforementioned rope crossings in this model, greater sandwiching lesser to host giving more hitching force against sandwiched layer, lesser over greater giving more firmly pasting the sandwiched layer to host.  In all same rope, the lesser is softer than the greater(i call tensioned rigidity); the more rigid greater (tension) does not dent the lesser even if bent around for more force than tension and deforming more rigid greater.  The lower tension softer would dent if any.  Pinch into ropePart deforms so more positive 'lock' than pinch onto more topical rider just trapping/pasting where no to deforming/denting.  Kinda a model of can lengthen bolt cutter leverage for more power, but softer jaws to deliver that force are still softer jaws; arc can increase leverage pull beyond tension, but still tension sets the rigidity.
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All the references to adjsuting nip to top in ABoK,  the 3 Half Hitches of different nip at start of tinyurl.com/abok-chap21 and the way the forces change in Sailor Hitch as adjust nip helped me to my radial view of forces maintaining until swap from external linear input to internal radial input(against radial swell already inside controlling arcs/needs no SPart pull in usage).

.
Family of Constrictor, Bag, Groundline show no matter what name or position in landscape given, the forces i try to follow define the same, except differently if used as a Hitch as linear external force thru SPart to machine(s) of arc(s) vs used for binding against diffused radial spread as internal force spread to same machine(s) of arc(s).   The rope in this imagery thus, is passive material; that force to watch is ported thru is the pivotal game changer quantity.  The focused linear input of Hitch usage gives decreasing tension (and thus rigidities) thru the rope, and a focused aspect to where greatest nip is.  By contrast, thru same architecture  radial bind against swell gives same tension evenly around to nip, and all nip positions thus equal from the evenly diffused radial input to radial arcs.  The difference is autonomous and cares not where the coordinate system of rope is for the differences, just the matters of force.
.

[agent_smith]

Hello KC,
I have attached an image of a knot within a 3D coordinate system.
I have set the y axis to be 'up', and the z axis to come out of the screen (for depth).
I did this to keep the familiar x axis in the horizontal direction.
I have also attached an image of a loaded #1010 Simple Bowline.

Trig functions require a coordinate system - you cannot perform calculations using Sine and Cosine without defining a coordinate system (either 2D 'xy', or 3D 'xyz').
Here is a link to another youtube channel that defines Sine and Cosine in relation to a coordinate system.
Link: https://www.radfordmathematics.com/functions/circular-functions/definition-cosine-sine-tangent/definition-cosine-sine-functions.html

I am defining a 'hitch' to be a structure that requires a separate 'host' object.
If you remove the 'host' - a hitch loses structural integrity (and usually collapses).
The 'host' can be external or internal (most hitches form around an external host).
There are sub-categories of hitches, eg:
[ ] slide and grip hitches (eg Prusik hitch)
[ ] binder hitches (eg Clove hitch, Constrictor hitch)
[ ] noose hitches (eg Gnat hitch)
[ ] load control (eg Italian / Munter / HMS hitch at #206)
[ ] termination hitches (eg 'tensionless hitch' #2047, and 'round turn and 2 half hitches')

All require a separate external 'host', with the only exception being 'noose hitches' which supply their own internal 'host' - usually the S.Part
This is a special case where the internal host (S.Part) is linear - allowing the core of the hitch to slide along a axial/linear pathway - which enables the hitch to act as a noose.

...

I define a knot to be a fixed self supporting structure that requires no host.
I am careful to make distinction with the subset of hitches that are nooses.
A knot is a fixed geometric structure that retains its form under load.
Knots can be distinguished as being either eye knots (aka loop knots) or bends (aka end to end joining knots).
Example 1:
I define #1010 Simple Bowline to be a type of knot (in fact an 'eye knot').
It is not a hitch.

Example 2:
I define the Zeppelin bend to be a type of knot (in fact a bend, or 'end to end joining knot').
It is not a hitch.

...

I had hoped that you would provide information about how you apply Sine and Cosine to knots (eg eye knots or bends).
To that end, I have attached images to assist you for that purpose.

[KC]

i remember trying to put this together for force in a vertically loaded world with the horiz cosine references connected to other researchings etc.  Then found some that seemed use cosine for vert force (pendulum, rigs etc.) to confuse more;.  It is not the intention to share the confusion part.
The cosine pattern of change for extending a length, force against, wind, light, shadowing, sound fade etc. exist natively all the time; that we chart to track/show.
Trying to keep up with your references, guess you are still reading mine.
For this ask/point i made this:

Should as like show store is just as far from house if map from house, store, local, city, state, national level; same story/relationship told.
i don't dictate the organic cos position, but rather read it/set by it; don't inset Bowline in graph table but table to Bowline.
Bowline will try to pull SPart inline to center eye, but Bowline build is off balanced as swells to only 1 side un-mirrored.
Continuing/lengthwise/longitudinal Half Hitch in Bowline is more advanced topic, but then more firmly proving the linear force premise too i think.  Wasn't going to jump straight to it. 
.
i borrowed from other things have known, what i consider most expressive words, models etc.
One of these is displacement, displacement volume from engines.
>>Things displace against open space as travel or displace against each other as force(or to some quantity of each), so same math as force is reciprocal of distance lost/traded/displaced against. 
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But also from drawing, animation, virtual reality etc. term :Organic
>>virtual reality etc. looks, sounds, feels fakey, etc.unless correct easeIn/easeOut is used for sound, size, speed, shadowing etc. changes.
>>these are cosine/sine scales, that i played with programing into stuff before was a push button service/all super precalc'd filter.
All the ropes in drawings are raw, flat white.  The shadowing is an easeIn/easeOut logic they call out as correct as is organic.
This only echo'd out even more to me how pervasive the cos scale was organically in these things as rate of change; matching rigging experience etc.  For have seen cos/sine in waveforms, engineering diagrams, circles etc.
So borrowed their word 'organic', for when is naturally correct; accepted by animal/survival eye, as to bypass brain filter drag you in at deeper depth/trick brain guardian and make feel like are in the action etc. 
The hang of the pendulum became the organic position to me, farthest reach down and fastest position(even standing still!) as what i'd show as cos1, benchmark and revealing these vertical reach even statically; and in motion force, speed potentials @ cos1.  cos0 at breathless stall of Zer0 speed, force, vertical reach down within the motion domain.  Wrench pendulum from balanced, stabilized cos=1 position to any other direction makes cos <1 + sine>0.  For there is only 1 unique, organic, mid span axis is cos, but can choose many sine axises around it.  Center of spring or piston travel cos1 organic mid balanced position, projected travel of bullet is cos1 of sine0, any drop reduces cos factor as takes on sine etc.  Been reading it like that for years, using clock face for instant guesstimation on cos and sine, some tan to plan, jump out of the way, diagnose etc.  Cosine scale(and reverse antagonistic/displacing scale of sine) patterns are how they are played out, and things make a certain logical sense of flow if read/decoded as such.  Well enough to predict, adjust and then re-predict as if to command the happening; thus quote in sig.

[KC]

Bowline example kinda jumps ahead past simplers to start, as has the HH 'nipping loop' that i think doesn't follow normal expectations of force path reading/decoding.


So part of my linear imagery/logic proof is one that seems to break traditional thinking of tension degrading thru the friction path of a knot. It starts at one end at full bull and recedes to Zer0 in sequential pattern.
Also in building my own understandings, an arc90 changed the force flow from present reign of loaded axis across w/'Samson Angle' at cos0 /perpendicular angle , then continues to a new, now restated loaded axis for rest of following ropeParts.
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Not so i say with the continuous/both ends loaded form of Half-Hitch(HH) as put on nose of spar to pull to convert Timber to Killick etc., also in Bowline and Sheepshank as possible fail point of HH rolling out. The force instead seems to stay inline more from input of 1st arc90 passing to output of 2nd arc90. What seems to break the rules, instead screams the same ones back louder, the tension is least midway here, only peripherally loading grip around spar, as most tension favors the simpler, lesser as if 'electrical' short path. This is lost if the 2 hooks of arc90s un-hook; then we see the more expected gradual progression of greatest, mid and then least tensions working end2end like most knots. This then changes from a 1x pass of linear force bypass the arc180 mostly, to then fully loading 2 now arc180s, w/ 2 xTension and bypassing to 'crunch' thru a rope type host.



i think the force flow pattern changes here to more sequential/normal; only after unhook/rollout of the HH where loses the more direct path end2end before unhooks.
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i have many electrical path/logic theories for rope etc., under theory of force is force/electrical or not and it is all force flow conducted thru a device, even if not a wire conducting device.

[agent_smith]

Hello KC,
I have attached what I think is an improved xyz coordinate reference frame.
The depicted knot is #1010 Simple Bowline.
I am defining this structure to be a 'knot' - it isn't a hitch.

I have centred the zero point of the axes where the S.Part enters the core of the knot.
I have oriented the xyz coordinates so that the y axis points 'up' and the x axis is 'horizontal.
The z axis comes out of the screen.

This (in my view) should make it easier to conceptualise that all force is aligned in the +x / -x axis direction. The 100N on the +x axis is balanced against the 100N on the -x axis - so that the knot is in equilibrium. 100N force on -X axis is evenly divided into 50N on each eye leg.
The tail segment (the portion of rope that exits from the core and terminates at an unloaded end) has zero load.

Trig functions (eg Sine and Cosine) can only have meaning within a defined coordinate reference frame.

Experimentation and empirically derived knowledge tells us that the Bowline does not slip and fall apart under constant unchanging load - it remains intact.
Tension force is distributed through the core, a significant portion acting on the 'nipping loop'.
Given:
[ ] there is 100N load on the S.Part
[ ] there is 50N load on each eye leg
[ ] there is zero load on the tail segment.

Proposal: Model how force is distributed through the knot core.

EDIT NOTE:
I have added another image with knot components labelled.

[KC]

Thank-you Agent Smith; to me that would be fair to show the equal/opposites balance at SPart instead of carabiner/eye, but still the same unique loaded axis reveals to same maths; and deformities from this singular, sole, unique benchmark like nipping loop path deformity etc.  These same structures, in any other material could be shown as cos/sine in their weight bearing for a home etc.
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i have used the cosine pattern etc. just as pure alignment relationship % vs. crossing relationship % most my life, it simply works.  For me the lesson of the Ancients is these relationships are in anything that 'displaces' 'against' totally open/freespace as travel, or if stopped as force against travel(displacer vs. displacer instances).  To be applied even in parts of so much travel and so much resistance at same time as force and travel concurrently incurred.  They found these in star/planet path as a universal 'organic' pattern, so also applying to us here.  They showed these cycles , seasons/calendar, clock and circle itself all on clock in their base60 maths. 
.
i can go on and call these things something else besides cosine etc. because not on coordinate system, but same scale of change (cos) universally applys same maths by any other name too. 
In equation cos xTension = Load 
cos is just another math factor we can solve for if know the other 2 factors, even w/o co-ordinates.
>>cos is the efficiency of alignment that xTension is used to perform work to hold/move Load
A point of balance in a relationship between parts, = as center pivot of the balance showing needle to Zer0 favoring neither side.
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If can see everything in the cos/sine aligned/crossing perpendicular pairs sense of total influences, the load would pull down the inner length of a rope equal to or less than xTension and 90degree deflection to outer side as then less than xTension.  Linear can only use this lesser xTension, more nominal  sideForce byproduct of deforming from pure line as nip etc. and then only in 1 direction at individual point(s), so nip must be (generally much)less than xTension. Just as when sweat/swig a line, the line tension is the greater force, not the (baby by comparison)cross force you impose. Radial nip by contrast can also use the greater load force with the more nominal side force for nip, generally to a sum greater than xTension itself, and also over a range in multiple directions.  Literally the radial nip is as like the linear nip strategy on steroids.  Same for linear vs radial frictions and grips.  Radial is the machine in ropework of power to change, linear is more transfer at perhaps some nominal cost if deformed.  In linears cost byproduct/tax over and above normal loading can then be capitalized on as nips, grips, frictions w/host seating sideForce byproduct, but not the primary load force orchestrated into host seating factors until corner or arc deformities i think.  Corner at a point to linear faces for greater nip at apex of corner deformity only or arc over a range.
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i always thought the ship peg lessons of tinyurl.com/abok-chap20 'Belaying & Making Fast' showed this best/even as a conniving fight for it.  As to even turn 2 perpendicular linear faces(face and 90degree shoulder/shelf to/on) to then extrude radial nip, and the fails if don't size properly etc. showed this best, cleanest/simplest(fewest ropeParts and functions); and even with arc90's of lesser than arc180 nipPotential.  These simplest rigid host (as opposed to soft/rope host) single, simple Hitches, are real base examples i think.  Ones that calls Slippery Hitch only after makes Bitter End a bight as to slip; to remotely dance/shake/flirt an unloaded rope free.  Working  End (WE) does not bypass SPart but serves immediately under it's greatest primary arc, w/only an arc90.  But orchestrates secure nip in minimalist setting of fewest players; as study starting point for me.  In next chapter defines right angle grab around host as 2 types:  bypass SPart comes back around it to secure/nip around SPart as 'loose' Hitch or nip off around host instead as 'snug' Hitch, of which these Single Hitches are neither.  So is another class in this chapter_20 set apart just before the right angle chapter_21 that leads to the lengthwise chapter_22 in logical flow.
i believe these to be some of the most simple, minimal examples to openly see the most basic hitching requirements of hook, tension rigidity reduction and nip.

Basic 3, all lacing wise match, but sizing and positioning/alignment of the ropeParts is still critical

Other simple, single Hitch on rigid host would be tree and hook(Blackwall), but must be sized right ratios, especially hook to work.  Then takes strategy to soft host of rope in Sheetbend etc.
i like groups of 3 that can cover all, like :arc0,90,180 (binary choice + the non)and
for knot class, while loaded i go by opposing SParts to feed focused linear force
>>then into machinery of arcs of dispersed radial force, after conversion (loaded/in use for simple models):
0:Binding(already force contained inside arcs),
1:Hitch(termination of rope force),
2:Bend(2 opposing in continuance of rope force/coupling)
(again ternary/3)
>>in node deformities from pure linear rope line.

[KC]

To me the cosine pattern of change is universally organic, graph paper tool helps see what is already there, as a 2D ruler; but lends no change to scenario.
So cos is in all things, even these knots/rope workings.  i mite not have applied something right in knots Agent Smith?/but the pattern itself must hold i must maintain.  This post is how i envision going from simpler, set, hopefully more accepted, focused linear 'path' to radial 'path' to same math, cos:sin pair and logics of why/how .
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Below pic is what i see from the standard linears of sideForce per tangents xLoad. In linears only the sideForce powers host seating to then get frictions, nips and grips controls to use i think.  But in the alternative, more organic radial path can also use the xLoad axis force dimension with the sideForce axis dimension for greater host seating force to power the same rope controls.

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More theory i guess:
Travels to a same geometry of cos:sin value pair can only be either linear or radial. 
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problems posing past here
preview is fine.
A-ha, oh know, does not like the hold alt down and type 0176 on keypad for degrees symbol, used much, am fixing.  If add one, to this, and save will preview fine but give err on save..

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Linear is fixed cos:sin vaule pair per flow.
Another linear must be different cos:sin value pair, with harsh, artificial reset/corner to this linear from the other in between them.
Full Radial path between crossing dimensions gives organic range of ALL cos:sin pairs per flow by extreme contrast; always and all ways try to show as focused linear vs dispersed radial.  This covers all paths for the value pairs are simply either fixed(linear), or not(radial) in a given 'flow', simple organic path.