General > Knotting Concepts & Explorations

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

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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.

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:
Where color changing fibres are sued.

Here is a link to some research papers that you may find very interesting:
From the above link - there are some nice research papers as follows:
[ ] Clove hitch research paper:
[ ] Frictional response of knots:

There is another research paper that I am currently reading here:

And I just found another interesting paper here:


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

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:

Many times in the blurring flood of rope forms/knots urges to verify "nip adjusted to bear at the top of the spar"(lesson#1663, fave chapter so also have registered /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  ( i believe these points are directly to this radial logic.  "longitudinal or perpendicular" i can go with as Right Angle Hitches and Lengthwise as one real basis of my studying.
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.
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.

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:
Here is a link for inverse sines, cosines, and tangents:
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:
Also here:
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:

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')?

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.
All the references to adjsuting nip to top in ABoK,  the 3 Half Hitches of different nip at start of 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.


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