TiraMissU

Torus Knots (Planar)

As long as the world is turning and spinning, we're gonna be dizzy and we're gonna make mistakes. --Mel Brooks 

Rotational Matrix Take II

Tommy's Interwoven Paths In His Past and Present Life
In His Home
Raiding His Father's Refrigerator
I See Him Standing by the Fridge
His Father Said the Pot Flew Out of a Pantry Kitty-Corner to the Fridge During a Crisis Some Time Later
His Kitchen Is Just About Square
And Tommy's Path Seems to Be On a Diagonal , His Orientation Turned Around

I take it Yous guys are less than impressed with my arithmetic derring-do, since yer all so focused on her looks (not my looks, her looks, because I'm not supposed to take it personally).

Based on the above Scenario, our brave young soldier Tommy goes into his kitchen (at his father's house) just days before he was supposed to report to boot camp for deployment to some vacation spot.

Like any other young man, he goes through the kitchen door and heads due East to raid the refrigerator.

Surely there must be something to eat!

Some time after, Tommy dies of dehydration because his drill Sergeant couldn't allow a break for water during a strenuous exercise before ever getting deployed. Which causes Tommy's father a great deal of distress, and makes his good friend wonder 'what is it about men that makes them act like accomplished women?'

And during a pivotal moment in his father's coping mechanism failure, a sauce pot comes flying out of the NW pantry of the kitchen (no earthquakes reported in that part of town on that particular occasion) and lands right in the middle of the square kitchen (6 ' x 6').

Determine the rotational matrix based on the above scenario.

 (A bad drawing to kind o' confuse the issue (the planes are offset along a central axis, much like a helix/screw threads/or augers), think 3D and not certain the time).

{"Sit on it...and Rotate!"--The Fonz (Henry Winkler), popular 1970'a sitcom Happy Days

The Rotational Matrix

In Euclidean geometry, a rotation is an example of an isometry, a transformation that moves points without changing the distances between them. 

Rotations are distinguished from other isometries by two additional properties: 

1. they leave (at least) one point fixed, and 
2. they leave "handedness" unchanged. 

By contrast, a translation moves every point, a reflection exchanges left- and right-handed ordering, and a glide reflection does both.

A rotation that does not leave "handedness" unchanged is an improper rotation or a rotoinversion.

A geometric rotation transforms lines to lines, and preserves ratios of distances between points. A rotation is a linear transformation of the vectors, and can be written in matrix form, Qp. 

The fact that a rotation preserves, not just ratios, but distances themselves, it follows that...

Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, all these properties may be summarize by saying that the n×n rotation matrices form a group, which for n > 2 is non-abelian. 

Called a special orthogonal group, and denoted by SO(n), SO(n,R), SOn, or SOn(R), the group of n×n rotation matrices is isomorphic to the group of rotations in an n-dimensional space. 

This means that multiplication of rotation matrices corresponds to composition of rotations, applied in left-to-right order of their corresponding matrices.

To rotate points in the xy-Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system use a rotation matrix R; each point is represented by a column vector v, of each point's coordinates (position). The rotated vector is then obtained by matrix multiplication Rv. 

Rotation matrices are square matrices (nxn), with real entries; characterized as orthogonal matrices with determinant 1.

The special orthogonal group SO(n) is the set of all such matrices of size n, generally not commutative.

Improper rotations, characterized by orthogonal matrices with determinant -1 (instead of +1) combine proper rotations with reflections (which invert orientation). 

The direction of vector rotation is counterclockwise if θ is positive (e.g. 90°), and clockwise if θ is negative (e.g. -90°).

Basic rotations:

Three basic (gimbal-like) rotation matrices rotate vectors about the x, y, or z axis... 

All other rotation matrices can be obtained from these three using matrix multiplication. 

...to represent a rotation whose yaw, pitch, and roll are α, β, and γ, respectively. 

...whose Euler angles are α, β, and γ (using the y-x-z convention for Euler angles).

Every rotation in three dimensions is defined by its axis — a direction that is left fixed by the rotation — and its angle — the amount of rotation about that axis (Euler rotation theorem).

There are several methods to compute an axis and an angle from a rotation matrix one is based on the computation of the eigenvectors and eigenvalues of the rotation matrix. It is also possible to use the trace of the rotation matrix.

...which shows that  is the null space of... 

Viewed another way, [] is an eigenvector of R corresponding to the eigenvalue; every rotation matrix must have this eigenvalue.

To find the angle of a rotation, once the axis of the rotation is known, select a vector  perpendicular to the axis. Then the angle of the rotation is the angle between__and__.

It is much easier to calculate the trace (i.e. the sum of the diagonal elements of the rotation matrix)...

In three dimensions, for any rotation matrix , where a is a rotation axis and θ a rotation angle,

 (i.e.,  is an orthogonal matrix)

 (i.e, the determinant of  is 1)

 (where  is the identity matrix)

The eigenvalues...

The trace of  is  equivalent to the sum of its eigenvalues.

Some of these properties can be generalized to any number of dimensions. In other words, they hold for any rotation matrix...

Ambiguities:  Alias and alibi rotations

The interpretation of a rotation matrix can be subject to many ambiguities.

The change in a vector's coordinates can be due to a turn of the coordinate system (alias) or a turn of the vector (alibi). Any rotation can be legitimately described both ways, as vectors and coordinate systems actually rotate with respect to each other. Here, the alibi approach is used to describe rotations. 

The vector can be pre-multiplied by a rotation matrix (Rv, where v is a column vector), or post-multiplied by it (vR, where v is a row vector). Here rotations are produced by means of a pre-multiplication.

The vector space has a dual space of linear forms, and the matrix can act on either vectors or forms.

In most cases the effect of the ambiguity is equivalent to the effect of a transposition of the rotation matrix.

Two features are noteworthy: 

1. First, one of the roots (or eigenvalues) is 1, i.e., some direction is unaffected by the matrix. For rotations in three dimensions, this is the axis of the rotation (a concept that has no meaning in any other dimension). 
2. Second, the other two roots are a pair of complex conjugates, whose product is 1 (the constant term of the quadratic), and whose sum is 2 cos θ (the negated linear term). This factorization is of interest for 3×3 rotation matrices because the same thing occurs for all of them. (As special cases, for a null rotation the "complex conjugates" are both 1, and for a 180° rotation they are both −1.) Furthermore, a similar factorization holds for any n×n rotation matrix. If the dimension, n, is odd, there will be a "dangling" eigenvalue of 1; and for any dimension the rest of the polynomial factors into quadratic terms like the one here (with the two special cases noted). Its a given that the characteristic polynomial will have degree n and therefore n eigenvalues. And since a rotation matrix commutes with its transpose, it is a normal matrix, so can be diagonalized. 

Every rotation matrix, when expressed in a suitable coordinate system, partitions into independent rotations of two-dimensional subspaces, at most n⁄2 of them...

The sum of the entries on the main diagonal of a matrix is called the trace...

The large number of options is possible because rotations in three dimensions (and higher) do not commute. 

Reversing a given sequence of rotations results in a different outcome. 

This implies that two rotations cannot be composed by simply adding their corresponding angles; i.e., Euler angles are not vectors, despite a similar representation as a triple of numbers. 

} 

(The rest of us just spin on our heels and walk a distance about 6Ö2 along the diagonal heading WNW if we think we are at the fridge or ESE if we think we are at the pantry and want something out of the fridge.

Or become really well acquainted with quaternions to get a sense for what Tommy was seen doing in his 'spare time'. )

Like I said, some people say, 'Thank You,' and some people can be quite demonstrative of it.

Just like those 'I love you' tokens I been hearing of late.

If You Love Me, you would stop this (You Know Exactly what I mean).

Surah 2 Al.Baqarah (The Calf)
وَلاَ تَقُولُواْ لِمَنْ يُقْتَلُ فِي سَبيلِ اللّهِ أَمْوَاتٌ
 بَلْ أَحْيَاء وَلَكِن لاَّ تَشْعُرُونَ (2:154

2:154 And do not say of those slain in God's Cause that they are dead;
Nay, they Live and yet you perceive it not.

I know, I know, it must be End of the World because i am the smartest girl in the room; don't anybody panic because God can make another dysfunctional one just like it!

Of Maps and Legends, Innies and Outies

"There must be a proper theory of place other than 'To Find Kalamazoo, look in F3'."--from Euclid's Window, The Story of Geometry from Parallel Lines to Hyperspace, by Leonard Mlodinow

You Are Here...There...Everywhere

To determine Latitude, imagine a Little Earth, and 3 Little Points. {This should sound familiar to You, since You dreamt it when You were 8}.

Take one point P1 and put it at the North Pole, and the second point P2 and put it at the Center of the Little Earth, and use the third point P3 to Mark where you are on the Little Earth.

Now draw a line from the North Pole straight through the point at the Center, to the other end of the Little Earth. This First line, Line 1, forms the Little Earth's axis of Rotation.

Then draw a line from the Point where You are standing through the point at the Center of the Little Earth, Line 2,  to form an angle, α , with the Little Earth's axis of rotation, that first line You drew.

The two lines act like a Protractor and that angle α is a generic way to determine Your Latitude.

Determine Latitude Using
The Little Earth
A Portion of the Fano Plane

Which is a great way of thinking of where You are, but will not help Us find each other.

For that We have to imagine a Big Earth (Way Way Bigger than Our Little Earth). Something on the Cosmic scale, that would be the Celestial Sphere, the map of Stars that circles the globe.

Imagine the Big Earth around the Little Earth, the globe that is now the Core (where You are standing & spinning) of the Celestial Sphere.

This will help to determine Longitude, sinceYou are not a Flatlander and need at least these 2 coordinates to determine where You are exactly in 3D.

Remember, since the Earth Rotates 360 degrees in 24 hours, the Stars over Your head are going to move about 15 degrees every hour, so that someOne standing West of You will See the Same Sky an hour later. Which has direct associations to the Fano Plane (diagram below. see also 'kissing circles' and Appolonian Network):

If You are at the Equator (to help me do the math) that translates to distance of about 1000 miles in that much time.

So, You look up at the Sky over where You are standing, and pick a star--any star, that You may be familiar with and note the local time it rises above the horizon, its Right Ascension (RA).  The reference, or baseline is the star γ Aries at 0 hours RA, the First Star in Aries where it crosses the Celestial Equator on the First Day of Spring.

Since there are Apps for this, just point Your PDA at Triangulum Australis, I think He Said 'The Second Star on the Left' and make a note of its RA in Hours-Minutes-Seconds (this presumes that You may be somewhere in the Southern Hemisphere of the Little Earth). {But if I got that wrong, since I heard this while under duress, and wish I had written it down to be sure so many years ago, make a note of the Second Star on the Right RA , too.}

Triangulum Australis
A Constellation in the Southern Hemisphere

No matter what, He Will Find You. You already know His Voice, so don't think too much about what He may look like, since there is usually some type of distortion due to Projection, but I can assure You, He is either Your Heavenly Father or Your Brother, and Either One will take You where You belong--to be with the Ones who love You.{Knowing that no matter who or what it is You think is going on around You, and what ever disinformation or misinformation is propagated through rumor mills or media channels, they are collectively  Nothing but Remnants, Effigies, Residue of things and people gone by, because they lie in order to sever Our Ties of Kinship and in so doing work corruption on the Little Earth

فَهَلْ عَسَيْتُمْ إِن تَوَلَّيْتُمْ أَن تُفْسِدُوا فِي الْأَرْضِ وَتُقَطِّعُوا أَرْحَامَكُمْ (47:22  
47:22 Would you, perchance if You were given Dominion in the Earth, work corruption and Sever Your Maternal Bonds (bonds of kinship, literally, 'sever/cut off Your wombs/matrices/'Places of Mercy')?

Which is exactly what they do when they spread their conspiracy theories of UFOs  and aliens.

Recall that God made everything first, then He Created the Human Being--meaning that the Human Being was to be the Ultimate Creation--meant to Evolve or Reach the Highest form of Creation, but now what have We?}

And the reason that no matter how far You wander off or how big the Universe gets, it's no big deal for Him to find You?--The simplest, 2D planar point group, the Fano Plane PG(3,2)!
{I just answered my own question, must be another from of insanity}.

Can You imagine how it feels to an ant when it ventures off that ant hill? Apparently, even the ants know that As Above, So Is Below...{oops, did it again.}

And He Probably Triangulates with nothing more sophisticated than using 2 Dimensional finite geometry to wend His Way throughout the 'flat, not moving Earth' and the infinite (Multi-Dimensional) Cosmos--->

What is the simplest, smallest, ...perfect universe...a complex universe of breathtaking, abstract, beauty...consisting of only 7 points, 7 lines and their dual? 

{I'll let someone who knows what He's talking about answer that one..} 

According to one Mathematician, (paraphrasing): 

'The 2D planar point group, the Fano Plane PG(3,2) is......the smallest, perfect universe...a complex universe of breathtaking, abstract, beauty...consisting of only 7 points and 7 lines; the smallest project space over the field Z2  with 15 points,  35 lines and 15 planes.  

A space whose overall design incorporates and improves many of the standard features of the 3D Euclidean space we think we live in..the smallest 3D projective space...that plays an important and core role in many fields of mathematics, inlcuding combinatorics, group theory, and geometry'--Burkard Polster, 2001

In non-Euclidian geometry parallel lines meet (intersect) at a point (just like when drawing a perspective of train tracks on paper, they seem to meet at the 'vanishing point'), and that is why the Fano plane is a good 2D small-scale model of the larger 3D Celestial Sphere with the Little Earth at its Core (Your 'Big Earth/Little Earth' dream).

In fact, the Fano plane is the smallest projective plane and probably why the Pyramids were built and are still standing. And it is no mere chance that the projective plane is sometimes called a 'twisted sphere'--'a surface without boundary derived from a usual plane by addition of a line at infinity. 

'Just as a straight line in projective geometry contains a single point at infinity at which the endpoints meet, a plane in projective geometry contains a single line at infinity at which the edges of the plane meet.  

Lio Comic Strip August 28, 2012
Escherisque 

A projective plane can be constructed by gluing both pairs of opposite edges of a rectangle together giving both pairs a half-twist (The Reverse would be Unfolding such a surface). It is a one-sided surface, but cannot be realized in three-dimensional space without crossing itself.'

Projective Plane From A Sphere

A finite projective plane of order  is formally defined as a set of  points with the properties that:
1. Any two points determine a line,
2. Any two lines determine a point,
3. Every point has  lines on it, and
4. Every line contains  points.
(Note that some of these properties are redundant.) 

A projective plane is therefore a symmetric (, , 1) block design. 

An affine plane of order  exists iff a projective plane of order  exists.

Bizarro  Comic August 21, 2012
Innie_ Audi Belly Buttons

The 3 points at each vertex of the triangle defining the Fano plane, represent these vanishing points (You took technical drawing classes so You get this).  They serve to define 'Outside' because, as You know, there is no way to define the 'Inside' without SomeThing or SomeOne being on the 'Outside'. {kind of like there is no way to know Up from Down in outer space; now, You know why You were so facinated by your belly button when You were 2 years old}.

But since we are still biding our Time in the 'Real' world, having missed our ride out of here so many years ago, I suspect we are to put this to more pragmatic, concrete use than just appreciating its abstract beauty---I mean, really, what do mathematicians know? --they, too, are still stuck on the ground.

Fano Plane, Point Group PG(3,2)
Planar Projection of 'Twisted Circle'
Low Order Projective Planes and Implications
Stereographic Projection When
Representing the Earth as a Flat Surface

The Lehmers (1974) found an application of the Fano plane for factoring integers via quadratic forms. Here, the triples of forms used form the lines of the projective geometry on seven points, whose planes are Fano configurations corresponding to pairs of residue classes mod 24 (Lehmer and Lehmer 1974, Guy 1975, Shanks 1985). The group of automorphisms (incidence-preserving bijections) of the Fano plane is the simple group of group order 168 (full automorphisms== 7 generators x  the 24 automorphisms of the tetrahedron symmetry group).--excerpt from Wolframalpha

Argyle Sweater August 24, 2012
Rand & McNally Cartogrophers  Extraordinaire 

A long time ago, Ptolomy published Geographia, and used a similar method, like stereographic projection, to represent the Earth as a Flat Surface, and he used Latitude and Longitude coordinates to locate about 8000 places he knew of at the time and gave instructions for map making (cartography).

It served as as standard reference for 100 years, but made no progress under Roman rule. When a Christian mob sacked the library at Alexandria, Geographia disappeared along with other math and Greek works (never mind how they disposed of the Greek worker Hypatia).

With the fall of the Roman Empire, "the new age found a civilization as much in the dark about describing their place in space as was about the theorems and relations among spatial objects."--That's why We take care of books...

Anyhow, using 'irrational' numbers was also lost and this 'bad math' did not resurface until Cantor and Dedekind rediscovered it in the late 19th century (that's probably the punishment they deserved for what they did to Hypatia).

From the Middle Ages until that time scientists and mathematicians had been using them "happily, if awkwardly, anyway for the simple reward of getting the right answer outweighing having to wield unpleasant numbers 'that did not exist'."

This 'illegal' math is quite common today and we take it for granted. 

There was also Dirac's delta function which posed a quandary for mathematicians; one that is simply =0 everywhere except at 1 point, where the value is infinity, but when applied in calculus or certain math operations typically gives us finite non-zero answers. Then the mathematician Laurent Schwartz (one more thing we can thank the French for) came along and showed that 'the rules of math can be redefined to allow using the delta function, opening up a whole new discipline of math and later buttressing 'illegal' physics concepts like quantum field theories, as well.  

As of 2001, no one had yet successfully shown that, 'mathematically speaking, such theories legally exist.'

Which is just as well, since from God's Point of View (wayy out there on the Outside, at Infinity), Nothing Really Happens; according to Mom and Psalm 93:1 "...for God hath established the world, which shall not be moved." What most people don't realize, is the tacit caveat, 'unless God moves it'. 

An antipode to the stance taken by the general population, and teenagers (who think they know everything) without stopping to consider that even the greatest minds throughout history had reason to know better. 

Socrates having said, 'I need know nothing except that I know nothing.' 

When we next see each other, You can let me know if You had chanced on anyone of that ilk.

And for practical purposes, geometry found more inroads as people tried to describe in comprehensive terms what they saw in the world around them. One instance being that of motion in different frames of reference, like a deck hand on a boat who slides his hand down a mast appears to the seaman as though his hand is moving in a vertical direction (perpendicular to the deck), but to someone watching from the shore the deck hand's motion is running on a diagonal while the boat traverses the water.

The story continues, that Decartes translated space into numbers (quantitatively, not simply by construction), or phrased geometry in terms of algebra. Decartes distance formula being deeply tied to Euclidean Geometry and key to understanding both Euclidean and non-Euclidean geometries; serving as the basis for much of his contribution to physics.

Yet, Rene delayed publishing for 19 years--in fact, "...he delayed publishing anything until he was 40. What was he afraid of? The usual suspect. The Catholic Church."

Then came Gauss, who contributed to the development of non-Euclidean hyperbolic space.

"Through the centuries, mathematicians who attempted to prove the parallel postulate (Euclid's 'parallel lines never meet') as a theorem came close to discovering a strange and exciting new kind of space, but each of them was hampered by a simple belief: that the postulate was a true and necessary property of space."

...everyone but Gauss, who basically acceded to his own observation that a line is a curve of some sort, rather than clinging to Euclid's postulate, which in turn led to the rediscovery of curved spaces. A property that Gauss had to propose when applying the basic tenet that "superposition is a way to check for congruence or equivalence of geometric shapes."

There are other examples of non-Euclidean geometries.

In turn, Poincare developed his model of hyperbolic space, which was not just the 'model', but hyperbolic space in 2D, concluding that all possible mathematical descriptions of hyperbolic planes are isomorphic, in other words, 'the same.'

Poincare's lines are analogous to great circles, a rediscovery of what the Ancients knew about geodesics. The Equator being a curve of constant latitude.

"The view from outer space is not the view of the indigenous people of Noneuclid (a fictional professor sending her students out to measure distance across the globe). To her, there is no 'center of the earth', and as Gauss showed possible, there is no 'outer space.'...the space...is non-Euclidean, not hyperbolic space, but the space appropriate to the surface of the globe: elliptical space."

Meaning, the great circles intersect, and the sum of the angles of the triangle on such a surface is greater than 180 degrees, (as compared to that for hyperbolic space being less than 180 degrees, and for Euclid in a plane it is equal to 180). The Geo-metrics that define what we think of  as 'hyperspace', 'subspace', and 'flat earth'.

And these concepts laid the foundation for Riemann's differential geometry in 1915, in the vernacular having to define 'place vs. space', where place is 'part of a space' occupied by a specific object:

"The question of the valididty of geometry...is related to the question of the internal basis of metric [distance] relationships of the space...we must seek the ground of its metric relations outside it, in the binding forces which act on it..."--Georg Friedrich Bernhardt Riemann

And then breakthroughs by Klein, Piola, and Hilbert following soon after, applying tensor mathematics and algebraic curves.

Godel's theorem proved that in a system of sufficient complexity, such as the theory of numbers, there must exist a statement that cannot be proved either true of false.

A corollary of which: 

There must exist a true statement that cannot be proved. 

Example: There Is A God.

And all this ground-breaking, thought-provoking, formulations, rediscoveries and challenges swung like a pendulum back to the Flat Earth, again, considering what Clifford had proposed in 1870:

William Kingdon Clifford maintained:

"I hold in fact:

1. That small portions of space are of a nature analogous to little hills on a surface which is on average flat.
2. That the property of being curved or distorted is continually passed on from one portion of space to another after the manner of a wave.
3. That this variation of the curvature of space is really what happens in that phenomenon which we call the motion of matter..." 

And while Clifford professed his position based on instinct alone, he got it right! Which in turn inspired Einstien's space-time and warpage ideology encapsulated in his Relativity theories.

Eventually leading to concepts like supersymmetry and string theory.

As for them-thar-hills Clifford was intuitively talking about-- be they mounds, mountains or dunes...Quran anticipated someone wondering about them.

Surah 20 TaHa (طه )
طه 20:1

20:105 وَيَسْأَلُونَكَ عَنِالْجِبَا
فَقُلْ يَنسِفُهَا رَبِّي نَسْفًا فَيَذَرُهَا قَاعًا صَفْصَفًا 20:106
20:107 لَا تَرَى فِيهَا عِوَجًا وَلَا أَمْتًا

20:108يَوْمَئِذٍ يَتَّبِعُونَ الدَّاعِيَ لَا عِوَجَ لَهُ وَخَشَعَت الْأَصْوَاتُ لِلرَّحْمَنِ فَلَا تَسْمَعُ إِلَّا هَمْسًا


20:1 Tta.Ha
20:105 They'll ask You about the Mountains/Hills/Highlands/Barrows
20:106 Say, "God Will Level them Flat (Blast/Torpedo them Flat/To Sinking); Leaving it a level plane (Leaving it bottoming (simply) wasting)
20:107  You will not see in her (the earth) any Crookedness (Bend/Curvature) or Ruggedness
(i.e, no surface features, it will appear smooth, the flat earth metric).
20:108 On a Day they will follow the Caller/Summoner, without Crookedness/Dishonestly/Evasion to Him, and the Voices will be Hushed/Overawed by the Merciful, and you'll not hear anything except murmurs/whispers/soughing sound.


Surah 46 Al-Ahqaf ( الْأَحْقَافِ , Curved Sand Dunes)
وَاذْكُرْ أَخَا عَادٍ إِذْ أَنذَرَ قَوْمَهُ بِالْأَحْقَافِ وَقَدْ خَلَتْالنُّذُرُ مِن بَيْنِ يَدَيْهِ
 وَمِنْ خَلْفِهِ أَلَّا تَعْبُدُوا إِلَّا اللَّهَ
إِنِّي أَخَافُ عَلَيْكُمْ عَذَابَ يَوْمٍ عَظِيمٍ 46:21

46:21 And remember 'Aad's bretheren when He warned his people among the wind-Curved Sand Dunes, and indeed warners came and went before and after him, saying: 'Serve none but God. Truly, I fear for you the suffering of a Momentous/Significant/Great/Extraordinary Day'.

{Aside:You're asking me to do this and that's a steep apathy curve You expect me to get over after all these years---I Rest In God; and, since You been fronting me so long--after You my dear Alfonse!  Besides, what's my motivation?

Pantograph

He Says: Insolence!
She Says: I've just been talkin' to my Brothers Michael.
He Says: Every time I turn around I see you talking to one of them. How many brothers named Michael does a woman need?
She Says: All of''em!
He Says: All...ah? So, You're that One.}

Ok, Ok... SO, Let's try a W-Curve...

A Pantograph is a jointed device for copying a plane figure to any desired scale. It incorporates a pair of face bows fixed to the jaws, used for inscribing centrically related points and arcs leading to the points on segments relatable to the three craniofacial planes.

2x Pentograph
Scaling Instrument Used In Drafting
فَكَانَ قَابَ قَوْسَيْنِ أَوْ أَدْنَى (53:9) 
53:9 Until He (Gabriel) was 2 Bow's Lengths or Lower (Closer) 

 

A throw –  in German ‘wurf’, refers to a series of four points on a line which trace the 1-Dimensional motion of a point on a projective line, the W-curve.

The etymology for the ‘W-curve’derives from the German "Weg-Kurve" meaning "pathcurve", which include conics, logarithmic spirals, exponentials (i.e., y = x³ ), logarithms and the helix, but not sinusoidal functions.  

The term Throw (Wurf) was first introduced to avoid the metrical implication of a ratio. The Algebra of Throws provides an approach to numerical propositions, usually taken as axioms, but proven in projective geometry.

In the diagram ..., it turns out that the cross-ratio of a series of 4 points, taken in afixed order, does not depend on the choice of a line L, and is said to be invariant of the 4-tuple of lines {L_i}.This can be understood as follows: if L and L′are two lines not passing through Q then the perspective transformationfrom L to L′with the center Q is a projective transformation thattakes the quadruple {Pi} of points on L into the quadruple {Pi′}of points on L′. Therefore, the invariance of the cross-ratio under projective automorphisms of the line implies (in fact, is equivalent to) the independence of the cross-ratio of the four collinear points {P_i} on the lines {L_i} from the choice of the line that contains them. 

For points on a projective line (like the W-curve), the cross-ratio, (aka, double ratio and anharmonic ratio), is a special number associated with such a series of an ordered quadruple of collinear points.

If the four points are represented in homogeneous coordinates by vectors (ie., objects that transform like first-rank tensors) a,b,c,d such that c=a+b and d=ka+b, then their cross-ratio is k.

Dual concepts exist for a quadrupleof concurrent lines on the projective plane and a quadruple of points on the Riemann sphere.

The cross-ratio is essentially the only projective invariant of a quadruple of points, it being preserved by fractional linear transformations, which underlies its importance for projective geometry. If four points lie on a straight line L in R² then their cross-ratio is a well-defined quantity, because any choice of the origin andeven of the scale on the line will yield the same value of the cross-ratio.

In the Cayley–Klein model of hyperbolic geometry, the distance between points is expressed in terms of a certain cross-ratio.

Since the cross-ratio is invariant under projective transformations, it follows that the hyperbolic distance is invariant under the projective transformations that preserve the conic C. Conversely, the group G acts transitively on the set of pairs of points (p,q) in the unit disk at a fixed hyperbolic distance.

{ For a generalized conic, in this case a unit circle, any two points in the unit disk, p, q, the line connecting them intersects the circle in two points, a and b. The points are, in order,a, p, q, b.  The distance between p and q in the Cayley–Klein model of the plane hyperbolic geometry can be expressed as

... (the factor 1/2 is needed to make the curvature −1).}

                       

In the language of group theory, the symmetric group S₄ acts on the cross-ratio by permuting coordinates. The kernel of this action is isomorphic to the Klein four-group K. This group consists of 2-cycle permutations of type  (in addition to the identity), which preserve the cross-ratio. The effective symmetry group is then the quotient group , which is isomorphic to S₃.

There are a number of definitions of the cross-ratio. However, they all differ from each other by a suitable permutation of the coordinates. In general, there are six possible different values the cross-ratio ( z₁ , z₂:  z₃ , z₄) can take depending on the order in which the points z_i are given.

Since there are 24 possiblepermutations of the four coordinates, some permutations must leave the cross-ratio unaltered. In fact, exchanging any 2 pairs of coordinates preserves the cross-ratio:

Cross-Ratio Symmetries, Equivalent Permutations

Using these symmetries, there can then be 6 possible values of the cross-ratio, depending on the order in which the points are given. These are: 

The 6 Possible Values of the Cross-Ratio

For certain values of  λ there are fewer than 6 possible values of the 6 possible values of cross-ratio due to  enhanced symmetry. These values of λ correspond to fixed points of the action of S₃ on the Riemann sphere (given by the above 6 functions); in other words, those points with a non-trivial stabilizer in this permutation group.

The first set of fixed points is {0, 1, ∞}. However, the cross-ratio can never take on these values if the points {z_i} are all distinct. These values are limit values as one pair of coordinates approach each other:

Limit Values of Cross-Ratio
Due to Enhanced Symmetry Corresponding to Fixed Points

The second set of fixed points is {−1, 1/2, 2}--what is classically called the harmonic cross-ratio, and arises in projective harmonic conjugates. 

In the real case, there are no other exceptional orbits (See About the Myth-tery, projection/buruj/zodiac).

The most symmetric cross-ratio occurs when 

l = e^±ip/3

...the only possible values of the cross-ratio.

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The addition of points and multiplication of points, as one would do when adding distances, comprises von Stuadt's work ‘wurftheorie’ or the "algebra of points".

Given the same starting point, find the point midway between their terminal points, aka the harmonic conjugate of infinity with regard to their terminal points, and then find the harmonic conjugate of the initial point with regard to this mid-point and infinity. Generalizing this to add throws (CA,BD) and (CA,BD' ), then finding M the harmonic conjugate of C with regard to D and D' , and then S the harmonic conjugate of A with regard to C and M :

A definition of the product of two throws is derived in a similar fashion.

A summary statement is given by Veblen & Young given by Theorem 10: "The set of points on a line, with P_∞ removed, forms a field with respect to the operations previously defined". 

Another affirmation harmonic conjugates given by the theorem:

The only one-to-one correspondence between the real points on a line which preserves the harmonic relation betweenfour points is a non-singular projectivity.

Horizontal Reflection
Retrograde Motion Outer Planet
Hand Drawn Freeform

~ ~~ ~

Outer Planet
Retrograde Motion

Retrograde Motion Outer Planet
Superimposed Hand
Drawn by Enzo @ DeviantArt

لَقَدْ أَرْسَلْنَا رُسُلَنَا بِالْبَيِّنَاتِ وَأَنزَلْنَا مَعَهُمُ الْكِتَابَ وَالْمِيزَانَ لِيَقُومَ النَّاسُ بِالْقِسْطِ وَأَنزَلْنَا الْحَدِيدَ فِيهِ بَأْسٌ شَدِيدٌ وَمَنَافِعُ لِلنَّاسِ وَلِيَعْلَمَ اللَّهُ مَن يَنصُرُهُ وَرُسُلَهُ بِالْغَيْبِ إِنَّ اللَّهَ قَوِيٌّ عَزِيزٌ (57:25) 

 57:25 We Sent Our Messengers With Clarifying Signs (Evidence) And We Sent With Them The Scripture (The Book) and The Scale (Metric, Measure) Such that People May Judge Justly; and We Sent Iron in  which there is awesome strength and benefits for People, and so that God May Determine who helps Him and His Messenger Unseen (Without Knowing, On Faith alone); Indeed, God is Powerful, Almighty.

Below is one Measure of Music Based on the Retrograde Motion of An Outer Planet (As Above).

One Way to Set The Scale and Determine the Measure
Based on Retrograde Motion of an Outer Planet