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Presents, a Life with a Plan. My name is Karen Anastasia Placek, I am the author of this Google Blog. This is the story of my journey, a quest to understanding more than myself. The title of my first blog delivered more than a million views!! The title is its work as "The Secret of the Universe is Choice!; know decision" will be the next global slogan. Placed on T-shirts, Jackets, Sweatshirts, it really doesn't matter, 'cause a picture with my slogan is worth more than a thousand words, it's worth??.......Know Conversation!!!

Wednesday, April 26, 2023

Mane And Tail: Heads And Tales: Feathers And Withers, Dock And Pole

 


The evolution of creation at the nature of caveman to what is considered for the growth of a world would have to understand size.  The animal on land and the mammal in ocean would deliver as the Unicorn would suffice to both species to pronunciation of said beast.  

The original picture of the unicorn countered the pictures of dragons and still are able to deliver this point.  As the Unicorn has a forelock from where this would show that size is growth it is the understanding of the pull to increase the diameter of size itself throughout time.  The dragon backwards would deliver to the named Komoto to equal the delivery at known origin.

Albert Einstein had ventured into the direction that delivered a the black hole:  The tusk of the Narwal.  To this and what is applicable for the study it is the Friesian horse that truly can compass the Unicorn to now as the inverse  on horn to known now would truly compass the line.

reference for tact: Horns in the kjv

Friesian horse

From Wikipedia, the free encyclopedia
Friesian horse
Friesian Stallion.jpg
Friesian horse
Other namesBelgian Black (UK)
Country of originNetherlands
Traits
Height
  • 15 to 17 hands (60 to 68 inches, 152 to 173 cm)
ColourBlack
Distinguishing featuresBlack, powerfully muscled, agile with elegant action, thick mane and tail, feather on lower legs.
Breed standards

The Friesian (also Frizian) is a horse breed originating in Friesland, in the Netherlands. Although the conformation of the breed resembles that of a light draught horse, Friesians are graceful and nimble for their size. It is believed that during the Middle Ages, ancestors of Friesian horses were in great demand as war horses throughout continental Europe. Through the Early Middle Ages and High Middle Ages, their size enabled them to carry a knight in armour. In the Late Middle Ages, heavier, draught type animals were needed. Though the breed nearly became extinct on more than one occasion, the modern day Friesian horse is growing in numbers and popularity, used both in harness and under saddle. Most recently, the breed is being introduced to the field of dressage, causing the decline of the draught-type with its sturdy legs and back.

Breed characteristics[edit]

The Friesian breed is most often recognised by its black coat colour. However, colour alone is not the only distinguishing characteristic; Friesians are occasionally chestnut as some bloodlines do carry the "red" ('e") gene.[1] In the 1930s, chestnuts and bays were seen.[2] Friesians rarely have white markings of any kind; most registries allow only a small star on the forehead for purebred registration. To be accepted as breeding stock by the FPS studbook (Friesch Paarden Stamboek), a stallion must pass a rigorous approval process.

Friesian stallion

The Friesian stands on average about 15.3 hands (63 inches, 160 cm), although it may vary from 14.2 to 17 hands (58 to 68 inches, 147 to 173 cm) at the withers, and mares or geldings must be at least 15.2 hands (62 inches, 157 cm) to qualify for a "star-designation" pedigree.[3] Horses are judged at an inspection, or keuring, by Dutch judges, who decide whether the horse is worthy of star designation. The breed has powerful overall conformation and good bone structure, with what is sometimes called a "Baroque" body type. Friesians have long, arched necks and well-chiseled, short-eared, "Spanish-type" heads. They have powerful, sloping shoulders, compact, muscular bodies with strong, sloping hindquarters and low-set tails. Their limbs are comparatively short and strong. A Friesian horse also has a long, thick mane and tail, often wavy, and "feather"—long, silky hair on the lower legs—deliberately left untrimmed. The breed is known for a brisk, high-stepping trot. The Friesian is considered willing, active, and energetic, but also gentle and docile. A Friesian tends to have great presence and to carry itself with elegance.[4] Today, there are two distinct conformation types—the "baroque" type, which has the more robust build of the classical Friesian, and the modern, "sport horse" type, which is finer-boned. Both types are common, though the modern type is currently more popular in the show ring than is the baroque Friesian. However, conformation type is considered less important than correct movement.[5]

Closeup of the head

The chestnut colour is generally not accepted for registration for stallions, though it is sometimes allowed for mares and geldings.[1][6] A chestnut-coloured Friesian that competes is penalised. However, discoloration from old injuries or a black coat with fading from the sun is not penalised.[1] The chestnut allele, a recessive genetic trait in the Friesian, does exist; in the 1990s, two mares gave birth to chestnut foals.[2] The Friesch Paarden Stamboek began to attempt breeding out the chestnut colour in 1990, and today stallions with genetic testing indicating the presence of the chestnut or "red" gene, even if heterozygous and masked by black colour, are not allowed registration with the FPS.[7] The American Friesian Association, which is not affiliated to the KFPS, allows horses with white markings and/or chestnut colour to be registered if purebred parentage can be proven.[8] In 2014 there were eight stallion lines known to still carry the chestnut gene.[7]

There are four genetic disorders acknowledged by the industry that may affect horses of Friesian breeding: dwarfismhydrocephalus, a tendency for aortic rupture, and megaesophagus. There are genetic tests for the first two conditions. The Friesian is also among several breeds that may develop equine polysaccharide storage myopathy.[9] Approximately 0.25% of Friesians are affected by dwarfism, which results in horses with a normal-sized head, a broader chest than normal, an abnormally long back and very short limbs. It is a recessivecondition.[10] Additionally, the breed has a higher-than-usual rate of digestive system disorders, and a greater tendency to have insect bite hypersensitivity.[11] Like some other draught breeds, they are prone to a skin condition called verrucous pastern dermatopathy and may be generally prone to having a compromised immune system.[12] Friesian mares have a very high 54% rate of retained placenta after foaling. Some normal-sized Friesians also have a propensity toward tendon and ligament laxity which may or may not be associated with dwarfism. The relatively small gene pool and inbreeding are thought to be factors behind most of these disorders.[11]

History[edit]

The Friesian originates in the province of Friesland in the northern Netherlands, where there is evidence of thousands of years of horse populations.

Statue honouring the 100th anniversary of the modern Friesian studbook

As far back in history as the 4th century there are mentions of Friesian troops which rode their own horses. One of the most well-known sources of this was by an English writer named Anthony Dent[13] who wrote about the Friesian mounted troops in Carlisle. Dent, amongst others, wrote that the Friesian horse was the ancestor of both the British Shire, and the Fell pony. However, this is just speculation. It was not until the 11th century, that there were illustrations of what appeared to be Friesans. Many of the illustrations found depict knights riding horses which resembled the breed, with one of the most famous examples being William the Conqueror.[14][15]

These ancestors of the modern Friesians were used in medieval times to carry knights to battle. In the 12th and 13th centuries, some eastern horses of crusaders were mated with Friesian stock. During the 16th and 17th centuries, when the Netherlands were briefly linked with Spain, there was less demand for heavy war horses, as battle arms changed and became lighter. Andalusian horses were crossbred with Friesians, producing a lighter horse more suitable (in terms of less food intake and waste output) for work as urban carriage horses.

Historian Ann Hyland wrote of the Friesian breed:

The Emperor Charles (reigned 1516 -56) continued Spanish expansion into the Netherlands, which had its Frisian warhorse, noted by Vegetius and used on the continent and in Britain in Roman times. Like the Andalusian, the Frisian bred true to type. Even with infusions of Spanish blood during the sixteenth century, it retained its indigenous characteristics, taking the best from both breeds. The Frisian is mentioned in 16th and 17th century works as a courageous horse eminently suitable for war, lacking the volatility of some breeds or the phlegm of very heavy ones. Generally black, the Frisian was around 15hh with strong, cobby conformation, but with a deal more elegance and quality. The noted gait was a smooth trot coming from powerful quarters. Nowadays, though breed definition is retained, the size has markedly increased, as has that of most breeds due to improved rearing and dietary methods.[16]

The breed was especially popular in the 18th and 19th centuries, when they were in demand not only as harness horses and for agricultural work, but also for the trotting races so popular then. The Friesian may have been used as foundation stock for such breeds as the Dole Gudbrandsdal, the Norfolk Trotter(ancestor of the Hackney), and the Morgan.[17] In the 1800s, the Friesian was bred to be lighter and faster for trotting, but this led to what some owners and breeders regarded as inferior stock, so a movement to return to pureblood stock took place at the end of the 19th century. 

Friesian horses are sometimes referred to as "Belgian Blacks"

A studbook society was founded in 1879 by Frisian farmers and landowners who had gathered to found the Fries Rundvee Stamboek (FRS)[18] The Paardenstamboek ("horse stud book") was published in 1880 and initially registered both Friesian horses and a group of heavy warmblood breeds, including Ostfriesen and Alt-Oldenburgers, collectively known as "Bovenlanders".[19] At the time, the Friesian horse was declining in numbers, and was being replaced by the more fashionable Bovenlanders, both directly, and by crossbreeding Bovenlander stallions on Friesian mares. This had already virtually exterminated the pure Friesian in significant parts of the province in 1879, which made the inclusion of Bovenlanders necessary. While the work of the society led to a revival of the breed in the late 19th century, it also resulted in the sale and disappearance of many of the best stallions from the breeding area, and Friesian horse populations dwindled. By the early 20th century, the number of available breeding stallions was down to three.[20] Therefore, in 1906, the two parts of the registry were joined, and the studbook was renamed the Friesch Paarden Stamboek (FPS) in 1907."[19]

In 1913 a society, Het Friesch Paard, was founded to protect and promote the breed. By 1915 it had convinced FPS to split registration into two groups. By 1943, the breeders of non-Friesian horses left the FPS completely to form a separate association, which later became the Koninklijk Warmbloed Paardenstamboek Nederland (Royal Warmblood Studbook of the Netherlands (KWPN).[19]

Displacement by petroleum-powered farm equipment on dairy farms also was a threat to the survival of Friesian horse. The last draught function performed by Friesians on a significant scale was on farms that raised dairy cattle. World War II slowed the process of displacement, allowing the population and popularity of the breed to rebound. Important in the initial stage of the recovery of the breed was due to the family owned Circus Strassburger, who, having fled Nazi Germany for the Low Countries, discovered the show qualities of the breed and demonstrated its abilities outside of its local breeding area during and after the Nazi occupation[21]

Uses[edit]

A Friesian in surcingle, showing at the trot

As use in agricultural pursuits declined, the Friesian became popular for recreational uses.[22] Today, about seven percent of the horses in the Netherlands are Friesians.[12]

The Friesian horse today is used both in harness and under saddle, particularly in the discipline of dressage. In harness, they are used for competitive and recreational driving, both singly and in teams. A traditional carriage seen in some events designed for Friesian horses is a high-wheeled cart called a sjees.[23][24] Friesians are also used in ventures such as pulling vintage carriages at assorted ceremonial events.[25]

Because of their color and striking appearance, Friesian horses are a popular breed in movies and television, particularly in historic and fantasy dramas. They are viewed as calm in the face of the activity associated with filmmaking, but also elegant on-camera.[25]


Monday, April 24, 2023

The Claw!!




The introduction of what evolution has presented in word to Cantore arithmetic leaves the available word to text.  This aspect left the parameter for the King James Version to the avenue of what is known from origin to origin.  As the paragraph to the text begins a sentence I will choose to set another parameter for the C.D.C. and applicable agencies to compare to their current defcon numbers as represented in that portion to understand that currently the sign to DO NOT ENTER represents(see below).


The parameter on this set will not note Nostradamus and the quatrain as the restrictions set parameter and would have to be opened into physics via comprehension of what this will set as explained.  To depth is to currently known and origin of to said in the KJV will be set at Satan:  Expansion.  The works of will follow as the defcon representation for science and the most frigid conditions bringing no man to that sealed representation of very dangerous and or explosive.  To continue to set parameters based on arithmetic will further the checkpoint and increase the gates to cross thresholds with safety and knowledge without the danger before the crisis as many laboratories are already in function(Feng Shui) and I have a working government:  United States of America.


The banyan tree to the elephant represent water to collection of thirst as the earth represents landing and this draw will be curious to the waterhole bringing a route to the router and a model to the chart as the roads, trails and ball system of both living and dead can be found in a fossil to depth of ocean in coral, the sand being a hold as salt is it’s base to breadth for the kill to represent death.  To engage such will open the parameter for the mile as the ivory and long-lasting decomposer of the leave.  This rough draft is as the elephant will not trumpet rather drop the trunk(boom operator(BOOM)) to engage the crawl as seen with an anteater.  To average the vacuum to the mouth will be held in reprieve for the air and breadth.




It prescribes five graduated levels of readiness (or states of alert) for the U.S. military. It increases in severity from DEFCON 5 (least severe) to DEFCON 1 ( ...




Saturday, April 22, 2023

The Outline To Pressures And Weight To A Rotary Phone? What Of The Moon To The Oar!

 




The terraforming of what is a planet to the rotation of the sun(s) is of great importance as the cone is of gravity to atmospheric interruption to the current values.  As this is comprehended it is best to employ that available goods to envelope the letter to Cantore arithmetic to measure the mathematics at physics in gallery.  These base forms to what is practical is as schooling, the horses application to saddle and girth is only of story and, in order to bridle arena currently have been left to what is that.  That is not the Earth, that is Jupiter.

To average the pyramid at the protractor to square at the Aztec by basis allows the wave on the stairs as the sun rises and sets.  The dust to the moon and the balance to the planet is the beginning of understanding the pull.  As an equator to be the keel and the education to the movement it is left to the measurement via the infinite balance of the picture or window verbiage of just Fibonacci

Rotation

From Wikipedia, the free encyclopedia
A sphere rotating (spinning) about an axis

Rotation, or spin, is the circular movement of an object around a central axis. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional object has an infinite number of possible central axes and rotational directions.

If the rotation axis passes internally through the body's own center of mass, then the body is said to be autorotating or spinning, and the surface intersection of the axis can be called a pole. A rotation around a completely external axis, e.g. the planet Earth around the Sun, is called revolving or orbiting, typically when it is produced by gravity, and the ends of the rotation axis can be called the orbital poles.

Mathematics[edit]

Rotation (angular displacement) of a planar figure around a point
Rotational Orbit v Spin
Relations between rotation axis, plane of orbit and axial tilt (for Earth).

Mathematically, a rotation is a rigid body movement which, unlike a translation, keeps a point fixed. This definition applies to rotations within both two and three dimensions (in a plane and in space, respectively.)

All rigid body movements are rotations, translations, or combinations of the two.

A rotation is simply a progressive radial orientation to a common point. That common point lies within the axis of that motion. The axis is perpendicular to the plane of the motion. 

If a rotation around a point or axis is followed by a second rotation around the same point/axis, a third rotation results. The reverse (inverse) of a rotation is also a rotation. Thus, the rotations around a point/axis form a group. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, e.g. a translation.

Rotations around the xy and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis, and followed by a rotation around the zaxis. That is to say, any spatial rotation can be decomposed into a combination of principal rotations.

In flight dynamics, the principal rotations are known as yawpitch, and roll (known as Tait–Bryan angles). This terminology is also used in computer graphics.

Astronomy[edit]

Star trails caused by the Earth's rotation during the camera's long exposure time.[1]

In astronomy, rotation is a commonly observed phenomenon. Starsplanets and similar bodies all spin around on their axes. The rotation rate of planets in the solar system was first measured by tracking visual features. Stellar rotation is measured through Doppler shift or by tracking active surface features.

This rotation induces a centrifugal acceleration in the reference frame of the Earth which slightly counteracts the effect of gravitation the closer one is to the equatorEarth's gravity combines both mass effects such that an object weighs slightly less at the equator than at the poles. Another is that over time the Earth is slightly deformed into an oblate spheroid; a similar equatorial bulge develops for other planets.

Another consequence of the rotation of a planet is the phenomenon of precession. Like a gyroscope, the overall effect is a slight "wobble" in the movement of the axis of a planet. Currently the tilt of the Earth's axis to its orbital plane (obliquity of the ecliptic) is 23.44 degrees, but this angle changes slowly (over thousands of years). (See also Precession of the equinoxes and Pole star.)

Revolution[edit]

While revolution is often used as a synonym for rotation, in many fields, particularly astronomy and related fields, revolution, often referred to as orbital revolution for clarity, is used when one body moves around another while rotation is used to mean the movement around an axis. Moons revolve around their planet, planets revolve about their star (such as the Earth around the Sun); and stars slowly revolve about their galaxial center. The motion of the components of galaxies is complex, but it usually includes a rotation component.

Retrograde rotation[edit]

Most planets in the Solar System, including Earth, spin in the same direction as they orbit the Sun. The exceptions are Venus and Uranus. Venus may be thought of as rotating slowly backward (or being "upside down"). Uranus rotates nearly on its side relative to its orbit. Current speculation is that Uranus started off with a typical prograde orientation and was knocked on its side by a large impact early in its history. The dwarf planet Pluto (formerly considered a planet) is anomalous in several ways, including that it also rotates on its side.

Physics[edit]

The speed of rotation is given by the angular frequency (rad/s) or frequency (turns per time), or period (seconds, days, etc.). The time-rate of change of angular frequency is angular acceleration (rad/s2), caused by torque. The ratio of torque to the angular acceleration is given by the moment of inertia.

The angular velocity vector (an axial vector) also describes the direction of the axis of rotation. Similarly, the torque is an axial vector.

The physics of the rotation around a fixed axis is mathematically described with the axis–angle representation of rotations. According to the right-hand rule, the direction away from the observer is associated with clockwise rotation and the direction towards the observer with counterclockwise rotation, like a screw.

Cosmological principle[edit]

The laws of physics are currently believed to be invariant under any fixed rotation. (Although they do appear to change when viewed from a rotating viewpoint: see rotating frame of reference.)

In modern physical cosmology, the cosmological principle is the notion that the distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throughout the universe and have no preferred direction, and should, therefore, produce no observable irregularities in the large scale structuring over the course of evolution of the matter field that was initially laid down by the Big Bang.

In particular, for a system which behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally invariant. According to Noether's theorem, if the action (the integral over time of its Lagrangian) of a physical system is invariant under rotation, then angular momentum is conserved.

Euler rotations[edit]

Euler rotations of the Earth. Intrinsic (green), Precession (blue) and Nutation (red)

Euler rotations provide an alternative description of a rotation. It is a composition of three rotations defined as the movement obtained by changing one of the Euler angles while leaving the other two constant. Euler rotations are never expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture. They constitute a mixed axes of rotation system, where the first angle moves the line of nodes around the external axis z, the second rotates around the line of nodes and the third one is an intrinsic rotation around an axis fixed in the body that moves.

These rotations are called precessionnutation, and intrinsic rotation.

Flight dynamics[edit]

The principal axes of rotation in space

In flight dynamics, the principal rotations described with Euler angles above are known as pitchroll and yaw. The term rotation is also used in aviation to refer to the upward pitch (nose moves up) of an aircraft, particularly when starting the climb after takeoff.

Principal rotations have the advantage of modelling a number of physical systems such as gimbals, and joysticks, so are easily visualised, and are a very compact way of storing a rotation. But they are difficult to use in calculations as even simple operations like combining rotations are expensive to do, and suffer from a form of gimbal lock where the angles cannot be uniquely calculated for certain rotations.

Amusement rides[edit]

Many amusement rides provide rotation. A Ferris wheel has a horizontal central axis, and parallel axes for each gondola, where the rotation is opposite, by gravity or mechanically. As a result, at any time the orientation of the gondola is upright (not rotated), just translated. The tip of the translation vector describes a circle. A carouselprovides rotation about a vertical axis. Many rides provide a combination of rotations about several axes. In Chair-O-Planes the rotation about the vertical axis is provided mechanically, while the rotation about the horizontal axis is due to the centripetal force. In roller coaster inversions the rotation about the horizontal axis is one or more full cycles, where inertia keeps people in their seats.

Sports[edit]

Rotation of a ball or other object, usually called spin, plays a role in many sports, including topspin and backspin in tennisEnglishfollow and draw in billiards and poolcurve balls in baseballspin bowling in cricketflying disc sports, etc. Table tennis paddles are manufactured with different surface characteristics to allow the player to impart a greater or lesser amount of spin to the ball.

Rotation of a player one or more times around a vertical axis may be called spin in figure skatingtwirling (of the baton or the performer) in baton twirling, or 360540720, etc. in snowboarding, etc. Rotation of a player or performer one or more times around a horizontal axis may be called a fliprollsomersaultheli, etc. in gymnasticswaterskiing, or many other sports, or a one-and-a-halftwo-and-a-halfgainer (starting facing away from the water), etc. in diving, etc. A combination of vertical and horizontal rotation (back flip with 360°) is called a möbius in waterskiing freestyle jumping.

Rotation of a player around a vertical axis, generally between 180 and 360 degrees, may be called a spin move and is used as a deceptive or avoidance manoeuvre, or in an attempt to play, pass, or receive a ball or puck, etc., or to afford a player a view of the goal or other players. It is often seen in hockeybasketballfootball of various codes, tennis, etc.

Fixed axis vs. fixed point[edit]

The combination of any sequence of rotations of an object in three dimensions about a fixed point is always equivalent to a rotation about an axis (which may be considered to be a rotation in the plane that is perpendicular to that axis). Similarly, the rotation rate of an object in three dimensions at any instant is about some axis, although this axis may be changing over time.

In other than three dimensions, it does not make sense to describe a rotation as being around an axis, since more than one axis through the object may be kept fixed; instead, simple rotations are described as being in a plane. In four or more dimensions, a combination of two or more rotations about in a plane is not in general a rotation in a single plane.

Axis of 2-dimensional rotations[edit]

2-dimensional rotations, unlike the 3-dimensional ones, possess no axis of rotation, only a point about which the rotation occurs. This is equivalent, for linear transformations, with saying that there is no direction in the plane which is kept unchanged by a 2 dimensional rotation, except, of course, the identity.

The question of the existence of such a direction is the question of existence of an eigenvector for the matrix A representing the rotation. Every 2D rotation around the origin through an angle  in counterclockwise direction can be quite simply represented by the following matrix:

A standard eigenvalue determination leads to the characteristic equation

which has

as its eigenvalues. Therefore, there is no real eigenvalue whenever , meaning that no real vector in the plane is kept unchanged by A.

Rotation angle and axis in 3 dimensions[edit]

Knowing that the trace is an invariant, the rotation angle  for a proper orthogonal 3×3 rotation matrix  is found by

Using the principal arc-cosine, this formula gives a rotation angle satisfying . The corresponding rotation axis must be defined to point in a direction that limits the rotation angle to not exceed 180 degrees. (This can always be done because any rotation of more than 180 degrees about an axis  can always be written as a rotation having  if the axis is replaced with .)

Every proper rotation  in 3D space has an axis of rotation, which is defined such that any vector  that is aligned with the rotation axis will not be affected by rotation. Accordingly, , and the rotation axis therefore corresponds to an eigenvector of the rotation matrix associated with an eigenvalue of 1. As long as the rotation angle  is nonzero (i.e., the rotation is not the identity tensor), there is one and only one such direction. Because A has only real components, there is at least one real eigenvalue, and the remaining two eigenvalues must be complex conjugates of each other (see Eigenvalues and eigenvectors#Eigenvalues and the characteristic polynomial). Knowing that 1 is an eigenvalue, it follows that the remaining two eigenvalues are complex conjugates of each other, but this does not imply that they are complex—they could be real with double multiplicity. In the degenerate case of a rotation angle , the remaining two eigenvalues are both equal to −1. In the degenerate case of a zero rotation angle, the rotation matrix is the identity, and all three eigenvalues are 1 (which is the only case for which the rotation axis is arbitrary).

A spectral analysis is not required to find the rotation axis. If  denotes the unit eigenvector aligned with the rotation axis, and if  denotes the rotation angle, then it can be shown that . Consequently, the expense of an eigenvalue analysis can be avoided by simply normalizing this vector if it has a nonzero magnitude. On the other hand, if this vector has a zero magnitude, it means that . In other words, this vector will be zero if and only if the rotation angle is 0 or 180 degrees, and the rotation axis may be assigned in this case by normalizing any column of  that has a nonzero magnitude.[2]

This discussion applies to a proper rotation, and hence . Any improper orthogonal 3x3 matrix  may be written as , in which  is proper orthogonal. That is, any improper orthogonal 3x3 matrix may be decomposed as a proper rotation (from which an axis of rotation can be found as described above) followed by an inversion (multiplication by −1). It follows that the rotation axis of  is also the eigenvector of  corresponding to an eigenvalue of −1.

Rotation plane[edit]

As much as every tridimensional rotation has a rotation axis, also every tridimensional rotation has a plane, which is perpendicular to the rotation axis, and which is left invariant by the rotation. The rotation, restricted to this plane, is an ordinary 2D rotation.

The proof proceeds similarly to the above discussion. First, suppose that all eigenvalues of the 3D rotation matrix A are real. This means that there is an orthogonal basis, made by the corresponding eigenvectors (which are necessarily orthogonal), over which the effect of the rotation matrix is just stretching it. If we write A in this basis, it is diagonal; but a diagonal orthogonal matrix is made of just +1s and −1s in the diagonal entries. Therefore, we don't have a proper rotation, but either the identity or the result of a sequence of reflections.

It follows, then, that a proper rotation has some complex eigenvalue. Let v be the corresponding eigenvector. Then, as we showed in the previous topic, is also an eigenvector, and  and  are such that their scalar product vanishes:

because, since  is real, it equals its complex conjugate , and  and  are both representations of the same scalar product between  and .

This means  and  are orthogonal vectors. Also, they are both real vectors by construction. These vectors span the same subspace as  and , which is an invariant subspace under the application of A. Therefore, they span an invariant plane.

This plane is orthogonal to the invariant axis, which corresponds to the remaining eigenvector of A, with eigenvalue 1, because of the orthogonality of the eigenvectors of A.

An Independent Mind, Knot Logic

An Independent Mind, Knot Logic

This is word Difficult as it word Tries time[Time[Thyme] equated word Plant

  File:Statue of a Toga-clad Roman Dignitary - Archaeological Museum of Epidaurus by Joy of Museums.jpg:   : Original file   (2,664 × 5,081 ...

Karen A. Placek, aka Karen Placek, K.A.P., KAP

My photo
Presents, a Life with a Plan. My name is Karen Anastasia Placek, I am the author of this Google Blog. This is the story of my journey, a quest to understanding more than myself. The title of my first blog delivered more than a million views!! The title is its work as "The Secret of the Universe is Choice!; know decision" will be the next global slogan. Placed on T-shirts, Jackets, Sweatshirts, it really doesn't matter, 'cause a picture with my slogan is worth more than a thousand words, it's worth??.......Know Conversation!!!

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