Андрей Смирнов
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Rotation перевод на русский язык и транскрипция произношения

Euler angles and quaternions

Euler angles

Euler angles are represented by three angle values for X, Y and Z that are applied sequentially. To apply a Euler rotation to a particular GameObjectThe fundamental object in Unity scenes, which can represent characters, props, scenery, cameras, waypoints, and more. A GameObject’s functionality is defined by the Components attached to it. More infoSee in , each rotation value is applied in turn, as a rotation around its corresponding axis.

  • Benefit: Euler angles have an intuitive “human readable” format, consisting of three angles.
  • Benefit: Euler angles can represent the rotation from one orientation to another through a turn of more than 180 degrees
  • Limitation: Euler angles suffer from Gimbal Lock. When applying the three rotations in turn, it is possible for the first or second rotation to result in the third axis pointing in the same direction as one of the previous axes. This means a “degree of freedom” has been lost, because the third rotation value cannot be applied around a unique axis.

Quaternions

Quaternions can be used to represent the orientation or rotation of a GameObject. This representation internally consists of four numbers (referenced in Unity as x, y, z & w) however these numbers don’t represent angles or axes and you never normally need to access them directly. Unless you are particularly interested in delving into the mathematics of Quaternions, you only really need to know that a Quaternion represents a rotation in 3D space and you never normally need to know or modify the x, y & z properties.

In the same way that a Vector can represent either a position or a direction (where the direction is measured from the origin), a Quaternion can represent either an orientation or a rotation — where the rotation is measured from the rotational “origin” or “Identity”. It is because the rotation is measured in this way — from one orientation to another — that a quaternion can’t represent a rotation beyond 180 degrees.

  • Benefit: Quaternion rotations do not suffer from Gimbal Lock.
  • Limitation: A single quaternion cannot represent a rotation exceeding 180 degrees in any direction.
  • Limitation: The numeric representation of a Quaternion is not intuitively understandable.

Unity stores all GameObject rotations internally as Quaternions, because the benefits outweigh the limitations.

The Transform Inspector displays the rotation using Euler angles, because this is easier to understand and edit. Unity converts new values into the Inspector for the rotation of a GameObject into a new Quaternion rotation value for the GameObject.

The rotation of a GameObject is displayed and edited as Euler angles in the Inspector, but is stored internally as a Quaternion

As a side-effect, it is possible in the Inspector to enter a value of, say, X: 0, Y: 365, Z: 0 for a GameObject’s rotation. This value is not possible to represent as a quaternion, so when you enter Play mode, the GameObject’s rotation values change to X: 0, Y: 5, Z: 0. This is because Unity converts rotation to a Quaternion which does not have the concept of a full 360-degree rotation plus 5 degrees, and instead is set to orient the same way as the result of the rotation.

Flight dynamics

The principal axes of rotation in space

In flight dynamics, the principal rotations described with are known as pitch, roll 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 where the angles cannot be uniquely calculated for certain rotations.

Implications for Animation

Many 3D authoring packages, and Unity’s own internal Animation window, allow you to use Euler angles to specify rotations during an animation.

These rotations values can frequently exceed ranges expressable by quaternions. For example, if a GameObject rotates 720 degrees, this could be represented by Euler angles X: 0, Y: 720, Z:0. But this is not representable by a Quaternion value.

Unity’s Animation Window

Within Unity’s own animation window, there are options which allow you to specify how the rotation should be interpolated — using Quaternion or Euler interpolation. By specifying Euler interpolation you are telling Unity that you want the full range of motion specified by the angles. With Quaternion rotation however, you are saying you simply want the rotation to end at a particular orientation, and Unity uses Quaternion interpolation and rotate across the shortest distance to get there. See Using Animation Curves for more information on this.

External Animation Sources

When importing animation from external sources, these files usually contain rotational keyframeA frame that marks the start or end point of a transition in an animation. Frames in between the keyframes are called inbetweens. See in animation in Euler format. Unity’s default behaviour is to resample these animations and generate a new Quaternion keyframe for every frame in the animation, in an attempt to avoid any situations where the rotation between keyframes may exceed the Quaternion’s valid range.

For example, imagine two keyframes, 6 frames apart, with values for X as 0 on the first keyframe and 270 on the second keyframe. Without resampling, a quaternion interpolation between these two keyframes would rotate 90 degrees in the opposite direction, because that is the shortest way to get from the first orientation to the second orientation. However by resampling and adding a keyframe on every frame, there are now only 45 degrees between keyframes so the rotation works correctly.

There are still some situations where — even with resampling — the quaternion representation of the imported animation may not match the original closely enough, For this reason, in Unity 5.3 and onwards there is the option to turn off animation resampling, so that you can instead use the original Euler animation keyframes at runtime. For more information, see Animation Import of Euler Curve Rotations.

Scale Constraints

Lights

Sports

Rotation of a ball or other object, usually called spin, plays a role in many sports, including and backspin in tennis, English, follow and draw in billiards and pool, curve balls in baseball, spin bowling in cricket, flying 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 skating, twirling (of the baton or the performer) in baton twirling, or 360, 540, 720, etc. in snowboarding, etc. Rotation of a player or performer one or more times around a horizontal axis may be called a flip, roll, somersault, heli, etc. in gymnastics, waterskiing, or many other sports, or a one-and-a-half, two-and-a-half, gainer (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 .

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 maneuver, 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 hockey, basketball, football of various codes, tennis, etc.

Physics

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 the two (how heavy is it to start, stop, or otherwise change rotation) 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

The laws of physics are currently believed to be . (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

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 precession, nutation, and intrinsic rotation.

Implications for scripting

When dealing with handling rotations in your scriptsA piece of code that allows you to create your own Components, trigger game events, modify Component properties over time and respond to user input in any way you like. More infoSee in , you should use the Quaternion class and its functions to create and modify rotational values. There are some situations where it is valid to use Euler angles, but you should bear in mind:
— You should use the Quaternion Class functions that deal with Euler angles
— Retrieving, modifying, and re-applying Euler values from a rotation can cause unintentional side-effects.

Creating and manipulating quaternions directly

Unity’s Quaternion class has a number of functions which allow you to create and manipulate rotations without needing to use Euler angles at all. For example:

Creating:

  • Quaternion.LookRotation
  • Quaternion.AngleAxis
  • Quaternion.FromToRotation

Manipulating:

  • Quaternion.Slerp
  • Quaternion.Inverse
  • Quaternion.RotateTowards
  • Transform.Rotate & Transform.RotateAround

However sometimes it’s desirable to use Euler angles in your scripts. In this case it’s important to note that you must keep your angles in variables, and only use them to apply them as Euler angles to your rotation. While it’s possible to retrieve Euler angles from a quaternion, if you retrieve, modify and re-apply, problems are likely to arise.

Here are some examples of mistakes commonly made using a hypothetical example of trying to rotate a GameObject around the X axis at 10 degrees per second. This is what you should avoid:

And here is an example of using Euler angles in script correctly:

See documentation on Quaternion.Euler for more details.

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2. Usting @melton as my bitch for gueen
3. Trying to pack for LA
4. Fresh nails wha dis
5. Me once my motor cycle license is done this summer
6. So cute
7. The detail. . . even down to my snake rings. . . Amazing!

Английский-Русский

(8) 4 ÷ 2 are divided into 5 steps

(a) Correct quotient

(b) Over-quotient by once

(c) Over-quotient by twice or more

(d) Same first digits (2 scenarios)

(e) Both first digits are 1 over-quotient (129, 348, 567, 786, 95)

(f) Line to help memorize returning : minus 1 time, skip a space and add back Mr. Cat

Английский-Русский

In the UK and the USA, law degree programmes usually take three years to complete. In the UK, these programmes typically include core subjects such as criminal law, contract law, tort law, land law, equity and trusts, administrative law and constitutional law. In addition, students ar

Английский-Русский

Mendeleev was foreshadowed in his great generalization by De Chancourtois’s helix of elements of 1863, J.A.R. New-lands’s *law of octaves* (1864-5)-which uncovered periodicity in the 8th elements of his chemical groupings — and W. Odling’s work, which suggested that recurrent chemical properties in elements arranged according to atomic weight could not be accidental.

Английский-Русский

Mathematics

Rotation 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 90 degrees perpendicular to the plane of the motion. If the axis of the rotation lies external of the body in question then the body is said to orbit. There is no fundamental difference between a “rotation” and an “orbit” and or «spin». The key distinction is simply where the axis of the rotation lies, either within or outside of a body in question. This distinction can be demonstrated for both “rigid” and “non rigid” bodies.

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 x, y 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 z axis. 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 yaw, pitch, and roll (known as Tait–Bryan angles). This terminology is also used in computer graphics.

Rotation plane

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 +1’s and -1’s 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, v¯{\displaystyle {\bar {v}}} is also an eigenvector, and v+v¯{\displaystyle v+{\bar {v}}} and i(v−v¯){\displaystyle i(v-{\bar {v}})} are such that their scalar product vanishes:

i(vT+v¯T)(v−v¯)=i(vTv−v¯Tv¯+v¯Tv−vTv¯)={\displaystyle i(v^{T}+{\bar {v}}^{T})(v-{\bar {v}})=i(v^{T}v-{\bar {v}}^{T}{\bar {v}}+{\bar {v}}^{T}v-v^{T}{\bar {v}})=0}

because, since v¯Tv¯{\displaystyle {\bar {v}}^{T}{\bar {v}}} is real, it equals its complex conjugate vTv{\displaystyle v^{T}v}, and v¯Tv{\displaystyle {\bar {v}}^{T}v} and vTv¯{\displaystyle v^{T}{\bar {v}}} are both representations of the same scalar product between v{\displaystyle v} and v¯{\displaystyle {\bar {v}}}.

This means v+v¯{\displaystyle v+{\bar {v}}} and i(v−v¯){\displaystyle i(v-{\bar {v}})} are orthogonal vectors. Also, they are both real vectors by construction. These vectors span the same subspace as v{\displaystyle v} and v¯{\displaystyle {\bar {v}}}, 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.

Перевод «Rotation» на русский язык: «Вращение»

Rotation:   вращение
ротор
циклический сдвиг
переход
вихрь

Rotation starten.

 

Начинаю вращение.

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Ist die Rotation gleichmäßig?

 

Вращение сбалансировано?

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Dazu sagt man Rotation.

 

Это называется ротация.

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Shield rotation complete.

 

Вращение щита завершено.

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Hier herrscht eine strikte Rotation.

 

У нас тут строгая ротация.

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Die atmosphärische Super-Rotation?

 

Атмосферных аномалий?

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Aber Rotation reichte nicht,

 

Но вращения было недостаточно —

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Langcrowd.com

übertragen wir die Rotation des Motors.

 

для передачи вращения мотора.

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und schon haben wir Rotation.

 

и вот он крутится.

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die Winkelgeschwindigkeit der Rotation von Gedankenbildern maß.

 

который измерил угловую скорость вращения мысленных образов.

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Nein, ich meine, ist da eine Rotation?

 

Нет, я имею в виду, есть какое-то расписание?

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da können Sie die Rotation sehen,

 

Вот так оно вращается,

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Wir können dich in die Rotation einbeziehen.

 

Можем выделить тебе смену.

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Ich kann dich in die Rotation legen.

 

Знаешь, приходи, я тебя пущу в оборот, будешь проводить спарринги с ребятами.

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Boki, du hast meine Playliste noch auf Rotation?

 

Боки, у тебя еще есть мой плэйлист?

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Rotation, Zeit,

 

Чередование, время,

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Sie verschlafen ein Drittel Ihrer Rotation.

 

Вы спите треть вашего вращения.

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die Rotation des visuellen Musters mittels ihrer

 

она на самом деле самостоятельно контролирует вращение этого визуального паттерна

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Verbrachte seine Rotation als M.P. in Deutschland.

 

Прошел курсы повышения в военной полиции Германии.

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und eine langsame Rotation erlaubt.

 

и низкая скорость вращения.

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Направленная резина и ассиметричная

Если речь заходит о протекторе, все шины принято подразделять на основные виды:

  • ассиметричные шины
  • симметричные шины
  • направленная резина

Особенностью асимметричной  шины  является такой рисунок протектора, когда на внешней, а также на внутренней  части указанный рисунок отличается.  При установке важно понимать, что такая шина должна стоять только с учетом наружной и внутренней  стороны.  Как правило, обозначается соответствующей надписью на боковине (Outside — наружная, Inside -внутренняя, Facing Out и т.д.). Симметричная  шина имеет протектор с симметричным рисунком, конструкцией корда, а также боковины

Такие покрышки можно ставить любой стороной. Еще можно выделить направленные шины, где рисунок протектора  имеет направление вращения. Такие покрышки имеют обозначение стрелкой на боковине, указывающей на направление вращения, а также маркируются надписью Rotation.

Симметричная  шина имеет протектор с симметричным рисунком, конструкцией корда, а также боковины. Такие покрышки можно ставить любой стороной. Еще можно выделить направленные шины, где рисунок протектора  имеет направление вращения. Такие покрышки имеют обозначение стрелкой на боковине, указывающей на направление вращения, а также маркируются надписью Rotation.

Также можно выделить  шины, когда в обязательном порядке необходимо строгое соблюдение правил их установки  на машину. Речь идет о шинах c маркировкой на боковине «right» (правые) или «left» (левые).  Если есть такая маркировка, шины можно ставить только на левую или правую сторону автомобиля.

Отсутствие данных надписей, причем не зависимо от рисунка протектора (направленный или асимметричный), позволяет ставить покрышки  без привязки к одной стороне.

Обратите внимание, все особенности, рассмотренные выше, могут быть как особенностью конкретных шин, так и в отдельных случаях сочетаться в одной покрышке. Например, можно встретить асимметричную направленную резину, асимметричные  шины могут быть «левыми» или «правыми» и т.д

Добавим, что сегодня асимметричные шины с заданным направлением вращения не производятся, так как на практике применение «левых» и «правых» шин не дает особых преимуществ, однако производство такой резины дороже,  часто возникают сложности с их подбором по каталогам.

Что в итоге

С учетом приведенной выше информации при выборе шин нужно учитывать целый ряд особенностей

 В первую очередь, важно принимать во внимание условия эксплуатации ТС. Например, если владелец активно эксплуатирует автомобиль, ездит преимущественно по сухим трассам на дальние расстояния, выбор направленной резины не всегда будет лучшим вариантом. В этом случае оптимально остановиться на ассиметричном протекторе

В этом случае оптимально остановиться на ассиметричном протекторе.

При этом более комфортной будет езда по мокрой дороге с высокой скоростью  как раз на направленной резине по причине лучшего водоотвода.  Также нужно учитывать, что  в зависимости от стоящих на машине шин возможны проблемы при подборе и установке  «запаски» или другой покрышки в случае повреждения или прокола  уже имеющегося колеса.

Так или иначе, главное условие, соблюдать все правила и рекомендации по установке направленных и ассиметричных шин.  В этом случае езда на автомобиле будет комфортной и безопасной, а правильно установленная зимняя или летняя шина в полной мере будет способна реализовать весь потенциал, заложенный в нее инженерами и конструкторами.

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