如果說︰

需求為創造之母。

那麼『並矢張量』

Dyadics

In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra.

There are numerous ways to multiply two Euclidean vectors. The dot product takes in two vectors and returns a scalar, while the cross product returns a pseudovector. Both of these have various significant geometric interpretations and are widely used in mathematics, physics, and engineering. The dyadic product takes in two vectors and returns a second order tensor called a dyadic in this context. A dyadic can be used to contain physical or geometric information, although in general there is no direct way of geometrically interpreting it.

The dyadic product is distributive over vector addition, and associative with scalar multiplication. Therefore, the dyadic product is linear in both of its operands. In general, two dyadics can be added to get another dyadic, and multiplied by numbers to scale the dyadic. However, the product is not commutative; changing the order of the vectors results in a different dyadic.

The formalism of dyadic algebra is an extension of vector algebra to include the dyadic product of vectors. The dyadic product is also associative with the dot and cross products with other vectors, which allows the dot, cross, and dyadic products to be combined together to obtain other scalars, vectors, or dyadics.

It also has some aspects of matrix algebra, as the numerical components of vectors can be arranged into row and column vectors, and those of second order tensors in square matrices. Also, the dot, cross, and dyadic products can all be expressed in matrix form. Dyadic expressions may closely resemble the matrix equivalents.

The dot product of a dyadic with a vector gives another vector, and taking the dot product of this result gives a scalar derived from the dyadic. The effect that a given dyadic has on other vectors can provide indirect physical or geometric interpretations.

Dyadic notation was first established by Josiah Willard Gibbs in 1884. The notation and terminology are relatively obsolete today. Its uses in physics include continuum mechanics and electromagnetism.

In this article, upper-case bold variables denote dyadics (including dyads) whereas lower-case bold variables denote vectors. An alternative notation uses respectively double and single over- or underbars.

 

『記號法』之興，豈無由乎？

人們講︰

好記號法使概念表達簡練清晰！

比方從『分數表示』 ，通『割線斜率』 ，達『切線微分』 。

為什麼不覺得它的『定義』及『術語』︰

Definitions and terminology

Dyadic, outer, and tensor products

A dyad is a tensor of order two and rank one, and is the result of the dyadic product of two vectors (complex vectors in general), whereas a dyadic is a general tensor of order two (which may be full rank or not).

There are several equivalent terms and notations for this product:

• the dyadic product of two vectors  and   is denoted by  (no symbol; no multiplication signs, crosses, dots etc.)
• the outer product of two column vectors  and  is denoted and defined as  or  , where means transpose,
• the tensor product of two vectors and  is denoted ,

In the dyadic context they all have the same definition and meaning, and are used synonymously, although the tensor product is an instance of the more general and abstract use of the term.

Dirac’s bra–ket notation makes the use of dyads and dyadics intuitively clear, see Cahill (2013).

Three-dimensional Euclidean space

To illustrate the equivalent usage, consider three-dimensional Euclidean space, letting:

be two vectors where i, j, k (also denoted e₁, e₂, e₃) are the standard basis vectors in this vector space (see also Cartesian coordinates). Then the dyadic product of a and b can be represented as a sum:

or by extension from row and column vectors, a 3×3 matrix (also the result of the outer product or tensor product of a and b):

A dyad is a component of the dyadic (a monomial of the sum or equivalently an entry of the matrix) — the dyadic product of a pair of basis vectors scalar multiplied by a number.

Just as the standard basis (and unit) vectors i, j, k, have the representations:

(which can be transposed), the standard basis (and unit) dyads have the representation:

For a simple numerical example in the standard basis:

N-dimensional Euclidean space

If the Euclidean space is N–dimensional, and

where e_i and e_j are the standard basis vectors in N-dimensions (the index i on e_i selects a specific vector, not a component of the vector as in a_i), then in algebraic form their dyadic product is:

This is known as the nonion form of the dyadic. Their outer/tensor product in matrix form is:

A dyadic polynomial A, otherwise known as a dyadic, is formed from multiple vectors a_i and b_j:

A dyadic which cannot be reduced to a sum of less than N dyads is said to be complete. In this case, the forming vectors are non-coplanar,^{[dubious – discuss]} see Chen (1983).

Classification

The following table classifies dyadics:

	Determinant	Adjugate	Matrix and its rank
Zero	= 0	= 0	= 0; rank 0: all zeroes
Linear	= 0	= 0	≠ 0; rank 1: at least one non-zero element and all 2 × 2 subdeterminants zero (single dyadic)
Planar	= 0	≠ 0 (single dyadic)	≠ 0; rank 2: at least one non-zero 2 × 2 subdeterminant
Complete	≠ 0	≠ 0	≠ 0; rank 3: non-zero determinant

Identities

The following identities are a direct consequence of the definition of the tensor product:^[1]

 

有何動人之處？？

蓋『符號』構成『系統』，並非孤立存在。既經不同『學習過程』洗禮，往往習以為常，或難客觀論辨矣！！

且借

Dyadic algebra

Product of dyadic and vector

There are four operations defined on a vector and dyadic, constructed from the products defined on vectors.

	Left	Right
Dot product
Cross product

………

 

舉例而言

將之改寫為︰

到底有什麼益處耶？？！！

反思

，果不能還其『向量本色』嘛！！？？

或可推知

狄拉克符號

狄拉克符號或狄拉克標記（英語：Dirac notation）是量子力學中廣泛應用於描述量子態的一套標準符號系統。在這套系統中，每一個量子態都被描述為希爾伯特空間中的態向量，定義為括量（ket）：；每一個括量的共軛轉置定義為其包量（bra）：。

此標記法為狄拉克於1939年將「bracket」（括號）這個詞拆開後所造的。^[1]在中國方面，一些舊有的教科書和文獻中也將其譯為「刁矢」和「刃矢」、或「彳矢」和「亍矢」，現已棄用。

……

Bra–ket notation

In quantum mechanics, bra–ket notation is a standard notation for describing quantum states. It can also be used to denote abstract vectors and linear functionals in mathematics. The notation begins with using angle brackets, ⟨ and ⟩, and a vertical bar, |, to denote the scalar product of vectors or the action of a linear functional on a vector in a complex vector space. The scalar product or action is written as

The right part is called the ket /kɛt/; it is a vector, typically represented as a column vector and written

The left part is called the bra, /brɑː/; it is the Hermitian conjugate of the ket with the same label, typically represented as a row vector and is written

A combination of bras, kets, and operators is interpreted using matrix multiplication. A bra and a ket with the same label are Hermitian conjugates of each other.

Bra-ket notation was introduced in 1939 by Paul Dirac^[1][2] and is also known as the Dirac notation.

The bra-ket notation has a precursor in Hermann Grassmann‘s use of the notation   for his inner products nearly 100 years earlier.^[3]

………

 

『直覺性』之由來也◎

 

 

 

 

 

 

 

 

一路跟隨 PyDy tutorial-human-standing 範例文本之腳步，來到了『轉動慣量』的部份︰

An overview of rigid body dynamics

※ 參考

Moment of inertia

The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body is a tensor that determines the torque needed for a desired angular acceleration about a rotational axis. It depends on the body’s mass distribution and the axis chosen, with larger moments requiring more torque to change the body’s rotation. It is an extensive (additive) property: For a point mass the moment of inertia is just the mass times the square of perpendicular distance to the rotation axis. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). One of its definitions is the second moment of mass with respect to distance from an axis r, , integrating over the entire mass .

For bodies constrained to rotate in a plane, it is sufficient to consider their moment of inertia about an axis perpendicular to the plane. For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3 × 3 matrix; each body has a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around the axes act independently of each other.

Tightrope walkers use the moment of inertia of a long rod for balance as they walk the rope. Samuel Dixon crossing the Niagara River in 1890.

 

進入之前，請先確定 SymPy 版本夠新︰

Potential Issues/Advanced Topics/Future Features in Physics/Vector Module

This document will describe some of the more advanced functionality that this module offers but which is not part of the “official” interface. Here, some of the features that will be implemented in the future will also be covered, along with unanswered questions about proper functionality. Also, common problems will be discussed, along with some solutions.

>>> from sympy.physics.mechanics import ReferenceFrame, inertia
>>> N = ReferenceFrame('N')

Supply a reference frame and the moments of inertia if the object
is symmetrical:

>>> inertia(N, 1, 2, 3)
(N.x|N.x) + 2*(N.y|N.y) + 3*(N.z|N.z)

Supply a reference frame along with the products and moments of inertia
for a general object:

>>> inertia(N, 1, 2, 3, 4, 5, 6)
(N.x|N.x) + 4*(N.x|N.y) + 6*(N.x|N.z) + 4*(N.y|N.x) + 2*(N.y|N.y) + 5*(N.y|N.z) + 6*(N.z|N.x) + 5*(N.z|N.y) + 3*(N.z|N.z)

Notice that the inertia function returns a dyadic with each component represented as two unit vectors separated by a |. Refer to the Dyadic section for more information about dyadics.

Inertia is often expressed in a matrix, or tensor, form, especially for numerical purposes. Since the matrix form does not contain any information about the reference frame(s) the inertia dyadic is defined in, you must provide one or two reference frames to extract the measure numbers from the dyadic. There is a convenience function to do this:

>>> inertia(N, 1, 2, 3, 4, 5, 6).to_matrix(N)
Matrix([
[1, 4, 6],
[4, 2, 5],
[6, 5, 3]])

 

 

接著展開『Dyadics』之旅。

 

 

 

 

 

 

 

 

 

如同 SymPy mechanics 程式庫

Masses, Inertias, Particles and Rigid Bodies in Physics/Mechanics

This document will describe how to represent masses and inertias in mechanics and use of the RigidBody and Particle classes.

It is assumed that the reader is familiar with the basics of these topics, such as finding the center of mass for a system of particles, how to manipulate an inertia tensor, and the definition of a particle and rigid body. Any advanced dynamics text can provide a reference for these details.

>>> from sympy.physics.mechanics import Particle, Point
>>> from sympy import Symbol
>>> m = Symbol('m')
>>> po = Point('po')
>>> # create a particle container
>>> pa = Particle('pa', po, m)

The associated point contains the position, velocity and acceleration of the particle. mechanics allows one to perform kinematic analysis of points separate from their association with masses.

 

所言，不熟悉力學基本者，恐難以應用也。

作者嘗試起個頭，作此搭橋之舉，然不過掛一漏萬之說而已。

就像問『粒子』是什麼？強調『質量』時，又常稱之為『質點』。只講其核心概念是可視為『點』的『粒子』

Point particle

A point particle (ideal particle^[1] or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension: being zero-dimensional, it does not take up space.^[2] A point particle is an appropriate representation of any object whose size, shape, and structure is irrelevant in a given context. For example, from far enough away, any finite-size object will look and behave as a point-like object.

※ 參照

Galaxies are so large that stars can be considered particles relative to them

Particle

In the physical sciences, a particle (or corpuscule in older texts) is a small localized object to which can be ascribed several physical or chemical properties such as volume or mass.^[1][2] They vary greatly in size or quantity, from subatomic particles like the electron, to microscopic particles like atoms and molecules, to macroscopic particles like powders and other granular materials. Particles can also be used to create scientific models of even larger objects depending on their density, such as humans moving in a crowd or celestial bodies in motion.

The term ‘particle’ is rather general in meaning, and is refined as needed by various scientific fields. Something that is composed of particles may be referred to as being particulate.^[3] However, the noun ‘particulate‘ is most frequently used to refer to pollutants in the Earth’s atmosphere, which are a suspension of unconnected particles, rather than a connected particle aggregation.

 

In the theory of gravity, physicists often discuss a point mass, meaning a point particle with a nonzero mass and no other properties or structure. Likewise, in electromagnetism, physicists discuss a point charge, a point particle with a nonzero charge.^[3]

Sometimes, due to specific combinations of properties, extended objects behave as point-like even in their immediate vicinity. For example, spherical objects interacting in 3-dimensional space whose interactions are described by the inverse square law behave in such a way as if all their matter were concentrated in their centers of mass. In Newtonian gravitation and classical electromagnetism, for example, the respective fields outside a spherical object are identical to those of a point particle of equal charge/mass located at the center of the sphere.^[4][5]

In quantum mechanics, the concept of a point particle is complicated by the Heisenberg uncertainty principle, because even an elementary particle, with no internal structure, occupies a nonzero volume. For example, the atomic orbit of an electron in the hydrogen atom occupies a volume of ~10⁻³⁰ m³. There is nevertheless a distinction between elementary particles such as electrons or quarks, which have no known internal structure, versus composite particles such as protons, which do have internal structure: A proton is made of three quarks. Elementary particles are sometimes called “point particles”, but this is in a different sense than discussed above.

Property concentrated at a single point

When a point particle has an additive property, such as mass or charge, concentrated at a single point in space, this can be represented by a Dirac delta function.

Physical point mass

 

Point mass (pointlike mass) is the concept, for example in classical physics, of a physical object (typically matter) that has nonzero mass, and yet explicitly and specifically is (or is being thought of or modeled as) infinitesimal(infinitely small) in its volume or linear dimensions.

Application

A common use for point mass lies in the analysis of the gravitational fields. When analyzing the gravitational forces in a system, it becomes impossible to account for every unit of mass individually. However, a spherically symmetric body affects external objects gravitationally as if all of its mass were concentrated at its center.

 

，怕太虛無飄渺乎？？

還是藉『殼層定理』說明『均勻的球』之重力『作用可以等效』於『球心質點』好耶！！

Shell theorem

In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy.

Isaac Newton proved the shell theorem^[1] and stated that:

1. A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point at its centre.
2. If the body is a spherically symmetric shell (i.e., a hollow ball), no net gravitational force is exerted by the shell on any object inside, regardless of the object’s location within the shell.

A corollary is that inside a solid sphere of constant density, the gravitational force varies linearly with distance from the centre, becoming zero by symmetry at the centre of mass. This can be seen as follows: take a point within such a sphere, at a distance  from the centre of the sphere. Then you can ignore all the shells of greater radius, according to the shell theorem. So, the remaining mass is proportional to  , and the gravitational force exerted on it is proportional to   , so to   , so is linear in  .

These results were important to Newton’s analysis of planetary motion; they are not immediately obvious, but they can be proven with calculus. (Alternatively, Gauss’s law for gravity offers a much simpler way to prove the same results.)

In addition to gravity, the shell theorem can also be used to describe the electric field generated by a static spherically symmetric charge density, or similarly for any other phenomenon that follows an inverse square law. The derivations below focus on gravity, but the results can easily be generalized to the electrostatic force. Moreover, the results can be generalized to the case of general ellipsoidal bodies.^[2]

Note: As viewed from m, the shaded blue band appears as a thin annulus whose inner and outer diameters converge to R sin θ as dθ vanishes.

 

但是將它當成『剛體』時，卻又不能看作『質點』呦！！？？

所以終究不如聽一堂 MIT 古典力學的公開課來的好哩？？！！

Video Introduction by Prof. Deepto Chakrabarty and Dr. Peter Dourmashkin

Classical Mechanics Course Introduction

 

Course Meeting Times

Lectures: 2 sessions / week, 2 hours / session

Problem Solving: 1 session / week, 1 hour / session

Prerequisites

This course has no prerequisites. 18.01SC Single Variable Calculus is a corequisite.

Course Overview

This first course in the physics curriculum introduces classical mechanics. Historically, a set of core concepts — space, time, mass, force, momentum, torque, and angular momentum — were introduced in classical  mechanics in order to solve the most famous physics problem, the motion of the planets.

The principles of mechanics successfully described many other phenomena encountered in the world. Conservation laws involving energy, momentum and angular momentum provided a second parallel approach to solving many of the same problems. In this course, we will investigate both approaches: Force and conservation laws.

Our goal is to develop a conceptual understanding of the core concepts, a familiarity with the experimental verification of our theoretical laws, and an ability to apply the theoretical framework to describe and predict the motions of bodies.

Textbook

The textbook for this course is “Classical Mechanics: MIT 8.01 Course Notes” (PDF – 67.9MB) by Peter Dourmashkin. Specific readings for each assignment are provided in the Readings section.

Topics Covered

• Week 1: Kinematics
• Week 2: Newton’s Laws Circular Motion
• Week 3: Circular Motion Momentum and Impulse
• Week 4: Drag Forces, Constraints and Continuous Systems Work and Mechanical Energy
• Week 5: Momentum and Impulse Collision Theory
• Week 6: Continuous Mass Transfer Torque
• Week 7: Kinetic Energy and Work
• Week 8: Potential Energy and Energy Conservation
• Week 9: Collision Theory
• Week 10: Rotational Motion
• Week 11: Angular Momentum
• Week 12: Rotations and Translation – Rolling

 

 

 

 

 

 

 

 

閱讀古今科技歷史往往有很多驚訝！！

On Apollo 11

A well-known gimbal lock incident happened in the Apollo 11 Moon mission. On this spacecraft, a set of gimbals was used on an inertial measurement unit (IMU). The engineers were aware of the gimbal lock problem but had declined to use a fourth gimbal.^[5] Some of the reasoning behind this decision is apparent from the following quote:

“The advantages of the redundant gimbal seem to be outweighed by the equipment simplicity, size advantages, and corresponding implied reliability of the direct three degree of freedom unit.”

— David Hoag, Apollo Lunar Surface Journal

They preferred an alternate solution using an indicator that would be triggered when near to 85 degrees pitch.

“Near that point, in a closed stabilization loop, the torque motors could theoretically be commanded to flip the gimbal 180 degrees instantaneously. Instead, in the LM, the computer flashed a ‘gimbal lock’ warning at 70 degrees and froze the IMU at 85 degrees”

— Paul Fjeld, Apollo Lunar Surface Journal

Rather than try to drive the gimbals faster than they could go, the system simply gave up and froze the platform. From this point, the spacecraft would have to be manually moved away from the gimbal lock position, and the platform would have to be manually realigned using the stars as a reference.^[6]

After the Lunar Module had landed, Mike Collins aboard the Command Module joked “How about sending me a fourth gimbal for Christmas?”

 

比方好奇『萬向鎖』到底是鎖住了什麼？？

想來有時以普通話『解釋名詞』實在也並非容易︰

Gimbal lock

Gimbal lock is the loss of one degree of freedom in a three-dimensional, three-gimbal mechanism that occurs when the axes of two of the three gimbals are driven into a parallel configuration, “locking” the system into rotation in a degenerate two-dimensional space.

The word lock is misleading: no gimbal is restrained. All three gimbals can still rotate freely about their respective axes of suspension. Nevertheless, because of the parallel orientation of two of the gimbals’ axes there is no gimbal available to accommodate rotation along one axis.

Gimbal with 3 axes of rotation. A set of three gimbals mounted together to allow three degrees of freedom: roll, pitch and yaw. When two gimbals rotate around the same axis, the system loses one degree of freedom.

……

In three dimensions

Consider a case of a level sensing platform on an aircraft flying due north with its three gimbal axes mutually perpendicular (i.e., roll, pitch and yaw angles each zero). If the aircraft pitches up 90 degrees, the aircraft and platform’s yaw axis gimbal becomes parallel to the roll axis gimbal, and changes about yaw can no longer be compensated for.

Solutions

This problem may be overcome by use of a fourth gimbal, intelligently driven by a motor so as to maintain a large angle between roll and yaw gimbal axes. Another solution is to rotate one or more of the gimbals to an arbitrary position when gimbal lock is detected and thus reset the device.

Modern practice is to avoid the use of gimbals entirely. In the context of inertial navigation systems, that can be done by mounting the inertial sensors directly to the body of the vehicle (this is called a strapdown system)^[3] and integrating sensed rotation and acceleration digitally using quaternion methods to derive vehicle orientation and velocity. Another way to replace gimbals is to use fluid bearings or a flotation chamber.^[4]

 

故而仍舊回歸其本的好呀！！？？

In applied mathematics

The problem of gimbal lock appears when one uses Euler angles in applied mathematics; developers of 3D computer programs, such as 3D modeling, embedded navigation systems, and video games must take care to avoid it.

In formal language, gimbal lock occurs because the map from Euler angles to rotations (topologically, from the 3-torus T³ to the real projective space RP³) is not a covering map – it is not a local homeomorphism at every point, and thus at some points the rank (degrees of freedom) must drop below 3, at which point gimbal lock occurs. Euler angles provide a means for giving a numerical description of any rotation in three-dimensional space using three numbers, but not only is this description not unique, but there are some points where not every change in the target space (rotations) can be realized by a change in the source space (Euler angles). This is a topological constraint – there is no covering map from the 3-torus to the 3-dimensional real projective space; the only (non-trivial) covering map is from the 3-sphere, as in the use of quaternions.

To make a comparison, all the translations can be described using three numbers  ,   , and , as the succession of three consecutive linear movements along three perpendicular axes  ,   and  axes. The same holds true for rotations: all the rotations can be described using three numbers α ,  β , and γ , as the succession of three rotational movements around three axes that are perpendicular one to the next. This similarity between linear coordinates and angular coordinates makes Euler angles very intuitive, but unfortunately they suffer from the gimbal lock problem.

Loss of a degree of freedom with Euler angles

A rotation in 3D space can be represented numerically with matrices in several ways. One of these representations is:

An example worth examining happens when  . Knowing that  and  , the above expression becomes equal to:

Carrying out matrix multiplication:

And finally using the trigonometry formulas:

Changing the values of α and γ in the above matrix has the same effects: the rotation angle α+γ changes, but the rotation axis remains in the  direction: the last column and the first row in the matrix won’t change. The only solution for α and γ to recover different roles is to change β.

It is possible to imagine an airplane rotated by the above-mentioned Euler angles using the X-Y-Z convention. In this case, the first angle – α is the pitch. Yaw is then set to  and the final rotation – by γ – is again the airplane’s pitch. Because of gimbal lock, it has lost one of the degrees of freedom – in this case the ability to roll.

It is also possible to choose another convention for representing a rotation with a matrix using Euler angles than the X-Y-Z convention above, and also choose other variation intervals for the angles, but in the end there is always at least one value for which a degree of freedom is lost.

The gimbal lock problem does not make Euler angles “invalid” (they always serve as a well-defined coordinate system), but it makes them unsuited for some practical applications.

Alternate orientation representation

The cause of gimbal lock is representing an orientation as three axial rotations with Euler angles. A potential solution therefore is to represent the orientation in some other way. This could be as a rotation matrix, a quaternion (see quaternions and spatial rotation), or a similar orientation representation that treats the orientation as a value rather than three separate and related values. Given such a representation, the user stores the orientation as a value. To apply angular changes, the orientation is modified by a delta angle/axis rotation. The resulting orientation must be re-normalized to prevent floating-point error from successive transformations from accumulating. For matrices, re-normalizing the result requires converting the matrix into its nearest orthonormal representation. For quaternions, re-normalization requires performing quaternion normalization.

 

但因『形式語』常常難以領會，且此『借例說例』吧？？！！

如果已知一『旋轉矩陣』 ，將如何求解『歐拉角』呢？

假使依循前述文本之『約定』，可推導如下也︰

※ 注意不同的『慣例』︰

Euler angles

Complexity of conversion escalates with Euler angles (used here in the broad sense). The first difficulty is to establish which of the twenty-four variations of Cartesian axis order we will use. Suppose the three angles are θ₁, θ₂, θ₃; physics and chemistry may interpret these as

while aircraft dynamics may use

One systematic approach begins with choosing the rightmost axis. Among all permutations of (x,y,z), only two place that axis first; one is an even permutation and the other odd. Choosing parity thus establishes the middle axis. That leaves two choices for the left-most axis, either duplicating the first or not. These three choices gives us 3 × 2 × 2 = 12 variations; we double that to 24 by choosing static or rotating axes.

This is enough to construct a matrix from angles, but triples differing in many ways can give the same rotation matrix. For example, suppose we use the zyz convention above; then we have the following equivalent pairs:

(90°,	45°,	−105°)	≡	(−270°,	−315°,	255°)	multiples of 360°
(72°,	0°,	0°)	≡	(40°,	0°,	32°)	singular alignment
(45°,	60°,	−30°)	≡	(−135°,	−60°,	150°)	bistable flip

Angles for any order can be found using a concise common routine (Herter & Lott 1993; Shoemake 1994).

The problem of singular alignment, the mathematical analog of physical gimbal lock, occurs when the middle rotation aligns the axes of the first and last rotations. It afflicts every axis order at either even or odd multiples of 90°. These singularities are not characteristic of the rotation matrix as such, and only occur with the usage of Euler angles.

The singularities are avoided when considering and manipulating the rotation matrix as orthonormal row vectors (in 3D applications often named the right-vector, up-vector and out-vector) instead of as angles. The singularities are also avoided when working with quaternions.

 

這裡 眼見垂手可得。不過

，將有『兩解』哩！

要是 ，得出

以及

亦無困擾乎？★

※ 小心適用『角度範圍』︰

Atan2

在三角函數中，兩個參數的函數atan2是正切函數tan的一個變種。對於任意不同時等於0的實參數x和y，atan2(y,x)所表達的意思是坐標原點為起點，指向(x,y)的射線在坐標平面上與x軸正方向之間的角的角度。當y>0時，射線與x軸正方向的所得的角的角度指的是x軸正方向繞逆時針方向到達射線旋轉的角的角度；而當y<0時，射線與x軸正方向所得的角的角度指的是x軸正方向繞順時針方向達到射線旋轉的角的角度。

在幾何意義上，atan2(y, x) 等價於 atan(y/x)，但 atan2 的最大優勢是可以正確處理 x=0 而 y≠0 的情況，而不必進行會引發除零異常的 y/x 操作。

atan2函數最初在計算機程式語言中被引入，但是現在它的應用在科學和工程等其他多個領域十分常見。他的出現最早可以追溯到FORTRAN語言^[1]，並且可以在C語言的數學標準庫的math.h文件中找到，此外在Java數學庫、.NET的System.Math（可應用於C#、VB.NET等語言）、Python的數學模塊以及其他地方都可以找到atan2的身影。許多腳本語言，比如Perl，也包含了C語言風格的atan2函數^[2]。

函數定義

基於值域為   的反正切函數，該函數定義如下：

說明:

• 該函數的值域為 ，可以通過對負數結果加  的方法，將函數的結果映射到  範圍內。

 

其餘所謂  □ □ ○ ○ 可自得之的勒？☆

 

 

 

 

 

 

 

 

 

由於維基百科『軸角表示』詞條之

Rotating a vector

Rodrigues’ rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a Euclidean vector, given a rotation axis and an angle of rotation. In other words, Rodrigues’ formula provides an algorithm to compute the exponential map from (3) to SO(3) without computing the full matrix exponential.

If v is a vector in ℝ³ and e is a unit vector rooted at the origin describing an axis of rotation about which v is rotated by an angle θ, Rodrigues’ rotation formula to obtain the rotated vector is

For the rotation of a single vector it may be more efficient than converting e and θ into a rotation matrix to rotate the vector.

Relationship to other representations

There are several ways to represent a rotation. It is useful to understand how different representations relate to one another, and how to convert between them. Here the unit vector is denoted ω instead of e.

Exponential map from so(3) to SO(3)

The exponential map effects a transformation from the axis-angle representation of rotations to rotation matrices,

Essentially, by using a Taylor expansion one derives a closed-form relation between these two representations. Given a unit vector ω ∈   (3) = ℝ³ representing the unit rotation axis, and an angle, θ ∈ ℝ, an equivalent rotation matrix R is given as follows, where Kis the cross product matrix of ω, that is, Kv = ω × v for all vectors v ∈ ℝ³,

Because K is skew-symmetric, and the sum of the squares of its above-diagonal entries is 1, the characteristic polynomial P(t) of K is P(t) = det(K − tI) = −(t³ + t). Since, by the Cayley–Hamilton theorem, P(K) = 0, this implies that

As a result, K⁴ = –K², K⁵ = K, K⁶ = K², K⁷ = –K.

This cyclic pattern continues indefinitely, and so all higher powers of K can be expressed in terms of K and K². Thus, from the above equation, it follows that

that is,

This is a Lie-algebraic derivation, in contrast to the geometric one in the article Rodrigues’ rotation formula.^[1]

Due to the existence of the above-mentioned exponential map, the unit vector ω representing the rotation axis, and the angle θ are sometimes called the exponential coordinates of the rotation matrix R.

Log map from SO(3) to so(3)

Let K continue to denote the 3 × 3 matrix that effects the cross product with the rotation axis ω: K(v) = ω × v for all vectors v in what follows.

To retrieve the axis–angle representation of a rotation matrix, calculate the angle of rotation from the trace of the rotation matrix

and then use that to find the normalized axis,

Note also that the Matrix logarithm of the rotation matrix R is

An exception occurs when R has eigenvalues equal to −1. In this case, the log is not unique. However, even in the case where θ = π the Frobenius norm of the log is

Given rotation matrices A and B,

is the geodesic distance on the 3D manifold of rotation matrices.

For small rotations, the above computation of θ may be numerically imprecise as the derivative of arccos goes to infinity as θ → 0. In that case, the off-axis terms will actually provide better information about θ since, for small angles, R ≈ I + θK. (This is because these are the first two terms of the Taylor series for exp(θK).)

This formulation also has numerical problems at θ = π, where the off-axis terms do not give information about the rotation axis (which is still defined up to a sign ambiguity). In that case, we must reconsider the above formula.

At θ = π, we have

and so let

so the diagonal terms of B are the squares of the elements of ω and the signs (up to sign ambiguity) can be determined from the signs of the off-axis terms of B.

 

文本已將計算法及特殊情況解釋的十分清楚，此處不過拾遺而已。

通常旋轉軸 可藉

得出

，如果 。

再者因為矩陣『跡』

Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e.,

where a_ii denotes the entry on the ith row and ith column of A. The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis. This characterization can be used to define the trace of a linear operator in general. Note that the trace is only defined for a square matrix (i.e., n × n).

The trace (often abbreviated to “tr”) is related to the derivative of the determinant (see Jacobi’s formula).

The following three properties:

 ,

characterize the trace completely in the sense that follows. Let f be a linear functional on the space of square matrices satisfying f(x y) = f(y x). Then f and tr are proportional.^[2]

The trace is similarity-invariant, which means that A and P⁻¹AP have the same trace. This is because

 .

If A is symmetric and B is antisymmetric, then

 .

The trace of the identity matrix is the dimension of the space; this leads to generalizations of dimension using trace. The trace of an idempotent matrix A (for which A² = A) is the rank of A. The trace of a nilpotent matrix is zero.

More generally, if f(x) = (x − λ₁)^d₁···(x − λ_k)^d_k is the characteristic polynomial of a matrix A, then

 .

When both A and B are n-by-n, the trace of the (ring-theoretic) commutator of A and B vanishes: tr([A, B]) = 0; one can state this as “the trace is a map of Lie algebras  from operators to scalars”, as the commutator of scalars is trivial (it is an abelian Lie algebra). In particular, using similarity invariance, it follows that the identity matrix is never similar to the commutator of any pair of matrices.

Conversely, any square matrix with zero trace is a linear combinations of the commutators of pairs of matrices.^[3] Moreover, any square matrix with zero trace is unitarily equivalent to a square matrix with diagonal consisting of all zeros.

The trace of any power of a nilpotent matrix is zero. When the characteristic of the base field is zero, the converse also holds: if  for all , then  is nilpotent.

The trace of a Hermitian matrix is real, because the elements on the diagonal are real.

The trace of a projection matrix is the dimension of the target space.

 .

Note that is idempotent, and more generally the trace of any idempotent matrix equals its rank.

 

之性質，自可想像某一旋轉 將之變換到 軸之一︰

Basic rotations

A basic rotation (also called elemental rotation) is a rotation about one of the axes of a Coordinate system. The following three basic rotation matrices rotate vectors by an angle θ about the x-, y-, or z-axis, in three dimensions, using the right-hand rule—which codifies their alternating signs. (The same matrices can also represent a clockwise rotation of the axes.^{[nb 1]})

此時『跡』是 ，終將之轉回 ，故得

矣。

其餘自能玩轉深入呦◎