儘管為文主旨是想介紹『矩陣光學』實務應用，如果假設讀者已知 □ ○，這樣只需列出

Table of ray transfer matrices

for simple optical components

Element	Matrix	Remarks
Propagation in free space or in a medium of constant refractive index		d = distance
Refraction at a flat interface		n1 = initial refractive index n2 = final refractive index.
Refraction at a curved interface		R = radius of curvature, R > 0 for convex (centre of curvature after interface) n1 = initial refractive index n2 = final refractive index.
Reflection from a flat mirror		Only valid for mirrors perpendicular to the ray.
Reflection from a curved mirror		effective radius of curvature in tangential plane (horizontal direction)   effective radius of curvature in the sagittal plane (vertical direction) R = radius of curvature, R > 0 for concave, valid in the paraxial approximation is the mirror angle of incidence in the horizontal plane.
Thin lens		f = focal length of lens where f > 0 for convex/positive (converging) lens.Only valid if the focal length is much greater than the thickness of the lens.
Thick lens		n1 = refractive index outside of the lens. n2 = refractive index of the lens itself (inside the lens). R1 = Radius of curvature of First surface. R2 = Radius of curvature of Second surface. t = center thickness of lens.
Single right angle prism		k = (cos  /cos ϕ  ) is the beam expansion factor, where  is the angle of incidence,  is the angle of refraction, d = prism path length, n = refractive index of the prism material. This matrix applies for orthogonal beam exit.

 

就好。但思『事理不二』，要是『理』都不清，『事』能明乎？？那又何必曾講『勸學篇』也！！

訊︰☿ 把酒飛斝是同道，欲法荀子《勸學篇》趁年少︰

君子曰：學不可以已。青，取之於藍而青於藍；冰，水為之而寒於水。〈以喻學則才過其本性也。〉木直中繩，輮以為輪，其曲中規，雖有槁暴，不復挺者，輮使之然也。〈輮，屈。槁，枯。曓，乾。挻，宜也。《晏子春秋》作「不復贏也」。〉故木受繩則直，金就礪則利，君子博學而日參省乎己，則知明而行無過矣。〈參，三也。曾子曰︰「日三省吾身。」知，讀爲智。行，下孟反。〉故不登高山，不知天之高也；不臨深谿，不知地之厚也；不聞先王之遺言，不知學問之大也。〈大，謂有益於人。〉干、越、夷、貉之子，生而同聲，長而異俗，教使之然也。〈干、越，猶言吳、越。《呂氏春秋》「荊有次非得寶劍於干、越」，高誘曰︰「吳邑也。」貉，東北夷。同聲，謂啼聲同。貉，莫革反。〉《詩》曰：「嗟爾君子，無恆安息。靖共爾位，好是正直。神之聽之，介爾景福。」〈《詩》，《小雅‧小明》之篇。靖，謀。介，助。景，大也。無恆安息，戒之不使懷安也。言能謀恭其位，好正宜之道，則神聽而助之福，引此詩以喻勤學也。〉神莫大於化道，福莫長於無禍。〈爲學則自化道，故神莫大焉。修身則自無禍，故福莫長焉。〉吾嘗終日而思矣，不如須臾之所學也，吾嘗跂而望矣，不如登高之博見也 。〈跂，舉足也。〉登高而招，臂非加長也，而見者遠；順風而呼，聲非加疾也，而聞者彰。假輿馬者，非利足也，而致千里；假舟楫者，非能水也，而絕江河。〈能，善。絶，過。〉君子生非異也，善假於物也。〈皆以喻修身在假於學。生非異，言與衆人同也。〉南方有鳥焉，名曰蒙鳩，以羽為巢而編之以髮，繫之葦苕，風至苕折，卵破子死。巢非不完也 ，所繫者然也。〈蒙 鳩，鷦鷯也。苕，葦之秀也，今巧婦鳥之巢至精密，多繫於葦竹之上是也。「蒙」當爲「蔑」。《方言》雲︰「鷦鷯，關而西謂之桑飛，或謂之蔑雀。」或曰︰一名 蒙鳩，亦以其愚也。言人不知學問，其所置身亦猶繫葦之危也。《說苑》︰「客謂孟嘗君曰︰『鷦鷯巢於葦苕，箸之髮毛，可謂完堅矣，大風至則苕折卵破子死者何 也？其所託者然也。』〉西方有木焉，名曰射干，莖長四寸，生於高山之上，而臨百仞之淵，木莖非能長也，所立者然也 。〈《本草》藥名有射干，一名烏扇。陶弘景雲︰「花白莖長，如射人之執竿。」又引阮公詩云「射干臨層城」，是生於高處也。據《本草》在《草部》中，又生南陽川穀，此雲「西方有木」，未詳。或曰︰「長四寸」卽是草，雲木，誤也。蓋生南陽，亦生西方也。射音夜。〉蓬生麻中，不扶而直。蘭槐之根是爲芷。其漸之滫，君子不近，庶人不服，其質非不美也，所漸者然也。〈蘭槐，香草，其根是爲芷也。《本草》︰「白芷一名白茝。」陶弘景雲︰「卽《離騷》所謂蘭茝也。」葢苗名蘭茝，根名芷也。弱槐當是蘭茝別名，故云「蘭槐之根是爲芷」也。漸，漬也，染也。滫，溺也。言雖香草，浸漬於溺中，則可惡也。漸，子廉反。滫，思酒反。〉故君子居必擇鄉，遊必就士，所以防邪僻而近中正也。物類之起，必有所始。榮辱之來，必象其德。肉腐出蟲，魚枯生蠹。怠慢忘身，禍災乃作。強自取柱，柔自取束。〈凡物強則以爲柱而任勞，柔則見束而約急，皆其自取也。〉邪穢在身，怨之所構。〈構，結也。言亦所自取。〉施薪若一，火就燥也；〈布薪於地，均若一，火就燥而焚之矣。〉平地若一，水就溼也。草木疇生，禽獸羣焉，物各從其類也。〈疇與儔同，類也。〉是故質的張而弓矢至焉，林木茂而斧斤至焉，〈所謂召禍也。質，射矦。的，正鵠也。〉樹成蔭而衆鳥息焉，醯酸而蜹聚焉。〈喻有德則慕之者衆。〉故言有召禍也，行有招辱也，君子慎其所立乎！〈禍福如此，不可不慎所立。所立，卽謂學也。〉

積土成山，風雨興焉；積水成淵，蛟龍生焉；積善成德，而神明自得，聖心備焉。〈神明自得，謂自通於神明。〉故不積蹞步，無以致千里；〈半步曰蹞。蹞與跬同。〉不積小流，無以成江海。騏驥一躍，不能十步；駑馬十駕，〈言駑馬十度引車，則亦及騏驥之一躍。據下雲「駑馬十駕，則亦及之」，此亦當同，疑脫一句。〉功在不舍。鍥而舍之，朽木不折；鍥而不舍 ，金石可鏤。〈言立功在於不舍。舍與捨同。鍥，刻也，苦結反。《春秋傳》曰「陽虎借邑人之車，鍥其軸」也。〉螾無爪牙之利，筋骨之強，上食埃土，下飲黃泉，用心一也。〈螾與蚓同，蚯蚓也。〉蟹八跪而二螯，非虵蟺之穴無可寄託者，用心躁也。〈跪，足也。《韓子》以刖足爲刖跪。螫，蟹首上如鉞者。許叔重《說文》雲「蟹六足二螫」也。〉是故無冥冥之志者無昭昭之明 ，無惛惛之事者無赫赫之功。〈冥冥、惛惛，皆專默精誠之謂也。〉行衢道者不至，事兩君者不容。〈《爾雅》雲︰「四達謂之衢。」孫炎雲︰「衢，交道四出也。」或曰︰衢道，兩道也。不至，不能有所至。下篇有「楊朱哭衢塗」。今秦俗猶以兩爲衢，古之遺言歟？〉目不能兩視而明，耳不能兩聽而聰。螣蛇無足而飛，〈《爾雅》云：「螣，螣蛇。」郭璞雲「龍類，能興雲霧而遊其中」也。〉梧鼠五技而窮。〈「梧鼠」當爲「鼫鼠」，蓋本誤爲「鼯」字，傳寫又誤爲「梧」耳。技，才能也。言技能雖多而不能如螣蛇專一，故窮。五技，謂能飛不能上屋，能緣不能窮木，能游不能渡谷，能穴不能掩身，能走不能先人。〉《詩》曰 ：「屍鳩在桑，其子七兮。淑人君子，其儀一兮。其儀一兮，心如結兮。」故君子結於一也。〈《詩》，《曹風‧屍鳩》之篇。毛雲︰「屍鳩，鴶鞠也。屍鳩之養七子，旦從上而下，暮從下而上，平均如一。善人君子，其執義亦當如屍鳩之一。執義一則用心堅固。」故曰「心如結」也。

……… 摘自《M♪o 之學習筆記本《編者跋》》

 

豈非對好學者太失禮耶？？！！

聽聞楊百翰大學多年來開設光學教育課程︰

 

幸得 Justin Peatross 與 Michael Ware 先生們之慷慨︰

 

可以省憂少勞矣！！！故此還請讀者先讀

Optics Textbook

Physics of Light and Optics is a high-quality free textbook desiged for an advanced undergraduate optics course for physics majors. It is used at universities around the world. You are welcome to use the text for your course or personal study or for a formal course within the following guidelines:

• The text may not be sold for profit. If you would like to use it in a class, you may provide copies to the students for the cost of printing.
• Please do not publish solutions to the homework problems.
• Please let us know if you use the text for either personal or classroom study; we would like to know how widely the book is used!

The textbook is available in PDF format on this web site, and also as a printed and bound copy available for the cost of printing, available here:

• BYU Bookstore (Black and White)
• BYU Bookstore (Color)
• lulu.com

The authors (Justin Peatross and Michael Ware) welcome constructive feedback, which can be sent to opticsbook@byu.edu.

Why an online textbook?

While the authors retain the copyright to the book, we have made this book available free of charge as our contribution toward a future with free textbooks! We firmly believe in the free market system, and in the case of upper division physics textbooks, traditional publishing methods usually make little economic sense for the author or reader.  The small royalty from traditional publishing comes with a giant overhead that make the book nearly cost prohibitive to many students (especially those in developing countries), and the author gives up control of his work to the publishing company.  Electronic tools make it easy to produce and distribute a professional product (see our section on creating a textbook), and it can be revised, corrected, and enhanced on the authors’ timetable. If you are thinking about publishing a book this way, we highly recommend it.

 

之『第九章』

 

踏上旅程吧！！？？

 

 

 

 

 

 

 

 

 

 

仲夏入園中東陂‧唐‧儲光羲

方塘深且廣，伊昔俯吾廬。
環岸垂綠柳，盈澤發紅蕖。
上延北原秀，下屬幽人居。
暑雨若混沌，清明如空虛。
此鄉多隱逸，水陸見樵漁。
廢賞亦何貴，為歡良易攄。
且言重觀國，當此賦歸歟。

 

欲究 skimage 相似變換之理︰

SimilarityTransform

class skimage.transform.SimilarityTransform(matrix=None, scale=None, rotation=None, translation=None)

Bases: skimage.transform._geometric.EuclideanTransform

2D similarity transformation of the form:

X = a0 * x – b0 * y + a1 =

= s * x * cos(rotation) – s * y * sin(rotation) + a1

Y = b0 * x + a0 * y + b1 =

= s * x * sin(rotation) + s * y * cos(rotation) + b1

where s is a scale factor and the homogeneous transformation matrix is:

[[a0  b0  a1]
 [b0  a0  b1]
 [0   0    1]]

The similarity transformation extends the Euclidean transformation with a single scaling factor in addition to the rotation and translation parameters.

Parameters:	matrix : (3, 3) array, optional Homogeneous transformation matrix. scale : float, optional Scale factor. rotation : float, optional Rotation angle in counter-clockwise direction as radians. translation : (tx, ty) as array, list or tuple, optional x, y translation parameters.

Attributes

params

((3, 3) array) Homogeneous transformation matrix.

 

，如墜混沌中，莫非是因暑雨後，頭昏腦不清！這 何指？那 又是什麼 □ ○ ？？考之 draw 模組圓盤和線段之說明︰

circle

skimage.draw.circle(r, c, radius, shape=None)

Generate coordinates of pixels within circle.

Parameters:	r, c : double Centre coordinate of circle. radius : double Radius of circle. shape : tuple, optional Image shape which is used to determine the maximum extent of output pixel coordinates. This is useful for circles which exceed the image size. By default the full extent of the circle are used.
Returns:	rr, cc : ndarray of int Pixel coordinates of circle. May be used to directly index into an array, e.g. img[rr, cc] = 1.

Notes

This function is a wrapper for skimage.draw.ellipse()

……

line

skimage.draw.line()

Generate line pixel coordinates.

Parameters:	y0, x0 : int Starting position (row, column). y1, x1 : int End position (row, column).
Returns:	rr, cc : (N,) ndarray of int Indices of pixels that belong to the line. May be used to directly index into an array, e.g. img[rr, cc] = 1.

See also

line_aa

Anti-aliased line generator

 

明示『列』 row 與『行』 column ！！ 在 之前，難道是講︰ 耶？？既然無由計，方塘深且廣，怎不直搗黃龍，親驗座標系乎！！且以笛卡爾座標系── 橫平   、豎直 ，右手逆旋角度為正，左手順轉角度為負 ── 考察一番︰

pi@raspberrypi:~ YX$ ！！當真的呢鏡中定『右逆』、『左順』之轉向也！！為何這樣寫？？只覺太忽悠？？







倏的本意︰

 狗因杖打疾速逃窜。
《説文解字》：倏，走也。从犬，攸聲。
忽之語源︰

 未放在心上，不重视。
《説文解字》：忽，忘也。从心，勿聲。









《山海經‧西次三經》：
又西三百五十里曰天山，多金玉，有青雄黃，英水出焉，而西南流注於湯谷。有神焉，其狀如黃囊，赤如丹火，六足四翼，渾敦無面目，是識歌舞，實惟帝江也。
《莊子‧內篇‧應帝王》︰
南海之帝為倏，北海之帝為忽，中央之帝為混沌。

 倏與忽時相與遇於混沌之地，混沌待之甚善。

 倏與忽謀報混沌之德，曰：
『人皆有七竅，以視聽食息，此獨無有，嘗試鑿之。』
日鑿一竅，七日而混沌死。
不 知莊子是否在忽悠人的嗎？混沌和盤古有無關係的呢？？


─── 摘自《混沌理論》
 
總不會想教人將文件看清楚，不明白時請反覆實驗吧？？！！其後偶讀
 

A crash course on NumPy for images
……
Coordinate conventions
Because we represent images with numpy arrays, our coordinates must match accordingly. Two-dimensional (2D) grayscale images (such as camera above) are indexed by row and columns (abbreviated to either row, col or r, c), with the lowest element (0, 0) at the top- -left corner. In various parts of the library, you will also see rr and cc refer to lists of row and column coordinates. We distinguish this from (x, y), which commonly denote standard Cartesian coordinates, where x is the horizontal coordinate, y the vertical, and the origin is on the bottom right. (Matplotlib, for example, uses this convention.)
In the case of color (or multichannel) images, the last dimension contains the color information and is denoted channel or ch.
Finally, for 3D images, such as videos, magnetic resonance imaging (MRI) scans, or confocal microscopy, we refer to the leading dimension as plane, abbreviated as pln or p.
These conventions are summarized below:

Dimension name and order conventions in scikit-image


 


Image type
coordinates




2D grayscale
(row, col)


2D multichannel (eg. RGB)
(row, col, ch)


3D grayscale
(pln, row, col)


3D multichannel
(pln, row, col, ch)



 
只能哈哈大笑，何得清明如空虛矣！！！
 
 
 
 
 
 
 
 
 
 
 
 
 
	

送子由使契丹‧蘇軾

雲海相望寄此身，那因遠適更沾巾。
不辭驛騎淩風雪，要使天驕識鳳麟。
沙漠回看清禁月，湖山應夢武林春。
單於若問君家世，莫道中朝第一人。

 

既談派生科學計算，如果沒介紹『SciPy』套件集成，當真不道地也。無奈耶雲海無涯！不得不屢屢遠謫矣？

 

未必時宜的講︰

幾乎每個核心軟件都需要一本書之份量才能說得清楚，於是也只能請讀者自己找書來看，或是參考線上文件︰

 

借著導引指南嘗試摸索探究︰

Getting Started

This page is intended to help the beginner get a handle on SciPy and be productive with it as fast as possible.

 

自有一番樂趣乎？？！！

pi@raspberrypi:~ $ ipython3 --pylab
Python 3.4.2 (default, Oct 19 2014, 13:31:11) 
Type "copyright", "credits" or "license" for more information.

IPython 2.3.0 -- An enhanced Interactive Python.
?         -> Introduction and overview of IPython's features.
%quickref -> Quick reference.
help      -> Python's own help system.
object?   -> Details about 'object', use 'object??' for extra details.
Using matplotlib backend: TkAgg

In [1]: from scipy import special, optimize

In [2]: f = lambda x: -special.jv(3, x)

In [3]: sol = optimize.minimize(f, 1.0)

In [4]: x = linspace(0, 10, 5000)

In [5]: x
Out[5]: 
array([  0.00000000e+00,   2.00040008e-03,   4.00080016e-03, ...,
         9.99599920e+00,   9.99799960e+00,   1.00000000e+01])

In [6]: plot(x, special.jv(3, x), '-', sol.x, -sol.fun, 'o')
Out[6]: 
[<matplotlib.lines.Line2D at 0x722c6470>,
 <matplotlib.lines.Line2D at 0x722c6650>]

In [7]: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

廬山東林寺三笑庭名聯‧清‧唐蝸寄

橋跨虎溪，三教三源流，三人三笑語；
蓮開僧舍，一花一世界，一葉一如來。

曾經三人三笑語，聞得虎嘯，恍然大悟。何故三詠三抒懷︰

詠荊軻‧陶淵明

燕丹善養士，誌在報強嬴。
招集百夫良，歲暮得荊卿。
君子死知己，提劍出燕京；
素驥鳴廣陌，慷慨送我行。
雄發指危冠，猛氣衝長纓。
飲餞易水上，四座列群英。
漸離擊悲筑，宋意唱高聲。
蕭蕭哀風逝，淡淡寒波生。
商音更流涕，羽奏壯士驚。
心知去不歸，且有後世名。
登車何時顧，飛蓋入秦庭。
淩厲越萬裏，逶迤過千城。
圖窮事自至，豪主正怔營。
惜哉劍術疏，奇功遂不成！
其人雖已沒，千載有餘情。

晉時淵明，晉後名潛，已棄五斗米，不知姓字忘其何人，五柳先生『伍』『柳』吟誦耶？？果真二三子其志一也！！雖說是移時隔空得失不同，其人其心何其相似乎？？！！先生『菀柳』之『情』仍一樣吧！！？？

誰雲其人亡，久而道彌著。

良人不可贖，泫然沾我衣。

其人雖已沒，千載有餘情。

大暑已過，入秋之際，講此『春耕夏耘』之『心法』勒。古今中外『學問』縱有千百種，談起功夫『心法』則一矣。往往其『志一』其『人同』也！！只是『一花』『一葉』真誠對待歟？？

如果天地公平，天理能隨觀者變乎？？故知物理學之所以著意於

仿射變換之性質了！！

一仿射變換保留了：

1. 點之間的共線性，例如通過同一線之點 （即稱為共線點)在變換後仍呈共線。
2. 向量沿著一線的比例，例如對相異共線三點 與 的比例同於 及  。
3. 帶不同質量的點之質心。

一仿射變換為可逆的若且唯若A為可逆的。在矩陣表示中，其反元素為

可逆仿射變換組成仿射群，其中包含具n階的一般線性群為子群，且自身亦為一n+1階的一般線性群之子群。 當A為常數乘以正交矩陣時，此子集合構成一子群，稱之為相似變換。舉例而言，假如仿射變換於一平面上且假如A之行列式為1或-1，那麼該變換即為等面積變換。此類變換組成一稱為等仿射群的子集。一同時為等面積變換與相似變換之變換，即為一平面上保持歐幾里德距離不變之保距映射。 這些群都有一保留了原定向的子群，也就是其對應之A的行列式大於零。在最後一例中，即為三維中剛體運動之群（旋轉加平移）。 假如有一不動點，我們可以將其當成原點，則仿射變換被縮還到一線性變換。這使得變換更易於分類與理解。舉例而言，將一變換敘述為特定軸的旋轉，相較於將其形容為平移與旋轉的結合，更能提供變換行為清楚的解釋。只是，這取決於應用與內容。

 

若知『平移』與『旋轉』為『保距映射』，餘理可知矣︰

※矩陣乘法不具『交換性』，正是『平移』後『旋轉』往往不等於『旋轉』再『平移』也。

pi@raspberrypi:~  ipython3
Python 3.4.2 (default, Oct 19 2014, 13:31:11) 
Type "copyright", "credits" or "license" for more information.

IPython 2.3.0 -- An enhanced Interactive Python.
?         -> Introduction and overview of IPython's features.
%quickref -> Quick reference.
help      -> Python's own help system.
object?   -> Details about 'object', use 'object??' for extra details.

In [1]: from skimage import data, io, filter

In [2]: image = data.coins()

In [3]: edges = filter.sobel(image)

In [4]: io.imshow(edges)

In [5]: io.show()

In [6]: io.imshow(image)

In [7]: io.show()

In [8]: 文本 = data.text()

 

 

 

須知積極更新發展快速之程式庫，文件往往可能『前差後異』也？

In [9]: from skimage import transform as tf

In [10]: 旋轉文本 = tf.rotate(文本, 45, resize=True)

In [11]: io.imshow(文本)

In [12]: io.show()

In [13]: io.imshow(旋轉文本)

In [14]: io.show()

In [15]: import math

In [16]: 相似變換 = tf.SimilarityTransform( rotation=math.pi / 4, translation=( 文本.sape[0] / 2, -100))

In [17]: 相似變換文本 = tf.warp(文本, 相似變換)

In [18]: io.imshow(相似變換文本)

In [19]: io.show()

In [20]: 

 

 

 

 

難到山不會轉，竟然路也不能轉耶？？！！

 

 

 

 

 

 

 

 

 

 

 

 

 

 

歐式幾何裡並無『座標系』之概念，而是以『全等』關係︰

Congruence (geometry)

In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.^[1]

More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. So two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted.

In elementary geometry the word congruent is often used as follows.^[2] The word equal is often used in place of congruent for these objects.

• Two line segments are congruent if they have the same length.
• Two angles are congruent if they have the same measure.
• Two circles are congruent if they have the same diameter.

In this sense, two plane figures are congruent implies that their corresponding characteristics are “congruent” or “equal” including not just their corresponding sides and angles, but also their corresponding diagonals, perimeters and areas.

The related concept of similarity applies if the objects differ in size but not in shape.

An example of congruence. The two triangles on the left are congruent, while the third is similar to them. The last triangle is neither similar nor congruent to any of the others. Note that congruence permits alteration of some properties, such as location and orientation, but leaves others unchanged, like distance and angles. The unchanged properties are called invariants.

 

研究各類圖形具有種種性質之『不變性』。

參照之前文本所說︰

……

從物理上講，那個三角形就是『剛體』 rigid body ，它在『運動』中保持『形狀』的『不變性』。而且不同觀察者間的『座標變換』可以用

Rigid transformation

In mathematics, a rigid transformation (isometry) of a vector space preserves distances between every pair of points.^[1][2] Rigid transformations of the plane R², space R³, or real n-dimensional space Rⁿ are termed a Euclidean transformation because they form the basis of Euclidean geometry.^[3]

The rigid transformations include rotations, translations, reflections, or their combination. Sometimes reflections are excluded from the definition of a rigid transformation by imposing that the transformation also preserve the handedness of figures in the Euclidean space (a reflection would not preserve handedness; for instance, it would transform a left hand into a right hand). To avoid ambiguity, this smaller class of transformations is known as proper rigid transformations (informally, also known as roto-translations). In general, any proper rigid transformation can be decomposed as a rotation followed by a translation, while any rigid transformation can be decomposed as an improper rotation followed by a translation (or as a sequence of reflections).

Any object will keep the same shape and size after a proper rigid transformation.

All rigid transformations are examples of affine transformations. The set of all (proper and improper) rigid transformations is a group called the Euclidean group, denoted E(n) for n-dimensional Euclidean spaces. The set of proper rigid transformation is called special Euclidean group, denoted SE(n).

In kinematics, proper rigid transformations in a 3-dimensional Euclidean space, denoted SE(3), are used to represent the linear and angular displacement of rigid bodies. According to Chasles’ theorem, every rigid transformation can be expressed as a screw displacement.

Formal definition

A rigid transformation is formally defined as a transformation that, when acting on any vector v, produces a transformed vector T(v) of the form

T(v) = R v + t

where R^T = R⁻¹ (i.e., R is an orthogonal transformation), and t is a vector giving the translation of the origin.

A proper rigid transformation has, in addition,

det(R) = 1

which means that R does not produce a reflection, and hence it represents a rotation (an orientation-preserving orthogonal transformation). Indeed, when an orthogonal transformation matrix produces a reflection, its determinant is –1.

Distance formula

A measure of distance between points, or metric, is needed in order to confirm that a transformation is rigid. The Euclidean distance formula for Rⁿ is the generalization of the Pythagorean theorem. The formula gives the distance squared between two points X and Y as the sum of the squares of the distances along the coordinate axes, that is

where X=(X₁, X₂, …, X_n) and Y=(Y₁, Y₂, …, Y_n), and the dot denotes the scalar product.

Using this distance formula, a rigid transformation g:Rⁿ→Rⁿ has the property,

─── 摘自《勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧【專題】 PD‧箱子世界‧留白》

 

如果依據伽利略『相對性』原理之視角，那麼物理性質也不該因為『觀察者』或『座標系』之不同而有所改變？？因是古典力學就得符合『伽利略變換』矣！！

Galilean transformation

In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotations and translations in space and time form the inhomogeneous Galilean group (assumed throughout below). Without the translations in space and time the group is the homogeneous Galilean group. The Galilean group is the group of motions of Galilean relativity action on the four dimensions of space and time, forming the Galilean geometry. This is the passive transformation point of view. The equations below, although apparently obvious, are valid only at speeds much less than the speed of light. In special relativity the Galilean transformations are replaced by Poincaré transformations; conversely, the group contraction in the classical limit c → ∞ of Poincaré transformations yields Galilean transformations.

Galileo formulated these concepts in his description of uniform motion.^[1] The topic was motivated by his description of the motion of a ball rolling down a ramp, by which he measured the numerical value for the acceleration of gravity near the surface of the Earth.

Standard configuration of coordinate systems for Galilean transformations.

 

只是通常各自有『原點』的『座標系』並不方便以『矩陣』來表達『時空』之『等距同構』

Isometry

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving injective map between metric spaces.^[1]

Introduction

Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional Euclidean space, two geometric figures are congruent if they are related by an isometry;^[3] the isometry that relates them is either a rigid motion (translation or rotation), or a composition of a rigid motion and a reflection.

Isometries are often used in constructions where one space is embedded in another space. For instance, the completion of a metric space M involves an isometry from M into M’, a quotient set of the space of Cauchy sequences on M. The original space M is thus isometrically isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to a closed subset of some normed vector space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space.

An isometric surjective linear operator on a Hilbert space is called a unitary operator.

A composition of two opposite isometries is a direct isometry. A reflection in a line is an opposite isometry, like R ₁ or R ₂ on the image. Translation T is a direct isometry: a rigid motion.^[2]

 

的變換性質，因此借著

齊次坐標

在數學裡，齊次坐標（homogeneous coordinates），或投影坐標（projective coordinates）是指一個用於投影幾何裡的坐標系統，如同用於歐氏幾何裡的笛卡兒坐標一般。該詞由奧古斯特·費迪南德·莫比烏斯於1827年在其著作《Der barycentrische Calcul》一書內引入^[1][2]。齊次坐標可讓包括無窮遠點的點坐標以有限坐標表示。使用齊次坐標的公式通常會比用笛卡兒坐標表示更為簡單，且更為對稱 。齊次坐標有著廣泛的應用，包括電腦圖形及3D電腦視覺。使用齊次坐標可讓電腦進行仿射變換，並通常，其投影變換能簡單地使用矩陣來表示。

如一個點的齊次坐標乘上一個非零純量，則所得之坐標會表示同一個點。因為齊次坐標也用來表示無窮遠點，為此一擴展而需用來標示坐標之數值比投影空間之維度多一。例如，在齊次坐標裡，需要兩個值來表示在投影線上的一點，需要三個值來表示投影平面上的一點。

有理貝茲曲線－定義於齊次坐標內的多項式曲線（藍色），以及於平面上的投影－有理曲線（紅色）

 

來統整論述也就自然而然的了？？！！更別說由於雙眼之視覺現象還得走入『仿射變換』的哩！！？？