每日彙整: 2017-09-19

樹莓派、樹莓派之學習、樹莓派之教育

GoPiGo 小汽車︰格點圖像算術《投影幾何》【五‧線性代數】《導引八》觀察者《思辨》

並非特別在此鬼門將關之際，講這人所歸之處！

220px-Cangjie2

Ts'ang-Chieh-Lawrie-Highsmith
華盛頓
哥倫比亞特區

傳說倉頡生有『雙瞳四目』是黃帝史官，創造文字，故稱之為『倉頡先師』。

《淮南子‧本經》中記載：昔者倉頡作書，而天雨粟，鬼夜哭。

《春秋元命苞》裡講︰龍顏侈侈，四目靈光，實有睿德，生而能書。於是窮天地之變，仰觀奎星圓曲之勢，俯察龜文鳥羽山川，指掌而創文字，天為雨粟，鬼為夜哭，龍乃潛藏。

唐朝知名的國畫歷史家之祖，張彥遠之《歷代名畫記‧敘畫之源流》解釋說： 頡有四目，仰觀天象。因儷烏龜之跡，遂定書字之形。造化不能藏其秘，故天雨粟；靈怪不能 遁其形，故鬼夜哭。是時也，書畫同體而未分，象制肇創而猶略。無以傳其意故有書，無以見其形故有畫，天地聖人之意也。

《説文解字》：鬼，人所歸爲鬼。从人，象鬼頭。鬼陰气賊害，从厶。凡鬼之屬皆从鬼。

假使沒有『鬼』字，如何論『鬼』？即使已有『鬼』字，那麼『鬼』到底是什麼？？

甲骨文鬼

篆文鬼

『字』 『詞』是『符號』，承載『概念』，界定『事物』，具有『意義』。然而由於『字』『詞』的『意指』可能發生『含混』與『歧義』，或為『複雜錯綜』的萬象之 『命名』和『化約』描寫，也許正是『造新字』之時！其『大』者，一『字』竟然可以開拓『新視野』，創生多門『學術領域』！！

─── 摘自《M♪o 之 TinyIoT ︰ 《起合》※補充一︰造字》

 

恰好將寫『投影幾何』之『觀察者變換』也。想起『狹義相對論』曾因『光速的不變性』︰

十四世紀邏輯學家奥卡姆的威廉  William of  Occam 提出一個『剃刀原理』︰除非有必要，別多添假設。在西方，這一理念促進了經驗科學一步步擺脫神學的束縛，快速的發展進步，並為後來之邏輯經驗主義所特別重視。導致了今天科學的『美學』之理念，假使有多個理論能說明相同的事實，那就選個最簡潔的吧！！再經過五百年之後到了一九零五年── 物理奇蹟年 ──，此為愛因斯坦創造的奇蹟年，六篇劃時代的論文照亮天際。於此將說說顛覆『時空觀念』的狹義相對論。聲波的『都卜勒效應』是指當波源和觀察者有相對運動時，觀察者度量到的波頻率與波源發出 的頻率並不相同。就像遠方急駛過來的火車鳴笛聲頻率會變高變得尖銳；駛離我們而去時鳴笛聲頻率反之將會變低變得沉鬱一樣。Michelson–Morley 的實驗用著這個思路想要量測乙太風的相對運動，只不過自然卻『希聲』的回答『NO』，揭開了自牛頓之後『二次科學革命』的序幕。

150px-Albert_Einstein_(Nobel)

光子鐘

avyh2h

E=m‧c‧c

愛因斯坦的光偏折理論與澳大利亞觀測

愛因斯坦重新考察了『觀察者』如何『度量』『時間』和『空間』的問題，想像力之大能夠創講『想像實驗』乙事。也許科學的美學使他反對光波需要『乙太』之說，邁克耳遜‧莫雷的實驗『結果』，使他大膽假設『光速對一切等速運動的觀察者來說是不變的 c  』，然後提出了如何用『光子鐘』來量測『時空』中『事件』的辦法。那個『工』字形的『鐘』有兩個『反射鏡面』，光在其間來回，每個來回是一個『滴答』，一個『時距』的基本『單位』。光速的不變性，確保了對一個觀察者來講，只要『工』之『｜』的來回『距離』L 也不變，那一個滴答的時距就是不變的 2‧L / c  。這個『光子鐘』不論橫擺豎放都一樣的滴答，大概是完美巧奪『天工』的『計時器』了。現在如左圖三的宇宙飛船上有一座『豎放』垂直『飛船速度 V』的光子鐘，地上的人所見光之滴答『軌跡』並不是『↑↓』而是『↗↘』。既然等速運動對兩個觀察者是『相對的』，『垂直』運動方向的『尺』，兩者都會同意度量是相同的，於是地上的觀察者用『畢氏定理』計算『滴↗』得到︰

L^{'2} = c^{2}  ‧ T^{'2} = L^{2}  + V^{2}   ‧ T^{'2}
= c^{2}  ‧ T^{2} + V^{2}   ‧ T^{'2}

所以，

T^{'} =  {T / } \sqrt {1 - (V /c)^{2}}

然後結論說︰飛船上的光子鐘的滴答聲變慢了。這就是著名的『運動中的時鐘走得比較慢』，真是天上方三日，人間已千年的啊。

假使這次把那光子鐘橫擺著呢？這時地上的人觀察光的軌跡是『去→回←』，因為很難度量『移動中之尺』，也不知它會不會『改變』？但是光速是不變的，就量測時間吧︰

『→』光去花的時間 = L^{'} { / (c - V)}
『←』光回花的時間 = L^{'} { / (c + V)}
一滴答的總時間 T^{'} =  L^{'} { / (c - V)} + L^{'} { / (c + V)}
= {T / } \sqrt {1 - (V /c)^{2}}

由於在飛船上 T = 2‧L / c ，簡化後得到

L^{'} =  \sqrt {1 - (V /c)^{2}} ‧ L

也就是說『運動中的尺會縮短』，宇宙還真是妙得很呦！！

─── 摘自《觀測之『測天文』》

 

引發『同時性』破壞，產生難以理解的『勞侖茲變換』︰

如果從『伽利略變換』如何『觀察』這個『相對性』的意義的呢？假設以『□觀察者』 (x_{\Box}, t_{\Box}) 為『靜止』，『□觀察者』見『○觀察者』 (x_{\bigcirc}, t_{\bigcirc}) 以『速度』 v 向右運動，假使他們彼此能『交換資訊』，同意兩者的『原點』相同，那麼他們對『時空現象』或者說『事件』的『位置‧時間』描述滿足

\begin{bmatrix} x_{\bigcirc} \\ t_{\bigcirc} \end{bmatrix} = G_v \begin{bmatrix} x_{\Box} \\ t_{\Box} \end{bmatrix} = \begin{pmatrix} 1 & -v \\0 & 1 \end{pmatrix} \begin{bmatrix} x_{\Box} \\ t_{\Box} \end{bmatrix}

。『□觀察者』的『原點』 (0, t_{\Box}) 對『□觀察者』是『靜止』的，然而對『○觀察者』而言，是 x_{\bigcirc} = - v \cdot t_{\Box} 和 t_{\bigcirc} = t_{\Box} ，它以『速度』 v 『等速向左』 運動。其次對於『□觀察者』而言，所發生的『同時兩事件』 (x_{\Box}^1, t_{\Box})  與 (x_{\Box}^2, t_{\Box}) ，對『○觀察者』而言，是 (x_{\Box}^1 - v \cdot t_{\Box}, t_{\Box}) 和 (x_{\Box}^2 - v \cdot t_{\Box}, t_{\Box}) 也是『同時的』。既然『運動是相對的』，假使我們以『○觀察者』為『靜止』，來作個『伽利略變換』的『物理檢驗』， 那麼 \begin{bmatrix} x_{\Box} \\ t_{\Box} \end{bmatrix} = G_{-v} \begin{bmatrix} x_{\bigcirc} \\ t_{\bigcirc} \end{bmatrix} 當是應該的了。也就是說 G_{-v} = {G_v}^{-1} = \begin{pmatrix} 1 & v \\0 & 1 \end{pmatrix}，讀者自己可以『確證』 \begin{pmatrix} 1 & v \\0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & - v \\0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & - v \\0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & v \\0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\0 & 1 \end{pmatrix} 它的『正確性』。也可以說『物理之要求』不得不決定了『數學的表達式』的吧！，所謂的『自然律』並不『必須』要『滿足』這種或那種『數學』的耶！！如果說『○觀察者』觀測某一個『星辰』 (x_{\star}, t_{\star}) 用 w 的『速度』向右『直線運動』，那麼這一個『星辰』相對於『□觀察者』的『速度』是什麼的呢？『直覺上』我們認為既然『★ 對 ○ 是 w 向右，○ 對 □ 是 v 向右』，那麼『★ 對 ○ 該是 w + v 向右』的吧！我們可以用『伽利略變換』計算如下

\begin{bmatrix} x_{\bigcirc} \\ t_{\bigcirc} \end{bmatrix} = G_v \begin{bmatrix} x_{\Box} \\ t_{\Box} \end{bmatrix} = \begin{pmatrix} 1 & -v \\0 & 1 \end{pmatrix} \begin{bmatrix} x_{\Box} \\ t_{\Box} \end{bmatrix}，

\begin{bmatrix} x_{\star} \\ t_{\star} \end{bmatrix} = G_w \begin{bmatrix} x_{\bigcirc} \\ t_{\bigcirc} \end{bmatrix} = \begin{pmatrix} 1 & -w \\0 & 1 \end{pmatrix} \begin{bmatrix} x_{\bigcirc} \\ t_{\bigcirc} \end{bmatrix}

= \begin{pmatrix} 1 & -w \\0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & -v \\0 & 1 \end{pmatrix} \begin{bmatrix} x_{\Box} \\ t_{\Box} \end{bmatrix}

= \begin{pmatrix} 1 & -(w+v) \\0 & 1 \end{pmatrix} \begin{bmatrix} x_{\Box} \\ t_{\Box} \end{bmatrix}

= G_{(w+v)} \begin{bmatrix} x_{\Box} \\ t_{\Box} \end{bmatrix}

，果真是『符合直覺』的勒！！

假使這些『考察』改用『狹義相對論』的『勞侖茲變換』

\begin{bmatrix} x_{\bigcirc} \\ t_{\bigcirc} \end{bmatrix} = L_v \begin{bmatrix} x \\ t \end{bmatrix} = \frac{1}{\sqrt{1 - {(\frac{v}{c})}^2}} \begin{pmatrix} 1 & -v \\ -\frac{v}{c^2} & 1 \end{pmatrix} \begin{bmatrix} x_{\Box} \\ t_{\Box} \end{bmatrix} 來看的呢？

讀者自可『證實』除了『原點』之外，『同時性』因為有著 -\frac{v}{c^2} 『位置相關項』的『存在』而被『破壞』了；然而物理所要求的『相對性』 L_{- v} = L_{v}^{-1} 依然成立。那個『相對速度』之『加法』就顯然非常『違反直覺』的成了

L_w \cdot L_v = L_{w \bigoplus v} = \frac{1}{\sqrt{1 - {[\frac{(w+v)/c}{1+(wv/c^2)}]}^2}} \begin{pmatrix} 1 & -[\frac{(w+v)}{1+(wv/c^2)}] \\ -\frac{1}{c^2} {[\frac{(w+v)}{1+(wv/c^2)}]}^2 & 1 \end{pmatrix} \begin{bmatrix} x_{\Box} \\ t_{\Box} \end{bmatrix}

\neq L_{(w + v)} \begin{bmatrix} x_{\Box} \\ t_{\Box} \end{bmatrix}

。如果將『速度加法』 定義為 w \bigoplus v = \frac{w + v}{1 + (w v / c^2)}  的話，那麼 L_{w \bigoplus v} = \frac{1}{\sqrt{1 - {(\frac{(w \bigoplus v)}{c})}^2}} \begin{pmatrix} 1 & -(w \bigoplus v) \\ -\frac{(w \bigoplus v)}{c^2} & 1 \end{pmatrix} 這又能有什麼『不對』的嗎？因是之故，『狹義相對論』所帶來的『困惑』遠勝於『運動之不可能性』，反倒以為『運動』果真能是這種『現象』的嘛！！

─── 摘自《【Sonic π】電聲學之電路學《四》之《 !!!! 》下》

 

方有『所見收縮』困惑矣？

Length contraction

Visual effects

Length contraction refers to measurements of position made at simultaneous times according to a coordinate system. This could suggest that if one could take a picture of a fast moving object, that the image would show the object contracted in the direction of motion. However, such visual effects are completely different measurements, as such a photograph is taken from a distance, while length contraction can only directly be measured at the exact location of the object’s endpoints. It was shown by several authors such as Roger Penrose and James Terrell that moving objects generally do not appear length contracted on a photograph.[22] For instance, for a small angular diameter, a moving sphere remains circular and is rotated.[23] This kind of visual rotation effect is called Penrose-Terrell rotation.[24]

Formula on a wall in Leiden

Terrell rotation

Terrell rotation or Terrell effect is the name of a mathematical and physical effect. Specifically, Terrell rotation is the visual distortion that a passing object would appear to undergo, according to the special theory of relativity if it were travelling a significant fraction of the speed of light. This behaviour was described independently by both James Terrell and Roger Penrose in pieces published in 1959,[1][2] though the general phenomenon was noted already in 1924 by Austrian physicist Anton Lampa.[3]

Due to an early dispute about priority and correct attribution, the effect is also sometimes referred to as the Penrose–Terrell effect, the Terrell–Penrose effect or the Lampa–Terrell–Penrose effect, but not the Lampa effect.

Further detail

Terrell’s and Penrose’s papers pointed out that although special relativity appeared to describe an “observed contraction” in moving objects, these interpreted “observations” were not to be confused with the theory’s literal predictions for the visible appearance of a moving object. Thanks to the differential timelag effects in signals reaching the observer from the object’s different parts, a receding object would appear contracted, an approaching object would appear elongated (even under special relativity) and the geometry of a passing object would appear skewed, as if rotated.[1][2]

For images of passing objects, the apparent contraction of distances between points on the object’s transverse surface could then be interpreted as being due to an apparent change in viewing angle, and the image of the object could be interpreted as appearing instead to be rotated. A previously popular description of special relativity’s predictions, in which an observer sees a passing object to be contracted (for instance, from a sphere to a flattened ellipsoid), was wrong.[1][2]

Terrell’s and Penrose’s papers prompted a number of follow-up papers,[4][5][6][7][8] mostly in the American Journal of Physics, exploring the consequences of this correction. These papers pointed out that some existing discussions of special relativity were flawed and “explained” effects that the theory did not actually predict – while these papers did not change the actual mathematical structure of special relativity in any way, they did correct a misconception regarding the theory’s predictions.

Comparison of the measured length contraction of a cube versus its visual appearance. The view is from the front of the cube at a distance four times the length of the cube’s sides, three-quarters of the way from bottom to top, as projected onto a vertical screen (so that the vertical lines of the cube may initially be parallel).

 

才寫在前頭哩。

Plato was one of the first to discuss the problems of perspective.

“Thus (through perspective) every sort of confusion is revealed within us; and this is that weakness of the human mind on which the art of conjuring and of deceiving by light and shadow and other ingenious devices imposes, having an effect upon us like magic… And the arts of measuring and numbering and weighing come to the rescue of the human understanding – there is the beauty of them – and the apparent greater or less, or more or heavier, no longer have the mastery over us, but give way before calculation and measure and weight?”[15]

Satire on False Perspective by William Hogarth, 1753

 

Perspective images are calculated assuming a particular vanishing point. In order for the resulting image to appear identical to the original scene, a viewer of the perspective must view the image from the exact vantage point used in the calculations relative to the image. This cancels out what would appear to be distortions in the image when viewed from a different point. These apparent distortions are more pronounced away from the center of the image as the angle between a projected ray (from the scene to the eye) becomes more acute relative to the picture plane. In practice, unless the viewer chooses an extreme angle, like looking at it from the bottom corner of the window, the perspective normally looks more or less correct. This is referred to as “Zeeman’s Paradox”.[16] It has been suggested that a drawing in perspective still seems to be in perspective at other spots because we still perceive it as a drawing, because it lacks depth of field cues.[17]

For a typical perspective, however, the field of view is narrow enough (often only 60 degrees) that the distortions are similarly minimal enough that the image can be viewed from a point other than the actual calculated vantage point without appearing significantly distorted. When a larger angle of view is required, the standard method of projecting rays onto a flat picture plane becomes impractical. As a theoretical maximum, the field of view of a flat picture plane must be less than 180 degrees (as the field of view increases towards 180 degrees, the required breadth of the picture plane approaches infinity).

To create a projected ray image with a large field of view, one can project the image onto a curved surface. To have a large field of view horizontally in the image, a surface that is a vertical cylinder (i.e., the axis of the cylinder is parallel to the z-axis) will suffice (similarly, if the desired large field of view is only in the vertical direction of the image, a horizontal cylinder will suffice). A cylindrical picture surface will allow for a projected ray image up to a full 360 degrees in either the horizontal or vertical dimension of the perspective image (depending on the orientation of the cylinder). In the same way, by using a spherical picture surface, the field of view can be a full 360 degrees in any direction (note that for a spherical surface, all projected rays from the scene to the eye intersect the surface at a right angle).

Just as a standard perspective image must be viewed from the calculated vantage point for the image to appear identical to the true scene, a projected image onto a cylinder or sphere must likewise be viewed from the calculated vantage point for it to be precisely identical to the original scene. If an image projected onto a cylindrical surface is “unrolled” into a flat image, different types of distortions occur. For example, many of the scene’s straight lines will be drawn as curves. An image projected onto a spherical surface can be flattened in various ways:

  • An image equivalent to an unrolled cylinder
  • A portion of the sphere can be flattened into an image equivalent to a standard perspective
  • An image similar to a fisheye photograph