GoPiGo 小汽車︰格點圖像算術《投影幾何》【五‧線性代數】《導引四》

如果觀察者選擇 t ── 以 A 為『零點』之指定點 A,B 距離比 ── 當作 P 點投射之定位選擇

\frac{\overline{CA}}{\overline{CB}} = \frac{\overline{CA}}{\overline{CA} - \overline{AB}} = \frac{x}{x -1} = t, \ x =_{df} \frac{\overline{CA}}{\overline{AB}}

 

即使 t,x 所選『長度單位』是 \overline{AB} ,都是『距離比』 t = \frac{\overline{CA}}{\overline{CB}} \ vs \ x= \frac{\overline{CA}}{\overline{AB}} 。然而 x 卻是常見的『數線』。

Number line

In basic mathematics, a number line is a picture of a graduated straight line that serves as abstraction for real numbers, denoted by  \mathbb {R} . Every point of a number line is assumed to correspond to a real number, and every real number to a point.[1]

The integers are often shown as specially-marked points evenly spaced on the line. Although this image only shows the integers from −9 to 9, the line includes all real numbers, continuing forever in each direction, and also numbers not marked that are between the integers. It is often used as an aid in teaching simple addition and subtraction, especially involving negative numbers.

The number line

In advanced mathematics, the expressions real number line, or real line are typically used to indicate the above-mentioned concept that every point on a straight line corresponds to a single real number, and vice versa.

 

雖說 t, x 皆『無因次』︰

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在數學上的『』和物理中之『』,其實是兩種『不同概念』。『物理量』是從『度量』而來,所以會擁有『度量單位』,比方講『時間︰秒』、『長度︰米』與『質量︰公斤』等等;然而『數學數』卻是『沒有單位』dimensionless 的,它屬於『抽象』之『』的概念,就像說從『三朵花』、『三個人』和『三件事』 …,得出了『』這個『』,那裡頭並沒有『朵‧個‧件』這些的單位。一八六零年代英國科學家馬克士威 Maxwell 及克耳文 Kelvin 提出『公分‧克‧秒』的 CGS 制,是第一個有連貫性的公制系統。一八七四年英國科學促進會 BAAS 正式推動此公制系統。這一個公制系統的特點是『密度』為 \frac{g}{{cm}^3},『』是達因 dyne,與『機械能』稱爾格 erg,將『熱能』的『單位』叫做卡路里定義為一克的水由溫度 15.5 °C 加溫至 16.5 °C 所需的熱量。由於 CGS 制在電學上有二套不同的單位系統,一者是靜電單位 ESU 制 ,另一者是電磁單位 EMU 制,這就造成了使用上的不方便。一八九三年在芝加哥舉行的國際電工代表大會 International Electrical Congress 使用基於『米‧公斤‧秒』的定義,重新定義電流單位『國際安培』。一九零一年時,義大利科學家喬吉‧喬望尼 Giovanni Giorg 發現假使增加一個電學的單位為基礎單位,可以解決電學單位不一致的問題,比方說『米‧公斤‧秒‧庫侖』 MKSC 制或是『米‧公斤‧秒‧安培』 MKSA 制。現今 MKS 國際單位制』是使用最廣的單位系統,從喬吉所提出的 『MKSA 制』延伸而得,其基礎單位為『米‧公 斤‧秒‧安培‧熱力學溫標‧燭光及莫耳』。在二零一一年十月舉行的第二十四屆國際度量衡大會已經提議更改四個基礎單位的定義,即將成為新國際單位制之定義,不過上述的修改並不會影響一般人的『單位使 用』。怪哉!遽聞至今『美國』都沒有採用 MKS 『國際標準制』!!

為什麼要討論『物理量』的『單位』呢?因為一般物理定律都用著『數學式』表達,萬一所計算的物理量發生了『 1 公斤  + 6 米 – 8 秒 』,這可能要比南宋著名禪宗大師大慧宗杲,是臨濟宗楊岐派第五代傳人,所提倡的『看話禪』── 舉個例說︰『萬法歸一,一歸何處?』── 還要『難參無解』。據說物理量使用單位的『因次分析』dimensional analysis 始於牛頓之『相似性原理』;就建立因次分析的現代意義用法上講,馬克士威是位重要的推手,他區分『質量』、『長度』、『時間』的度量單位為『基礎單位』,將其它單位歸類為『衍生單位』。十九世紀時法國數學家約瑟夫‧傅立葉 Joseph Fourier 洞悉了『物理定律』的『數學方程式』應當與度量物理量的『單位無關』。難道說一個人用『 □□ 制』單位,另一個人用『○○制』單位,他們的牛頓第二運動定律 \mathbf{F}=m\mathbf{a} 就因此會是『兩種』的嗎?假使兩人描述『同一』自然現象,在彼此使用的『單位換算』後,竟然能夠是『答案不同』的嗎??

─── 摘自《【Sonic π】聲波之傳播原理︰原理篇《四上》

 

且能『一一對應』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 sympy import symbols  In [2]: from sympy.plotting import plot  In [3]: x = symbols('x')  In [4]: 座標對應圖 = plot(x/(x-1), (x, -2, 2), ylim=(-10,10))  In [5]:  </pre>    <img class="alignnone size-full wp-image-73897" src="http://www.freesandal.org/wp-content/uploads/x-t座標對應圖.png" alt="" width="652" height="553" />     <span style="color: #666699;">只不過</span>  <span style="color: #666699;">\lim \limits_{x \to 1^{+}} \  \frac{x}{x-1} = t^{+} \to \ \infty</span>  <span style="color: #666699;">\lim \limits_{x \to 1^{-}} \  \frac{x}{x-1} = t^{-} \to \ -\infty</span>  <span style="color: #666699;">這可能嗎??</span>  <span style="color: #666699;">那麼</span>  <span style="color: #666699;">\lim \limits_{x \to \infty} \  \frac{x}{x-1} = t \to \ 1</span>  <span style="color: #666699;">\lim \limits_{x \to - \infty} \  \frac{x}{x-1} = t \to \ 1$

又該怎麼說呢!!

如是『左向無窮』、『右往無限』,將會『聚首』嘛??!!

Cayley transform

In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by Cayley (1846), the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators (Nikol’skii 2001).

Real homography

The Cayley transform is an automorphism of the real projective line that permutes the elements of {1, 0, −1, ∞} in sequence. For example, it maps the positive real numbers to the interval [−1, 1]. Thus the Cayley transform is used to adapt Legendre polynomials for use with functions on the positive real numbers with Legendre rational functions.

As a real homography, points are described with homogeneous coordinates, and the mapping is

{\displaystyle (y,\ 1)=\left({\frac {x-1}{x+1}},\ 1\right)\sim (x-1,\ x+1)=(x,\ 1){\begin{pmatrix}1&1\\-1&1\end{pmatrix}}.}

Complex homography

In the complex projective plane the Cayley transform is:[1][2]

{\displaystyle f(z)={\frac {z-i}{z+i}}.}

Since {∞, 1, –1 } is mapped to {1, –i, i }, and Möbius transformations permute the generalised circles in the complex plane, f maps the real line to the unit circle. Furthermore, since f is continuous and i is taken to 0 by f, the upper half-plane is mapped to the unit disk.

In terms of the models of hyperbolic geometry, this Cayley transform relates the Poincaré half-plane model to the Poincaré disk model. In electrical engineering the Cayley transform has been used to map a reactance half-plane to the Smith chart used for impedance matching of transmission lines.

Cayley transform of upper complex half-plane to unit disk