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GoPiGo 小汽車︰格點圖像算術《投影幾何》【五‧線性代數】《導引七‧變換組合 VII‧B 》

奇門遁甲傳說

據煙波釣叟歌中記載，奇門遁甲起源於傳說時代，黃帝炎帝聯軍和蚩尤在涿鹿展開的一場大戰，蚩尤身高七尺，鐵頭銅身，刀槍不入 ，能呼風喚雨並在戰場上製造迷霧，使得炎黃聯軍陷入不利境地。黃帝於是向天祈禱，終於獲得九天玄女給的河圖洛書和彩鳳銜來的太乙、六壬、遁甲之書，黃帝以 此發明了指南車，逆轉了戰局，取得了勝利。黃帝令風后演繹天書，並最終演繹成三式之法：大六壬 、太乙神數、奇門遁甲一千零八十局（陽遁、陰遁各五百四十局） 。後來該術數為姜子牙所習得，由姜子牙刪減為七十二局（陽遁、陰遁各三十六局），再經過姜子牙傳給黃石公，再由黃石公傳給張良，最終由張良將其精簡為現今 的一十八局（陽遁、陰遁各九局） 。

天干首起『甲』，易曰︰用九，見群龍無首，吉。故『遁甲』也。在天有三光，日光乙乙萬物生，月光炳炳照大地，星光指向引路灯 ，乙丙丁三奇出矣。紫白飛星九宮八門，太上曰︰禍福無門，惟人自召。雖然，河圖洛書陰陽五行所以言生剋制化沖和之象，所以極其數，蓋揭露天地人三才生殺有 時乎？？

─── 時間序列︰生成函數‧漸近展開︰歐拉的天空《遁甲》
 

不知上帝是否有書房？裡面的藏書是否叫天書！不了那位歐拉是否有鑰匙？？為何數學之著述這麼地多！！

白露起秋意，早晚催知了。知了有五眼！晨昏聲聲報？

將如何看待『正則式』耶？

Normal form

Möbius transformations are also sometimes written in terms of their fixed points in so-called normal form. We first treat the non-parabolic case, for which there are two distinct fixed points.

Non-parabolic case:

Every non-parabolic transformation is conjugate to a dilation/rotation, i.e. a transformation of the form

  z\mapsto kz\,

(k ∈ C) with fixed points at 0 and ∞. To see this define a map

  g(z)={\frac {z-\gamma _{1}}{z-\gamma _{2}}}

which sends the points (γ1, γ2) to (0, ∞). Here we assume that γ1 and γ2 are distinct and finite. If one of them is already at infinity then g can be modified so as to fix infinity and send the other point to 0.

If f has distinct fixed points (γ1, γ2) then the transformation  gfg^{{-1}} has fixed points at 0 and ∞ and is therefore a dilation: gfg^{{-1}}(z)=kz. The fixed point equation for the transformation f can then be written

  {\frac {f(z)-\gamma _{1}}{f(z)-\gamma _{2}}}=k{\frac {z-\gamma _{1}}{z-\gamma _{2}}}.

Solving for f gives (in matrix form):

{\mathfrak {H}}(k;\gamma _{1},\gamma _{2})={\begin{pmatrix}\gamma _{1}-k\gamma _{2}&(k-1)\gamma _{1}\gamma _{2}\\1-k&k\gamma _{1}-\gamma _{2}\end{pmatrix}}

or, if one of the fixed points is at infinity:

{\mathfrak {H}}(k;\gamma ,\infty )={\begin{pmatrix}k&(1-k)\gamma \\0&1\end{pmatrix}}.

From the above expressions one can calculate the derivatives of f at the fixed points:

f'(\gamma _{1})=k\, and  f'(\gamma _{2})=1/k.\,

Observe that, given an ordering of the fixed points, we can distinguish one of the multipliers (k) of f as the characteristic constant of f. Reversing the order of the fixed points is equivalent to taking the inverse multiplier for the characteristic constant:

{\mathfrak {H}}(k;\gamma _{1},\gamma _{2})={\mathfrak {H}}(1/k;\gamma _{2},\gamma _{1}).

For loxodromic transformations, whenever |k| > 1, one says that γ1 is the repulsive fixed point, and γ2 is the attractive fixed point. For |k| < 1, the roles are reversed.

Parabolic case:

In the parabolic case there is only one fixed point γ. The transformation sending that point to ∞ is

g(z)={\frac {1}{z-\gamma }}

or the identity if γ is already at infinity. The transformation  gfg^{{-1}} fixes infinity and is therefore a translation:

gfg^{{-1}}(z)=z+\beta \,.

Here, β is called the translation length. The fixed point formula for a parabolic transformation is then

{\frac {1}{f(z)-\gamma }}={\frac {1}{z-\gamma }}+\beta .

Solving for f (in matrix form) gives

{\mathfrak {H}}(\beta ;\gamma )={\begin{pmatrix}1+\gamma \beta &-\beta \gamma ^{2}\\\beta &1-\gamma \beta \end{pmatrix}}

or, if γ = ∞:

{\mathfrak {H}}(\beta ;\infty )={\begin{pmatrix}1&\beta \\0&1\end{pmatrix}}

Note that β is not the characteristic constant of f, which is always 1 for a parabolic transformation. From the above expressions one can calculate:

  f'(\gamma )=1.\,

 

祇能盡力『考察字詞』

Homothetic transformation

In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends

  M\mapsto S+\lambda {\overrightarrow {SM}},

in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ.[1] In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if λ > 0) or reverse (if λ < 0) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line L is a line parallel to L.

In projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic involution) that leaves the line at infinity pointwise invariant.[2]

In Euclidean geometry, a homothety of ratio λ multiplies distances between points by |λ| and all areas by λ2. The first number is called the ratio of magnification or dilation factor or scale factor or similitude ratio. Such a transformation can be called an enlargement if the scale factor exceeds 1. The above-mentioned fixed point S is called homothetic center or center of similarity or center of similitude

wo similar geometric figures related by a homothetic transformation with respect to a homothetic center S. The angles at corresponding points are the same and have the same sense; for example, the angles ABC and A’B’C’ are both clockwise and equal in magnitude.

 

努力參悟『字詞意義』乎！

相似矩陣

在線性代數中，相似矩陣是指存在相似關係的矩陣。相似關係是兩個矩陣之間的一種等價關係。兩個n×n矩陣A與B為相似矩陣若且唯若存在一個n×n的可逆矩陣P，使得：

  \!P^{{-1}}AP=B

P被稱為矩陣A與B之間的相似變換矩陣。

相似矩陣保留了矩陣的許多性質，因此許多對矩陣性質的研究可以通過研究更簡單的相似矩陣而得到解決。

判斷兩個矩陣是否相似的輔助方法：

1.判斷特徵值是否相等； 2.判斷行列式是否相等； 3.判斷跡是否相等； 4.判斷秩是否相等； 以上條件可以作為判斷矩陣是否相似的必要條件，而非充分條件。

嚴格定義

兩個係數域為K的n×n的矩陣A與B為域L上的相似矩陣若且唯若存在一個係數域為L的n×n的可逆矩陣P，使得：

  \!P^{{-1}}AP=B

這時，稱矩陣A與B「相似」。B稱作A通過相似變換矩陣：P得到的矩陣。術語相似變換的其中一個含義就是將矩陣A變成與其相似的矩陣B。

 

忘卻誰說『真相』只有一個︰

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 *  In [2]: init_printing()  In [3]: γ1,γ2,k,z = symbols('γ1,γ2,k,z')  In [4]: 似位變換 = Matrix([[k,0],[0,1]])  In [5]: 似位變換 Out[5]:  ⎡k  0⎤ ⎢    ⎥ ⎣0  1⎦  In [6]: 保角變換 = Matrix([[1,-γ1],[1,-γ2]])  In [7]: 保角變換 Out[7]:  ⎡1  -γ1⎤ ⎢      ⎥ ⎣1  -γ2⎦  In [8]: 保角變換.inv() Out[8]:  ⎡     γ1           γ1  ⎤ ⎢- ─────── + 1  ───────⎥ ⎢  γ1 - γ2      γ1 - γ2⎥ ⎢                      ⎥ ⎢     -1           1   ⎥ ⎢   ───────     ───────⎥ ⎣   γ1 - γ2     γ1 - γ2⎦  In [9]: 相似變換 = 保角變換.inv()*似位變換*保角變換  In [10]: 相似變換 Out[10]:  ⎡  ⎛     γ1      ⎞      γ1           ⎛     γ1      ⎞    γ1⋅γ2 ⎤ ⎢k⋅⎜- ─────── + 1⎟ + ───────  - k⋅γ1⋅⎜- ─────── + 1⎟ - ───────⎥ ⎢  ⎝  γ1 - γ2    ⎠   γ1 - γ2         ⎝  γ1 - γ2    ⎠   γ1 - γ2⎥ ⎢                                                             ⎥ ⎢         k         1                  k⋅γ1       γ2          ⎥ ⎢    - ─────── + ───────             ─────── - ───────        ⎥ ⎣      γ1 - γ2   γ1 - γ2             γ1 - γ2   γ1 - γ2        ⎦  In [11]: 相似變換[0,0].simplify() Out[11]:  -k⋅γ2 + γ1 ──────────  γ1 - γ2    In [12]: 相似變換[0,1].simplify() Out[12]:  γ1⋅γ2⋅(k - 1) ─────────────    γ1 - γ2     In [13]: 相似變換[1,0].simplify() Out[13]:   -k + 1 ─────── γ1 - γ2  In [14]: 相似變換[1,1].simplify() Out[14]:  k⋅γ1 - γ2 ─────────  γ1 - γ2   In [15]:  </pre>    <span style="color: #666699;">或許『圖鑑』還有一張</span>  <img class="alignnone size-full wp-image-75430" src="http://www.freesandal.org/wp-content/uploads/Normal_form.png" alt="" width="800" height="600" />     <span style="color: #666699;">表白文字幾行︰</span>  <span style="color: #666699;">依上圖，三角形\Delta {\gamma}_1 {\gamma}_2} z^{'}與\Delta {\gamma}_1 {\gamma}_2} z面積比可以兩算︰</span>  <span style="color: #666699;">‧ 以z-{\gamma}_1}和z^{'}-{\gamma}_1}為底= \frac{z^{'} - {\gamma}_1}{z - {\gamma}_1}}；</span>  <span style="color: #666699;">‧ 以z-{\gamma}_2}和z^{'}-{\gamma}_2}  為底</span>  <span style="color: #666699;">= \frac{|{\gamma}_1 {\gamma}_2| \cdot \sin(\angle \phi)}{|{\gamma}_1 {\gamma}_2| \cdot \sin(\angle \theta)} \cdot \frac{z^{'} - {\gamma}_2}{z - {\gamma}_2}} = k \cdot \frac{z^{'} - {\gamma}_2}{z - {\gamma}_2}}</span>  <span style="color: #666699;">\therefore k = \frac{ \sin(\angle \phi)}{ \sin(\angle \theta)} </span>     <span style="color: #666699;">既『角不變』，『角比』能變嗎★</span>  <span style="color: #666699;">故而 \frac{z^{'} - {\gamma}_1}{z - {\gamma}_1}} = k \cdot \frac{z^{'} - {\gamma}_2}{z - {\gamma}_2}}，</span>  <span style="color: #666699;">且將兩邊乘上\frac{z - {\gamma}_1}{z^{'} - {\gamma}_2}$


果非所求的嘛☆