樹莓派、樹莓派之學習、樹莓派之教育

光的世界︰矩陣光學六庚

大哲學家亞里士多德認為自然界有一種『原因』Cause 關係，它用著『因為 Because 』回答了『為什麼』Why 之問題。他列舉了四種原因，故簡稱之為『四因說』︰

任何『事物』是由它所構成的『原料』、『組件』和『元素』，按著一套完整的『架構』、『組裝』與『結合』方式才形成的，這個『材質』的部份，就是『物質因』Material Cause ；而那個架構規劃的『藍圖』就是『形式因』Formal Cause，形式因也定義了『□之所以是□』。其次任何『事物』之『存在』總是有理由的，它因著『目的因』Final Cause 而能在時空中『存有』，又可能將隨著時流因之而被改變，這就推動著『動力因』Efficient Cause 去『改變什麼』？又會『如何將之改變』！！

商後期亞醜方觚

觚爵一套

西周初的饮酒器

那麼亞里士多德的四因說，能不能解說這個『忒修斯之船』的同一性問題呢？也許先讓我們聽聽孔老夫子的『觚之抱怨』吧！

《倫语‧雍也》：

子曰：觚不觚，觚哉！觚哉！

朱熹集注：觚，棱也；或曰酒器，或曰木簡，皆器之有棱者也。不觚者，蓋當時失其制而不為棱也。觚哉：觚哉！言不得為觚也。

從造字來講，『觚』字也有『棱角』的啊，竟然將『方』觚改為『圓』觚！無怪乎孔老夫子會喊著『這算是個觚嗎』？『這難到也算是個觚嗎』？？

因此如果問他老先生這個『忒修斯之船』的問題，也許他會說︰依其『形制』並沒有什麼被『改變』，所以還一樣是『那個』。或許說『形式因』定義著『什麼是什麼』，所以相較之下比它是用『什麼所構成』的『物質因』還來的重要的吧！！

─── 摘自《Thue 之改寫系統《三》》

 

前三篇文本中，我們談了一般『光學矩陣』

\left( \begin{array}{cc} A & B \\ C & D \end{array} \right)

只要 C \neq 0 ，都可借著『自由空間』

\left( \begin{array}{cc} 1 & t \\ 0 & 1 \end{array} \right)

化成一個等效之『薄透鏡』

\left( \begin{array}{cc} 1 & t_2 \\ 0 & 1 \end{array} \right) \left( \begin{array}{cc} A & B \\ C & D \end{array} \right) \left( \begin{array}{cc} 1 & t_1 \\ 0 & 1 \end{array} \right) = \left( \begin{array}{cc} 1 & 0 \\ - \frac{1}{f} & 1 \end{array} \right) 。

因此在『主平面』之參考系裡，分享著同樣的『成像公式』

\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}

，具有相同『成像條件』， B 參數為 0

\left( \begin{array}{cc} 1 & d_i \\ 0 & 1 \end{array} \right) \left( \begin{array}{cc} 1 & 0 \\ - \frac{1}{f} & 1 \end{array} \right) \left( \begin{array}{cc} 1 & d_o \\ 0 & 1 \end{array} \right) = \left( \begin{array}{cc} - \frac{d_i}{d_o}& 0 \\ - \frac{1}{f} & - \frac{d_o}{d_i} \end{array} \right) 。

甚至可以『串接成像』

\left( \begin{array}{cc} A_2 & 0 \\ C_2 & D_2 \end{array} \right) \left( \begin{array}{cc} A_1 & 0 \\ C_1 & D_1 \end{array} \right) = \left( \begin{array}{cc} A_2 A_1 & 0 \\ C_2 A_1 + D_2 C_1 & D_2 D_1 \end{array} \right)

的矣！如是就確定了參數 C 之『聚焦』地位，以及參數 A 的『影像縮放』性質！！若問為什麼『平面鏡』是理想成像系統的呢？難到原因在於『反射』與『折射』不同耶？？但思

【曲面折射】

\left( \begin{array}{cc} 1 & 0 \\ \frac{n_1 - n_2}{R n_2} & \frac{n_1}{n_2} \end{array} \right) 和

【曲面反射】

\left( \begin{array}{cc} 1 & 0 \\ - \frac{2}{R} & 1 \end{array} \right)

當 R \to \infty 時，參數 C 趨近於零﹐等同於『平面』

【平面折射】

\left( \begin{array}{cc} 1 & 0 \\ 0 & \frac{n_1}{n_2} \end{array} \right) 和

【平面反射】

\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) 的哩。

 

那為何維基百科特別註記『平面反射』之說明為

Only valid for mirrors perpendicular to the ray.

的呢！！？？問題在『小角度』近軸近似下，實在無法表象『任意角度』都能『完美成像』之『理想平面鏡』呀？？！！縱使想用著『等同矩陣』

Identity matrix

In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context. (In some fields, such as quantum mechanics, the identity matrix is denoted by a boldface one, 1; otherwise it is identical to I.) Less frequently, some mathematics books use U or E to represent the identity matrix, meaning “unit matrix”[1] and the German word “Einheitsmatrix”,[2] respectively.

I_{1}={\begin{bmatrix}1\end{bmatrix}},\ I_{2}={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\ I_{3}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}},\ \cdots ,\ I_{n}={\begin{bmatrix}1&0&0&\cdots &0\\0&1&0&\cdots &0\\0&0&1&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &1\end{bmatrix}}

When A is m×n, it is a property of matrix multiplication that

I_{m}A=AI_{n}=A.\,

In particular, the identity matrix serves as the unit of the ring of all n×n matrices, and as the identity element of the general linear group GL(n) consisting of all invertible n×n matrices. (The identity matrix itself is invertible, being its own inverse.)

Where n×n matrices are used to represent linear transformations from an n-dimensional vector space to itself, In represents the identity function, regardless of the basis.

The ith column of an identity matrix is the unit vector ei. It follows that the determinant of the identity matrix is 1 and the trace is n.

Using the notation that is sometimes used to concisely describe diagonal matrices, we can write:

I_{n}=\mathrm {diag} (1,1,...,1).\,

It can also be written using the Kronecker delta notation:

(I_{n})_{ij}=\delta _{ij}.\,

The identity matrix also has the property that, when it is the product of two square matrices, the matrices can be said to be the inverse of one another.

The identity matrix of a given size is the only idempotent matrix of that size having full rank. That is, it is the only matrix such that (a) when multiplied by itself the result is itself, and (b) all of its rows, and all of its columns, are linearly independent.

The principal square root of an identity matrix is itself, and this is its only positive definite square root. However, every identity matrix with at least two rows and columns has an infinitude of symmetric square roots.[3]

………

 

宣說此理，卻又礙於『物、像』有所不同，亦不可得『計算解析』之好處，不註記，且將如何言之哉？？？更別講，如果 C \neq 0 ，同樣『薄透鏡』等效方法也根本不可能使 C = 0 的乎！！！

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]: from sympy.physics.optics import FreeSpace, FlatRefraction, ThinLens, GeometricRay, CurvedRefraction, RayTransferMatrix

In [3]: init_printing()

In [4]: A, B, C, D, t1, t2 = symbols('A, B, C, D, t1, t2')

In [5]: 一般ABCD矩陣 = RayTransferMatrix(A, B, C, D)

In [6]: 一般ABCD矩陣
Out[6]: 
⎡A  B⎤
⎢    ⎥
⎣C  D⎦

In [7]: FreeSpace(t2) * 一般ABCD矩陣 * FreeSpace(t1)
Out[7]: 
⎡A + C⋅t₂  B + D⋅t₂ + t₁⋅(A + C⋅t₂)⎤
⎢                                  ⎥
⎣   C              C⋅t₁ + D        ⎦

In [8]: