每日彙整: 2017-08-23

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】《導引七‧變換組合 I 》

『 』 尺有所短，『 』寸有所長◎

※ 此處 x,y,z 都是本地座標系，皆如是賦值︰

\frac{\overline{C^{''}A^{''}}}{\overline{C^{''}B^{''}}} = \frac{\overline{C^{''}A^{''}}}{\overline{C^{''}A{''}} - \overline{A^{''}B^{''}}} = \frac{z}{z -1}, \ z =_{df} \frac{\overline{C^{''}A{''}}}{\overline{A{''}B{''}}} 。

\frac{\frac{\overline{CA}}{\overline{CB}}}{\frac{\overline{C^{'}A^{'}}}{\overline{C^{'}B{'}}} } = \frac{\frac{\overline{PA}}{\overline{PB}}}{\frac{\overline{PA^{'}}}{\overline{PB^{'}}}} = \frac{1}{k_{ll'}}

 

若問是否存在 k_{ll'} <0 之『透視』情況呢？

知道『孟氏定理』者

Menelaus’ theorem

Menelaus’ theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Given a triangle ABC, and a transversal line that crosses BC, AC and AB at points D, E and F respectively, with D, E, and F distinct from A, B and C, then

{\frac {AF}{FB}}\times {\frac {BD}{DC}}\times {\frac {CE}{EA}}=-1.

or simply

  AF\times BD\times CE=-FB\times DC\times EA.

This equation uses signed lengths of segments, in other words the length AB is taken to be positive or negative according to whether A is to the left or right of B in some fixed orientation of the line. For example, AF/FB is defined as having positive value when F is between A and B and negative otherwise.

The converse is also true: If points D, E and F are chosen on BC, AC and AB respectively so that

{\frac {AF}{FB}}\times {\frac {BD}{DC}}\times {\frac {CE}{EA}}=-1,

then D, E and F are collinear. The converse is often included as part of the theorem.

The theorem is very similar to Ceva’s theorem in that their equations differ only in sign.

Menelaus’ theorem, case 1: line DEF passes inside triangle ABC

 

情況1：直線LMN穿過三角形ABC

證明

如情況一，設  {\displaystyle \angle ANM=\alpha }，  {\displaystyle \angle AMN=\beta }，  {\displaystyle \angle MLC=\gamma }，則在  {\displaystyle \triangle AMN}中由正弦定理，有

  {\displaystyle {\frac {AN}{AM}}={\frac {\sin \beta }{\sin \alpha }},}

同理，因對頂角相等在  {\displaystyle \triangle NBL}和  {\displaystyle \triangle CLM}中有

  {\displaystyle {\frac {BL}{BN}}={\frac {\sin \alpha }{\sin \gamma }},}

及

  {\displaystyle {\frac {CM}{CL}}={\frac {\sin \gamma }{\sin \beta }}.}

三式相乘即得

  {\displaystyle {\frac {AN}{AM}}\cdot {\frac {BL}{BN}}\cdot {\frac {CM}{CL}}={\frac {\sin \beta }{\sin \alpha }}\cdot {\frac {\sin \alpha }{\sin \gamma }}\cdot {\frac {\sin \gamma }{\sin \beta }}=1,}

即

  {\displaystyle {\frac {AN}{NB}}\cdot {\frac {BL}{LC}}\cdot {\frac {CM}{MA}}=1.}

 

或可假借構圖也！

※ 這裡

A \ {\overset {P}{\doublebarwedge }} \ A^{'} 、 B \ {\overset {P}{\doublebarwedge }} \ B^{'} 、 C \ {\overset {P}{\doublebarwedge }} \ C^{'}

線 L 與 線 L' 相交於 X 點。

 

且由該定理推知固有『k = -1』之組態矣？！

其實從落於線段 \overline{AB} 之外的 C 點，被 P 投影至線段 \overline{A^{'}B{'}} 之內的 C^{'} 點，依『賦值約定』亦可知也！？

當是時

\frac{\frac{x}{x-1}}{\frac{y}{y-1}} = -1, \ \Rightarrow y = \frac{x}{2x -1} 。

它的矩陣表現為

\left( \begin{array}{cc} y \\ 1 \end{array} \right) \ {\overset {P}{\doublebarwedge }} \ \left( \begin{array}{cc} 1 & 0 \\ 1 - (-1) & -1 \end{array} \right) \left( \begin{array}{cc} x \\ 1 \end{array} \right)

= \left( \begin{array}{cc} 1 & 0 \\ 2 & -1 \end{array} \right) \left( \begin{array}{cc} x \\ 1 \end{array} \right) 。

為什麼特別談這個矩陣呢？？難道不是將 k=-1 代入

\left( \begin{array}{cc} y \\ 1 \end{array} \right) \ {\overset {P}{\doublebarwedge }} \ \left( \begin{array}{cc} 1 & 0 \\ 1 - \frac{1}{k} & \frac{1}{k} \end{array} \right) \left( \begin{array}{cc} x \\ 1 \end{array} \right)

一般『透視關係式』就得到了嘛！！

但思

\frac{x}{x-1} \ \equiv_{df} \left( \begin{array}{cc} 1 & 0 \\ 1 & -1 \end{array} \right)

雖然是『冪等』的︰

Idempotence

Idempotence (UK: /ˌɪdɛmˈpoʊtns/;[1] US: /ˌaɪdəmˈpoʊtəns/ EYE-dəm-POH-təns)[2] is the property of certain operations in mathematics and computer science, that can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency).

The term was introduced by Benjamin Peirce[3] in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means “(the quality of having) the same power”, from idem + potence (same + power).

There are several meanings of idempotence, depending on what the concept is applied to:

  • A unary operation (or function) is idempotent if, whenever it is applied twice to any value, it gives the same result as if it were applied once; i.e., ƒ(ƒ(x)) ≡ ƒ(x). For example, the absolute value function, where abs(abs(x)) ≡ abs(x), is idempotent.
  • Given a binary operation, an idempotent element (or simply an “idempotent”) for the operation is a value for which the operation, when given that value for both of its operands, gives that value as the result. For example, the number 1 is an idempotent of multiplication: 1 × 1 = 1.
  • A binary operation is called idempotent if all elements are idempotent elements with respect to the operation. In other words, whenever it is applied to two equal values, it gives that value as the result. For example, the function giving the maximum value of two equal values is idempotent: max(x, x) ≡ x.

Other examples

In Boolean algebra, both the logical and and the logical or operations are idempotent. This implies that every element of Boolean algebra is idempotent with respect to both of these operations. Specifically,  x \wedge x = x and  x \vee x = x for all  x. In linear algebra, projections are idempotent. In fact, the projections of a vector space are exactly the idempotent elements of the ring of linear transformations of the vector space. After fixing a basis, it can be shown that the matrix of a projection with respect to this basis is an idempotent matrix. An idempotent semiring (also sometimes called a dioid) is a semiring whose addition (not multiplication) is idempotent. If both operations of the semiring are idempotent, then the semiring is called doubly idempotent.[8]

 

卻無法用那個一般式得到哩？？！！

故而

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

實在有趣吧！！？？

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]: m = Matrix(([1,0],[2,-1]))

In [4]: m
Out[4]: 
⎡1  0 ⎤
⎢     ⎥
⎣2  -1⎦

In [5]: m*m
Out[5]: 
⎡1  0⎤
⎢    ⎥
⎣0  1⎦

In [6]: m.eigenvals()
Out[6]: {-1: 1, 1: 1}

In [7]: m.eigenvects()
Out[7]: 
⎡⎛-1, 1, ⎡⎡0⎤⎤⎞, ⎛1, 1, ⎡⎡1⎤⎤⎞⎤
⎢⎜       ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥⎥⎟⎥
⎣⎝       ⎣⎣1⎦⎦⎠  ⎝      ⎣⎣1⎦⎦⎠⎦

In [8]: