一個小小實驗︰辨識『印刷』的阿拉伯數字

 

，竟然『network.py』會誤以為是『3』！！難道說『印刷』字能比『手寫』更難辨認的嗎？？或許答案就落在那個『0』是有『位移』的吧？？而那個小程式之『訓練』時用之『手寫數字』資料庫可都是計算過『圖心』的哩！！

THE MNIST DATABASE
of handwritten digits
Yann LeCun, Courant Institute, NYU
Corinna Cortes, Google Labs, New York
Christopher J.C. Burges, Microsoft Research, Redmond

……

The original black and white (bilevel) images from NIST were size normalized to fit in a 20×20 pixel box while preserving their aspect ratio. The resulting images contain grey levels as a result of the anti-aliasing technique used by the normalization algorithm. the images were centered in a 28×28 image by computing the center of mass of the pixels, and translating the image so as to position this point at the center of the 28×28 field.

───

因此『訓練』的『條件』以及『環境』也就決定了『network.py』所擅長之『辨識零點』的了！！？？如是看來人類的『視覺』當真是厲害得很也！！否則如何能作『幾何論證』乎？？豈能發現『九點圓』的呢？？！！

─── 《W!o+ 的《小伶鼬工坊演義》︰神經網絡【MNIST】八》

 

為什麼有『位移』難以辨認？自然也用『平移』產生『視差』呢？

假設人類沒有『雙眼視覺』︰

Binocular vision

Binocular vision is vision in which creatures having two eyes use them together. The word binocular comes from two Latin roots, bini for double, and oculus for eye.^[1] According to Fahle (1987),^[2] having two eyes confers six advantages over having one.

1. It gives a creature a spare eye in case one is damaged.
2. It gives a wider field of view. For example, humans have a maximum horizontal field of view of approximately 190 degrees with two eyes, approximately 120 degrees of which makes up the binocular field of view (seen by both eyes) flanked by two uniocular fields (seen by only one eye) of approximately 40 degrees.^[3]
3. It can give stereopsis in which binocular disparity (or parallax) provided by the two eyes’ different positions on the head gives precise depth perception. This also allows a creature to break the camouflage of another creature.
4. It allows the angles of the eyes’ lines of sight, relative to each other (vergence), and those lines relative to a particular object (gaze angle) to be determined from the images in the two eyes.^[4] These properties are necessary for the third advantage.
5. It allows a creature to see more of, or all of, an object behind an obstacle. This advantage was pointed out by Leonardo da Vinci, who noted that a vertical column closer to the eyes than an object at which a creature is looking might block some of the object from the left eye but that part of the object might be visible to the right eye.
6. It gives binocular summation in which the ability to detect faint objects is enhanced.^[5]
7. It helps see and analyze 3 dimensional objects which are the ones having depth.

Other phenomena of binocular vision include utrocular discrimination (the ability to tell which of two eyes has been stimulated by light),^[6] eye dominance (the habit of using one eye when aiming something, even if both eyes are open),^[7] allelotropia (the averaging of the visual direction of objects viewed by each eye when both eyes are open),^[8] binocular fusion or singleness of vision (seeing one object with both eyes despite each eye’s having its own image of the object),^[9] and binocular rivalry (seeing one eye’s image alternating randomly with the other when each eye views images that are so different they cannot be fused).^[10]

Binocular vision helps with performance skills such as catching, grasping, and locomotion.^[11] It also allows humans to walk over and around obstacles at greater speed and with more assurance.^[12] Optometrists and/or Orthoptists are eyecare professionals who fix binocular vision problems.

Principle of binocular vision with horopter shown

世間會發明『立體眼鏡』

Stereoscope

A stereoscope is a device for viewing a stereoscopic pair of separate images, depicting left-eye and right-eye views of the same scene, as a single three-dimensional image.

A typical stereoscope provides each eye with a lens that makes the image seen through it appear larger and more distant and usually also shifts its apparent horizontal position, so that for a person with normal binocular depth perception the edges of the two images seemingly fuse into one “stereo window”. In current practice, the images are prepared so that the scene appears to be beyond this virtual window, through which objects are sometimes allowed to protrude, but this was not always the custom. A divider or other view-limiting feature is usually provided to prevent each eye from being distracted by also seeing the image intended for the other eye.

Most people can, with practice and some effort, view stereoscopic image pairs in 3D without the aid of a stereoscope, but the physiological depth cues resulting from the unnatural combination of eye convergence and focus required will be unlike those experienced when actually viewing the scene in reality, making an accurate simulation of the natural viewing experience impossible and tending to cause eye strain and fatigue.

Although more recent devices such as Realist-format 3D slide viewers and the View-Master are also stereoscopes, the word is now most commonly associated with viewers designed for the standard-format stereo cards that enjoyed several waves of popularity from the 1850s to the 1930s as a home entertainment medium.

Devices such as polarized, anaglyph and shutter glasses which are used to view two actually superimposed or intermingled images, rather than two physically separate images, are not categorized as stereoscopes.

Old Zeiss pocket stereoscope with original test image

Principles

A simple stereoscope is limited in the size of the image that may be used. A more complex stereoscope uses a pair of horizontal periscope-like devices, allowing the use of larger images that can present more detailed information in a wider field of view. The stereoscope is essentially an instrument in which two photographs of the same object, taken from slightly different angles, are simultaneously presented, one to each eye. This recreates the way which in natural vision, each eye is seeing the object from a slightly different angle, since they are separated by several inches, which is what gives humans natural depth perception. Each picture is focused by a separate lens, and by showing each eye a photograph taken several inches apart from each other and focused on the same point, it recreates the natural effect of seeing things in three dimensions.

A moving image extension of the stereoscope has a large vertically mounted drum containing a wheel upon which are mounted a series of stereographic cards which form a moving picture. The cards are restrained by a gate and when sufficient force is available to bend the card it slips past the gate and into view, obscuring the preceding picture. These coin-enabled devices were found in arcades in the late 19th and early 20th century and were operated by the viewer using a hand crank. These devices can still be seen and operated in some museums specializing in arcade equipment.

The stereoscope offers several advantages:

• Using positive curvature (magnifying) lenses, the focus point of the image is changed from its short distance (about 30 to 40 cm) to a virtual distance at infinity. This allows the focus of the eyes to be consistent with the parallel lines of sight, greatly reducing eye strain.
• The card image is magnified, offering a wider field of view and the ability to examine the detail of the photograph.
• The viewer provides a partition between the images, avoiding a potential distraction to the user.

A stereo transparency viewer is a type of stereoscope that offers similar advantages, e.g. the View-Master

Disadvantages of stereo cards, slides or any other hard copy or print are that the two images are likely to receive differing wear, scratches and other decay. This results in stereo artifacts when the images are viewed. These artifacts compete in the mind resulting in a distraction from the 3d effect, eye strain and headaches.

想拍照『視差』

Parallax

Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines.^[1][2] The term is derived from the Greek word παράλλαξις (parallaxis), meaning “alternation”. Due to foreshortening, nearby objects have a larger parallax than more distant objects when observed from different positions, so parallax can be used to determine distances.

To measure large distances, such as the distance of a planet or a star from the earth, astronomers use the principle of parallax. Here, the term “parallax” is the semi-angle of inclination between two sight-lines to the star, as observed when the Earth is on opposite sides of the Sun in its orbit.^[3] These distances form the lowest rung of what is called “the cosmic distance ladder“, the first in a succession of methods by which astronomers determine the distances to celestial objects, serving as a basis for other distance measurements in astronomy forming the higher rungs of the ladder.

Parallax also affects optical instruments such as rifle scopes, binoculars, microscopes, and twin-lens reflex cameras that view objects from slightly different angles. Many animals, including humans, have two eyes with overlapping visual fields that use parallax to gain depth perception; this process is known as stereopsis. In computer vision the effect is used for computer stereo vision, and there is a device called a parallax rangefinder that uses it to find range, and in some variations also altitude to a target.

A simple everyday example of parallax can be seen in the dashboard of motor vehicles that use a needle-style speedometer gauge. When viewed from directly in front, the speed may show exactly 60; but when viewed from the passenger seat the needle may appear to show a slightly different speed, due to the angle of viewing.

A simplified illustration of the parallax of an object against a distant background due to a perspective shift. When viewed from “Viewpoint A”, the object appears to be in front of the blue square. When the viewpoint is changed to “Viewpoint B”, the object appears to have moved in front of the red square.

This animation is an example of parallax. As the viewpoint moves side to side, the objects in the distance appear to move more slowly than the objects close to the camera.

照片嗎？

─── 摘自《GoPiGo 小汽車︰格點圖像算術《色彩空間》時中︰立體視覺【一】》

 

此乃『單眼』小汽車煩惱如何得『立體視覺』也！就像『平面國』的科學家思考怎麼樣『定位一點』矣！！

※ 註︰ 新版 raspbian stretch 測試。

pi@raspberrypi:~  geogebra

參考文本︰

 

然而所必須『邏輯』一致者，終究落於『假設』乎？？！！

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

 

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

 

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  has fixed points at 0 and ∞ and is therefore a dilation: . The fixed point equation for the transformation f can then be written

 

Solving for f gives (in matrix form):

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

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

and 

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:

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

or the identity if γ is already at infinity. The transformation  fixes infinity and is therefore a translation:

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

Solving for f (in matrix form) gives

or, if γ = ∞:

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:

 

─── 摘自《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 》》

 

故不得不思辨『變換』前、後耶！！？？

 

 

 

 

 

 

 

 

 

『透視』畢竟始於『眼』見『物像』也。『物』有形狀、遠近，且座落在世界中，『眼』感知為『觀念之像面』上的樣貌、景深也。

 

Rays of light travel from the object to the eye, intersecting with a notional picture plane.

 

Determining the geometry of a square floor tile on the perspective drawing

 

既然一條『視線』已經對應『像面』上的一點，一撇之見難以逆投『全貌』乎？況且『投影』可由幾個『透視』組成，故而終究專注『像面』與『像面』之『關係』哩。一如『投影線』和『投影線』之立論矣。

一張圖

 

借幾行文字

是平行四邊形， 是 、 兩線交點。

三角形 和 相似，

能否曲徑通幽處？身處高林觀日出！

得聞鐘磬聲◎

※ 此處 是 線之方向。

─── 摘自《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 VI‧XII 》》

 

如是同一『視野』下的景象，自可『辯證』耶！

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]: zα,zβ,za,zb = symbols('zα,zβ,za,zb')

In [4]: M1 = Matrix([[zβ,0],[1,-zα]])

In [5]: M1
Out[5]: 
⎡zβ   0 ⎤
⎢       ⎥
⎣1   -zα⎦

In [6]: M2 = Matrix([[zb,0],[1,-za]])

In [7]: M2
Out[7]: 
⎡zb   0 ⎤
⎢       ⎥
⎣1   -za⎦

In [8]: M2*M1
Out[8]: 
⎡ zb⋅zβ      0  ⎤
⎢               ⎥
⎣-za + zβ  za⋅zα⎦

In [9]: M1*M2
Out[9]: 
⎡ zb⋅zβ     0  ⎤
⎢              ⎥
⎣zb - zα  za⋅zα⎦

In [10]: M2*M1.inv()
Out[10]: 
⎡     zb         ⎤
⎢     ──       0 ⎥
⎢     zβ         ⎥
⎢                ⎥
⎢    za    1   za⎥
⎢- ───── + ──  ──⎥
⎣  zα⋅zβ   zβ  zα⎦

In [11]: M1*M2.inv()
Out[11]: 
⎡    zβ        ⎤
⎢    ──      0 ⎥
⎢    zb        ⎥
⎢              ⎥
⎢1      zα   zα⎥
⎢── - ─────  ──⎥
⎣zb   za⋅zb  za⎦

In [12]: M2.inv()*M1.inv()
Out[12]: 
⎡          1                 ⎤
⎢        ─────            0  ⎥
⎢        zb⋅zβ               ⎥
⎢                            ⎥
⎢     1          1        1  ⎥
⎢- ──────── + ────────  ─────⎥
⎣  za⋅zα⋅zβ   za⋅zb⋅zβ  za⋅zα⎦

In [13]: M1.inv()*M2.inv()
Out[13]: 
⎡         1                ⎤
⎢       ─────           0  ⎥
⎢       zb⋅zβ              ⎥
⎢                          ⎥
⎢   1          1        1  ⎥
⎢──────── - ────────  ─────⎥
⎣zb⋅zα⋅zβ   za⋅zb⋅zα  za⋅zα⎦

In [14]: 

 

故知『欲窮千里目，更上一層樓』的道理呦◎

Stereographic projections

Main article: Stereographic projection

It can be useful to think of the complex plane as if it occupied the surface of a sphere. Given a sphere of unit radius, place its center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole is “above” the plane.

We can establish a one-to-one correspondence between the points on the surface of the sphere minus the north pole and the points in the complex plane as follows. Given a point in the plane, draw a straight line connecting it with the north pole on the sphere. That line will intersect the surface of the sphere in exactly one other point. The point z = 0 will be projected onto the south pole of the sphere. Since the interior of the unit circle lies inside the sphere, that entire region (|z| < 1) will be mapped onto the southern hemisphere. The unit circle itself (|z| = 1) will be mapped onto the equator, and the exterior of the unit circle (|z| > 1) will be mapped onto the northern hemisphere, minus the north pole. Clearly this procedure is reversible – given any point on the surface of the sphere that is not the north pole, we can draw a straight line connecting that point to the north pole and intersecting the flat plane in exactly one point.

Under this stereographic projection the north pole itself is not associated with any point in the complex plane. We perfect the one-to-one correspondence by adding one more point to the complex plane – the so-called point at infinity – and identifying it with the north pole on the sphere. This topological space, the complex plane plus the point at infinity, is known as the extended complex plane. We speak of a single “point at infinity” when discussing complex analysis. There are two points at infinity (positive, and negative) on the real number line, but there is only one point at infinity (the north pole) in the extended complex plane.^[6]

Imagine for a moment what will happen to the lines of latitude and longitude when they are projected from the sphere onto the flat plane. The lines of latitude are all parallel to the equator, so they will become perfect circles centered on the origin z = 0. And the lines of longitude will become straight lines passing through the origin (and also through the “point at infinity”, since they pass through both the north and south poles on the sphere).

This is not the only possible yet plausible stereographic situation of the projection of a sphere onto a plane consisting of two or more values. For instance, the north pole of the sphere might be placed on top of the origin z = −1 in a plane that is tangent to the circle. The details don’t really matter. Any stereographic projection of a sphere onto a plane will produce one “point at infinity”, and it will map the lines of latitude and longitude on the sphere into circles and straight lines, respectively, in the plane.

Riemann sphere which maps all points on a sphere except one to all points on the complex plane

 

 

 

 

 

 

 

 

 

⊙	O	I	A	B	⊕	O	I	A	B
O	O	O	O	O	O	O	I	A	B
I	O	I	A	B	I	I	O	B	A
A	O	A	B	I	A	A	B	O	I
B	O	B	I	A	B	B	A	I	O

如果將它用複數改寫成 ，這不就是『上上篇』裡的『三次方程式』 的『解』 的嗎？再徵之以『相量』的『向量加法』和『旋轉乘法』，這個四個元素的『體』之『喻義』也許可以『想像』的了。假使人們對於『抽象思考』一再重複的『練習』，那麼『邏輯推演』也將會是『經驗』中的了！就像俗語說的︰熟能生巧；『抽象的』也就成了『直覺』上的了！！

─── 《【Sonic π】電路學之補充《四》無窮小算術‧下上》

 

先生，就算知道『透視』之形式可以表示為︰

但那可是『整個複數面』 對應到『整個複數面』 的啊！難到說不是『任一視線』，祇該『應合』一『投影點』嗎？

 

然而即使『固定』 ，可『選擇』的 可多了︰

。更別說『不共線』之 的『情況』哩？？

在此反問『透視』條件下，上圖 與 的『關係』是什麼呢 ？！  能不能表示一維『投影線』耶！？

特先請讀者觀此『神奇』也◎

 

“Möbius Transformations Revealed is a wonderful video clarifying a deep topic. This is amazing work by Douglas Arnold and Jonathan Rogness of the University of Minnesota.”

— Edward Tufte

Möbius Transformations Revealed is a short video by Douglas Arnold and Jonathan Rogness which depicts the beauty of Möbius transformations and shows how moving to a higher dimension reveals their essential unity. It was one of the winners in the 2007 Science and Engineering Visualization Challenge and was featured along with the other winning entries in the September 28, 2007 issue of journal Science. The video, which was first released on YouTube in June 2007, has been watched there by nearly two million viewers and classified as a “Top Favorite of All Time“ first in the Film & Animation category and later in the Education category. It was selected for inclusion in the MathFilm 2008 DVD, published by Springer.

From this web page you can:

• View the low resolution video on YouTube, or a version with  Portuguese subtitles
• View the medium resolution video in the IMA video library
• Download the high resolution wide-screen narrated video in QuickTime format. Warning: file is 130 MB.
• See some screenshoots (click any of the images below for higher resolution).
• Download some very high resolution renderings:
  2000×1600 px   3840×7800 px   6000×7800 px
• Read an article about the mathematical background (intended for mathematicians).

 

 

 

 

 

 

 

 

 

只聽 Mrphs 講︰ W!o+ 小時候很靦腆不喜交際。常獨自一人到郊外漫遊，因而結識了小伶鼬，自此成了莫逆。那時『自生成』機器的科學理論已經完備，礙於彼處技術之臨界極限尚且不能實用。不過當時 M♪o 之『 TinyIoT 踢呦ㄊㄜˋ』早就不祇是

Re: TinyIoT 踢呦ㄊㄜˋ From: M♪o@Tux.Anywhere

ㄊㄊㄊㄜˋ↓→ ♀♂

☯  

 

☿☹☺

───

文本中所指的︰深海經絡仿生探測儀了。應用『踢呦ㄊㄜˋ』技術製造之『聰明物件』種類極多，其中『玩具』更是小朋友的最愛。雖然概念上有點像先生貴處的

樂高

樂高（丹麥語：LEGO）是一家丹麥的玩具公司，亦指該公司出品的積木玩具，由五彩的塑膠積木、齒輪、迷你小人和各種不同其他零件，組成各種模型物件。

該公司與多個娛樂公司有合作，如迪士尼、時代華納。例如在哈利波特和星球大戰等電影在美國上映前後，樂高就會推出相應主題玩具。

───

。然而『聰明物件』是由彼此可辨識井通之『智慧組件』所構成，所以能夠不藉學習『無誤組合』，所謂『寫作程式』就像『點菜』一般指定『功能』，然後就能奇妙變化自動運作。不過  W!o+ 並不喜歡玩這種『聰明物件』，他自許為『創客』，想要深入了解那個『聰明』的由來。

─── 《W!o+ 的《小伶鼬工坊演義》︰ 聰明物件》

 

假使宇宙有『成』、『住』、『壞』、『空』諸階段，那麼萬物自有『生』、『老』、『病』、『死』等現象。故而 W!o+ 不把聰明物件當目的，未將仿生科技作宗旨，反倒投入生生傳承行列矣。

傳說『平面國』之『 』智典有言︰

智慧不囿於處境，無受限形式，能應事而用。

或早已知『一維影像』可積成『二維面』也！

Image scanner

In computing, an image scanner—often abbreviated to just scanner, although the term is ambiguous out of context (barcode scanner, CAT scanner etc.)—is a device that optically scans images, printed text, handwriting or an object and converts it to a digital image. Commonly used in offices are variations of the desktop flatbed scanner where the document is placed on a glass window for scanning. Hand-held scanners, where the device is moved by hand, have evolved from text scanning “wands” to 3D scanners used for industrial design, reverse engineering, test and measurement, orthotics, gaming and other applications. Mechanically driven scanners that move the document are typically used for large-format documents, where a flatbed design would be impractical.

Modern scanners typically use a charge-coupled device (CCD) or a contact image sensor (CIS) as the image sensor, whereas drum scanners, developed earlier and still used for the highest possible image quality, use a photomultiplier tube (PMT) as the image sensor. A rotary scanner, used for high-speed document scanning, is a type of drum scanner that uses a CCD array instead of a photomultiplier. Non-contact planetary scanners essentially photograph delicate books and documents. All these scanners produce two-dimensional images of subjects that are usually flat, but sometimes solid; 3D scanners produce information on the three-dimensional structure of solid objects.

Digital cameras can be used for the same purposes as dedicated scanners. When compared to a true scanner, a camera image is subject to a degree of distortion, reflections, shadows, low contrast, and blur due to camera shake (reduced in cameras with image stabilization). Resolution is sufficient for less demanding applications. Digital cameras offer advantages of speed, portability and non-contact digitizing of thick documents without damaging the book spine. As of 2010 scanning technologies were combining 3D scanners with digital cameras to create full-color, photo-realistic 3D models of objects.^[1]

In the biomedical research area, detection devices for DNA microarrays are called scanners as well. These scanners are high-resolution systems (up to 1 µm/ pixel), similar to microscopes. The detection is done via CCD or a photomultiplier tube.

The first image scanner developed for use with a computer was a drum scanner. It was built in 1957 at the US National Bureau of Standards by a team led by Russell A. Kirsch. The first image ever scanned on this machine was a 5 cm square photograph of Kirsch’s then-three-month-old son, Walden. The black and white image had a resolution of 176 pixels on a side.^[2]

Flatbed

This type of scanner is sometimes called reflective scanner because it works by shining white light onto the object to be scanned and reading the intensity and color of light that is reflected from it, usually a line at a time. They are designed for scanning prints or other flat, opaque materials but some have available transparency adapters, which for a number of reasons, in most cases, are not very well suited to scanning film.^[5]

CCD scanner

“A flatbed scanner is usually composed of a glass pane (or platen), under which there is a bright light (often xenon, LED or cold cathode fluorescent) which illuminates the pane, and a moving optical array in CCD scanning. CCD-type scanners typically contain three rows (arrays) of sensors with red, green, and blue filters.”^[6]

 

CIS scanner

Contact image sensor (CIS) scanning consists of a moving set of red, green and blue LEDs strobed for illumination and a connected monochromatic photodiode array under a rod lens array for light collection. “Images to be scanned are placed face down on the glass, an opaque cover is lowered over it to exclude ambient light, and the sensor array and light source move across the pane, reading the entire area. An image is therefore visible to the detector only because of the light it reflects. Transparent images do not work in this way, and require special accessories that illuminate them from the upper side. Many scanners offer this as an option.”^[6]

Scanner unit with CIS. A: assembled, B: disassembled; 1: housing, 2: light conductor, 3: lenses, 4: chip with two RGB-LEDs, 5: CIS

 

曉『兩點』可定『透視』，『三點』將決『觀察者變換』的耶？

 

 

 

 

 

 

 

 

 

 

我們必須謹慎區分『合理化』

Rationalization (psychology)

In psychology and logic, rationalization or rationalisation (also known as making excuses^[1]) is a defense mechanism in which controversial behaviors or feelings are justified and explained in a seemingly rational or logical manner to avoid the true explanation, and are made consciously tolerable—or even admirable and superior—by plausible means.^[2] It is also an informal fallacy of reasoning.^[3]

Rationalization happens in two steps:

1. A decision, action, judgement is made for a given reason, or no (known) reason at all.
2. A rationalization is performed, constructing a seemingly good or logical reason, as an attempt to justify the act after the fact (for oneself or others).

Rationalization encourages irrational or unacceptable behavior, motives, or feelings and often involves ad hoc hypothesizing. This process ranges from fully conscious (e.g. to present an external defense against ridicule from others) to mostly unconscious (e.g. to create a block against internal feelings of guilt or shame). People rationalize for various reasons—sometimes when we think we know ourselves better than we do. Rationalization may differentiate the original deterministic explanation of the behavior or feeling in question.^{[clarification needed][4][5]}

 

和『邏輯一致性』

Consistency

In classical deductive logic, a consistent theory is one that does not contain a contradiction.^[1][2] The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states a theory  is consistent if and only if there is no formula  such that both  and its negation  are elements of the set  . Let  be a set of closed sentences (informally “axioms”) and  the set of closed sentences provable from under some (specified, possibly implicitly) formal deductive system. The set of axioms  is consistent when  is.^[3]

If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive logic, the logic is called complete.^{[citation needed]} The completeness of the sentential calculus was proved by Paul Bernays in 1918^{[citation needed][4]} and Emil Post in 1921,^[5] while the completeness of predicate calculus was proved by Kurt Gödel in 1930,^[6] and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931).^[7] Stronger logics, such as second-order logic, are not complete.

A consistency proof is a mathematical proof that a particular theory is consistent.^[8] The early development of mathematical proof theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilbert’s program. Hilbert’s program was strongly impacted by incompleteness theorems, which showed that sufficiently strong proof theories cannot prove their own consistency (provided that they are in fact consistent).

Although consistency can be proved by means of model theory, it is often done in a purely syntactical way, without any need to reference some model of the logic. The cut-elimination (or equivalently the normalization of the underlying calculus if there is one) implies the consistency of the calculus: since there is obviously no cut-free proof of falsity, there is no contradiction in general.

 

之大不同，努力實證思合『 』符『 』契也◎

文心雕龍‧徵聖

夫鑒周日月，妙極機神；文成規矩，思合符契。或簡言以達旨，或博文以該情；或明理以立體，或隱義以藏用。故《春秋》一字以褒貶，「喪服」舉輕以包重：此簡言以達旨也。《邠詩》聯章以積句 ，《儒行》縟說以繁辭：此博文以該情也。書契斷決以象《夬》，文章昭晰以象《離》：此明理以立體也。「四象」精義以曲隱，「五例」微辭以婉晦：此隱義以藏用也。故知繁略殊形，隱顯異術；抑引隨時，變通會適。征之周、孔，則文有師矣。

 

故進一步再探『透視』關係之『分式線性變換』形式

設有 兩線，而且 與 是那兩線上共線和對應之三點。已知 ，那麼這兩線之間形成透視關係，同時滿足『分式線性變換』形式。

可得

也。

─── 摘自《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 VI‧VIII 》》

 

的『變換分解一致性』呦︰

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]: z1,z2,α,β,λ = symbols('z1,z2,α,β,λ')

In [4]: z = (1-λ)*z1 + λ*z2

In [5]: z
Out[5]: z₁⋅(-λ + 1) + z₂⋅λ

In [6]: H1 = Matrix([[z2-z,-z1*(z2-z)],[z2-z1,-z*(z2-z1)]])

In [7]: H1
Out[7]: 
⎡-z₁⋅(-λ + 1) - z₂⋅λ + z₂   -z₁⋅(-z₁⋅(-λ + 1) - z₂⋅λ + z₂) ⎤
⎢                                                          ⎥
⎣        -z₁ + z₂          (-z₁ + z₂)⋅(-z₁⋅(-λ + 1) - z₂⋅λ)⎦

In [8]: zp = (α*β*z*(z2-z1))/((α-β)*z + (β*z2-α*z1))

In [9]: zp
Out[9]: 
    α⋅β⋅(-z₁ + z₂)⋅(z₁⋅(-λ + 1) + z₂⋅λ)    
───────────────────────────────────────────
-z₁⋅α + z₂⋅β + (α - β)⋅(z₁⋅(-λ + 1) + z₂⋅λ)

In [10]: H2 = Matrix([[β*z2-zp,- α*z1*(β*z2-zp)],[β*z2-α*z1,- zp*(β*z2-α*z1)]])

In [11]: H2
Out[11]: 
⎡           α⋅β⋅(-z₁ + z₂)⋅(z₁⋅(-λ + 1) + z₂⋅λ)            ⎛           α⋅β⋅(-z
⎢z₂⋅β - ───────────────────────────────────────────  -z₁⋅α⋅⎜z₂⋅β - ───────────
⎢       -z₁⋅α + z₂⋅β + (α - β)⋅(z₁⋅(-λ + 1) + z₂⋅λ)        ⎝       -z₁⋅α + z₂⋅
⎢                                                                             
⎢                                                       -α⋅β⋅(-z₁ + z₂)⋅(-z₁⋅α
⎢                   -z₁⋅α + z₂⋅β                        ──────────────────────
⎣                                                           -z₁⋅α + z₂⋅β + (α 

₁ + z₂)⋅(z₁⋅(-λ + 1) + z₂⋅λ)    ⎞⎤
────────────────────────────────⎟⎥
β + (α - β)⋅(z₁⋅(-λ + 1) + z₂⋅λ)⎠⎥
                                 ⎥
 + z₂⋅β)⋅(z₁⋅(-λ + 1) + z₂⋅λ)    ⎥
──────────────────────────────   ⎥
- β)⋅(z₁⋅(-λ + 1) + z₂⋅λ)        ⎦

In [12]: H2反矩陣 = H2.inv()

In [13]: H2反矩陣
Out[13]: 
⎡                  ⎛           α⋅β⋅(-z₁ + z₂)⋅(z₁⋅(-λ + 1) + z₂⋅λ)    ⎞       
⎢z₁⋅(-z₁⋅α + z₂⋅β)⋅⎜z₂⋅β - ───────────────────────────────────────────⎟⋅(α⋅λ -
⎢                  ⎝       -z₁⋅α + z₂⋅β + (α - β)⋅(z₁⋅(-λ + 1) + z₂⋅λ)⎠       
⎢─────────────────────────────────────────────────────────────────────────────
⎢          2   ⎛  2  2                   2  2⎞                                
⎢         β ⋅λ⋅⎝z₁ ⋅α  - 2⋅z₁⋅z₂⋅α⋅β + z₂ ⋅β ⎠⋅(z₁⋅α⋅λ - z₁⋅α - z₂⋅β⋅λ + z₂⋅β)
⎢                                                                             
⎢                                                                            2
⎢                                              (-z₁⋅α + z₂⋅β)⋅(α⋅λ - β⋅λ + β) 
⎢                           ──────────────────────────────────────────────────
⎢                                 ⎛  2  2                   2  2⎞             
⎣                           α⋅β⋅λ⋅⎝z₁ ⋅α  - 2⋅z₁⋅z₂⋅α⋅β + z₂ ⋅β ⎠⋅(z₁⋅α⋅λ - z₁

         3                                          ⎛           α⋅β⋅(-z₁ + z₂)
 β⋅λ + β)                                       -z₁⋅⎜z₂⋅β - ──────────────────
                       α⋅λ - β⋅λ + β                ⎝       -z₁⋅α + z₂⋅β + (α 
────────── + ─────────────────────────────────  ──────────────────────────────
2            β⋅(z₁⋅α⋅λ - z₁⋅α - z₂⋅β⋅λ + z₂⋅β)         ⎛  2  2                
                                                   β⋅λ⋅⎝z₁ ⋅α  - 2⋅z₁⋅z₂⋅α⋅β +
                                                                              
                                                                              
                                                                            -(
───────────────────                                                ───────────
                                                                       ⎛  2  2
⋅α - z₂⋅β⋅λ + z₂⋅β)                                                α⋅λ⋅⎝z₁ ⋅α 

⋅(z₁⋅(-λ + 1) + z₂⋅λ)    ⎞                2 ⎤
─────────────────────────⎟⋅(α⋅λ - β⋅λ + β)  ⎥
- β)⋅(z₁⋅(-λ + 1) + z₂⋅λ)⎠                  ⎥
────────────────────────────────────────────⎥
   2  2⎞                                    ⎥
 z₂ ⋅β ⎠⋅(z₁⋅α⋅λ - z₁⋅α - z₂⋅β⋅λ + z₂⋅β)    ⎥
                                            ⎥
                                            ⎥
α⋅λ - β⋅λ + β)                              ⎥
────────────────────────                    ⎥
                   2  2⎞                    ⎥
 - 2⋅z₁⋅z₂⋅α⋅β + z₂ ⋅β ⎠                    ⎦

In [14]: H = H2反矩陣 * H1

In [15]: H
Out[15]: 
⎡                ⎛           α⋅β⋅(-z₁ + z₂)⋅(z₁⋅(-λ + 1) + z₂⋅λ)    ⎞         
⎢  z₁⋅(-z₁ + z₂)⋅⎜z₂⋅β - ───────────────────────────────────────────⎟⋅(α⋅λ - β
⎢                ⎝       -z₁⋅α + z₂⋅β + (α - β)⋅(z₁⋅(-λ + 1) + z₂⋅λ)⎠         
⎢- ───────────────────────────────────────────────────────────────────────────
⎢              ⎛  2  2                   2  2⎞                                
⎢          β⋅λ⋅⎝z₁ ⋅α  - 2⋅z₁⋅z₂⋅α⋅β + z₂ ⋅β ⎠⋅(z₁⋅α⋅λ - z₁⋅α - z₂⋅β⋅λ + z₂⋅β)
⎢                                                                             
⎢                                                                             
⎢                                                                        (-z₁ 
⎢                                                                 - ──────────
⎢                                                                       ⎛  2  
⎣                                                                   α⋅λ⋅⎝z₁ ⋅α

       2   ⎛                  ⎛           α⋅β⋅(-z₁ + z₂)⋅(z₁⋅(-λ + 1) + z₂⋅λ) 
⋅λ + β)    ⎜z₁⋅(-z₁⋅α + z₂⋅β)⋅⎜z₂⋅β - ────────────────────────────────────────
           ⎜                  ⎝       -z₁⋅α + z₂⋅β + (α - β)⋅(z₁⋅(-λ + 1) + z₂
──────── + ⎜──────────────────────────────────────────────────────────────────
           ⎜          2   ⎛  2  2                   2  2⎞                     
           ⎝         β ⋅λ⋅⎝z₁ ⋅α  - 2⋅z₁⋅z₂⋅α⋅β + z₂ ⋅β ⎠⋅(z₁⋅α⋅λ - z₁⋅α - z₂⋅
                                                                              
                                                                              
+ z₂)⋅(α⋅λ - β⋅λ + β)             (-z₁⋅α + z₂⋅β)⋅(-z₁⋅(-λ + 1) - z₂⋅λ + z₂)⋅(α
───────────────────────── + ──────────────────────────────────────────────────
2                   2  2⎞         ⎛  2  2                   2  2⎞             
  - 2⋅z₁⋅z₂⋅α⋅β + z₂ ⋅β ⎠   α⋅β⋅λ⋅⎝z₁ ⋅α  - 2⋅z₁⋅z₂⋅α⋅β + z₂ ⋅β ⎠⋅(z₁⋅α⋅λ - z₁

   ⎞                3                                    ⎞                    
───⎟⋅(α⋅λ - β⋅λ + β)                                     ⎟                    
⋅λ)⎠                              α⋅λ - β⋅λ + β          ⎟                    
───────────────────── + ─────────────────────────────────⎟⋅(-z₁⋅(-λ + 1) - z₂⋅
           2            β⋅(z₁⋅α⋅λ - z₁⋅α - z₂⋅β⋅λ + z₂⋅β)⎟                    
β⋅λ + z₂⋅β)                                              ⎠                    
                                                                              
             2                                                                
⋅λ - β⋅λ + β)                                                                 
───────────────────                                                           
                                                                              
⋅α - z₂⋅β⋅λ + z₂⋅β)                                                           

              ⎛                  ⎛           α⋅β⋅(-z₁ + z₂)⋅(z₁⋅(-λ + 1) + z₂⋅
              ⎜z₁⋅(-z₁⋅α + z₂⋅β)⋅⎜z₂⋅β - ─────────────────────────────────────
              ⎜                  ⎝       -z₁⋅α + z₂⋅β + (α - β)⋅(z₁⋅(-λ + 1) +
λ + z₂)  - z₁⋅⎜───────────────────────────────────────────────────────────────
              ⎜          2   ⎛  2  2                   2  2⎞                  
              ⎝         β ⋅λ⋅⎝z₁ ⋅α  - 2⋅z₁⋅z₂⋅α⋅β + z₂ ⋅β ⎠⋅(z₁⋅α⋅λ - z₁⋅α - 
                                                                              
                                                                              
                                                                              
                                                                              
                                                                              
                                                                              

λ)    ⎞                3                                    ⎞                 
──────⎟⋅(α⋅λ - β⋅λ + β)                                     ⎟                 
 z₂⋅λ)⎠                              α⋅λ - β⋅λ + β          ⎟                 
──────────────────────── + ─────────────────────────────────⎟⋅(-z₁⋅(-λ + 1) - 
              2            β⋅(z₁⋅α⋅λ - z₁⋅α - z₂⋅β⋅λ + z₂⋅β)⎟                 
z₂⋅β⋅λ + z₂⋅β)                                              ⎠                 
                                                                              
                                                                    2         
        z₁⋅(-z₁⋅α + z₂⋅β)⋅(-z₁⋅(-λ + 1) - z₂⋅λ + z₂)⋅(α⋅λ - β⋅λ + β)        (-
  - ───────────────────────────────────────────────────────────────────── - ──
          ⎛  2  2                   2  2⎞                                     
    α⋅β⋅λ⋅⎝z₁ ⋅α  - 2⋅z₁⋅z₂⋅α⋅β + z₂ ⋅β ⎠⋅(z₁⋅α⋅λ - z₁⋅α - z₂⋅β⋅λ + z₂⋅β)     

                                                 ⎛           α⋅β⋅(-z₁ + z₂)⋅(z
             z₁⋅(-z₁ + z₂)⋅(-z₁⋅(-λ + 1) - z₂⋅λ)⋅⎜z₂⋅β - ─────────────────────
                                                 ⎝       -z₁⋅α + z₂⋅β + (α - β
z₂⋅λ + z₂) - ─────────────────────────────────────────────────────────────────
                                    ⎛  2  2                   2  2⎞           
                                β⋅λ⋅⎝z₁ ⋅α  - 2⋅z₁⋅z₂⋅α⋅β + z₂ ⋅β ⎠⋅(z₁⋅α⋅λ - 
                                                                              
                                                                              
z₁ + z₂)⋅(-z₁⋅(-λ + 1) - z₂⋅λ)⋅(α⋅λ - β⋅λ + β)                                
──────────────────────────────────────────────                                
        ⎛  2  2                   2  2⎞                                       
    α⋅λ⋅⎝z₁ ⋅α  - 2⋅z₁⋅z₂⋅α⋅β + z₂ ⋅β ⎠                                       

₁⋅(-λ + 1) + z₂⋅λ)    ⎞                2⎤
──────────────────────⎟⋅(α⋅λ - β⋅λ + β) ⎥
)⋅(z₁⋅(-λ + 1) + z₂⋅λ)⎠                 ⎥
────────────────────────────────────────⎥
                                        ⎥
z₁⋅α - z₂⋅β⋅λ + z₂⋅β)                   ⎥
                                        ⎥
                                        ⎥
                                        ⎥
                                        ⎥
                                        ⎥
                                        ⎦

In [16]: H[0,0].simplify()
Out[16]: 
  2         2         2                                               2       
z₁ ⋅α⋅λ - z₁ ⋅β⋅λ + z₁ ⋅β - 2⋅z₁⋅z₂⋅α⋅λ + 2⋅z₁⋅z₂⋅β⋅λ - 2⋅z₁⋅z₂⋅β + z₂ ⋅α⋅λ - 
──────────────────────────────────────────────────────────────────────────────
                                  2  2                   2  2                 
                                z₁ ⋅α  - 2⋅z₁⋅z₂⋅α⋅β + z₂ ⋅β                  

  2         2  
z₂ ⋅β⋅λ + z₂ ⋅β
───────────────
               
               

In [17]: H[0,1].simplify()
Out[17]: 0

In [18]: H[1,0].simplify()
Out[18]: 
      2                               2         2       2                     
- z₁⋅α ⋅λ + 2⋅z₁⋅α⋅β⋅λ - z₁⋅α⋅β - z₁⋅β ⋅λ + z₁⋅β  + z₂⋅α ⋅λ - 2⋅z₂⋅α⋅β⋅λ + z₂⋅
──────────────────────────────────────────────────────────────────────────────
                                    ⎛  2  2                   2  2⎞           
                                α⋅β⋅⎝z₁ ⋅α  - 2⋅z₁⋅z₂⋅α⋅β + z₂ ⋅β ⎠           

          2         2
α⋅β + z₂⋅β ⋅λ - z₂⋅β 
─────────────────────
                     
                     

In [19]: H[1,1].simplify()
Out[19]: 
z₁⋅α⋅λ - z₁⋅β⋅λ + z₁⋅β - z₂⋅α⋅λ + z₂⋅β⋅λ - z₂⋅β
───────────────────────────────────────────────
               α⋅β⋅(z₁⋅α - z₂⋅β)               

In [20]: (H[1,1].simplify())/(H[1,0].simplify())
Out[20]: 
                 ⎛  2  2                   2  2⎞                              
                 ⎝z₁ ⋅α  - 2⋅z₁⋅z₂⋅α⋅β + z₂ ⋅β ⎠⋅(z₁⋅α⋅λ - z₁⋅β⋅λ + z₁⋅β - z₂⋅
──────────────────────────────────────────────────────────────────────────────
              ⎛      2                               2         2       2      
(z₁⋅α - z₂⋅β)⋅⎝- z₁⋅α ⋅λ + 2⋅z₁⋅α⋅β⋅λ - z₁⋅α⋅β - z₁⋅β ⋅λ + z₁⋅β  + z₂⋅α ⋅λ - 2

                                     
α⋅λ + z₂⋅β⋅λ - z₂⋅β)                 
─────────────────────────────────────
                         2         2⎞
⋅z₂⋅α⋅β⋅λ + z₂⋅α⋅β + z₂⋅β ⋅λ - z₂⋅β ⎠

In [21]: ((H[1,1].simplify())/(H[1,0].simplify())).simplify()
Out[21]: 
-z₁⋅α + z₂⋅β
────────────
   α - β    

In [22]: 

 

In [22]: ((H[0,0].simplify())/(H[1,0].simplify())).simplify()
Out[22]: 
-α⋅β⋅(z₁ - z₂) 
───────────────
     α - β     

 

In [26]: 分子 = - z1*α**2*λ + 2*z1*α*β*λ - z1*α*β - z1*β**2*λ + z1*β**2 + z2*α**2*λ - 2*z2*α*β*λ + z2*α*β + z2*β**2*λ - z2*β**2

In [27]: 分子
Out[27]: 
      2                               2         2       2                     
- z₁⋅α ⋅λ + 2⋅z₁⋅α⋅β⋅λ - z₁⋅α⋅β - z₁⋅β ⋅λ + z₁⋅β  + z₂⋅α ⋅λ - 2⋅z₂⋅α⋅β⋅λ + z₂⋅

          2         2
α⋅β + z₂⋅β ⋅λ - z₂⋅β 

In [28]: 分子.factor()
Out[28]: -(z₁ - z₂)⋅(α - β)⋅(α⋅λ - β⋅λ + β)

In [29]: 分母 = α*β*(z1**2*α**2 - 2*z1*z2*α*β + z2**2*β**2)

In [30]: 分母
Out[30]: 
    ⎛  2  2                   2  2⎞
α⋅β⋅⎝z₁ ⋅α  - 2⋅z₁⋅z₂⋅α⋅β + z₂ ⋅β ⎠

In [31]: 分母.factor()
Out[31]: 
                 2
α⋅β⋅(z₁⋅α - z₂⋅β) 

In [32]: