人們將如何『預約』未來，又怎麼『翻譯』過去呢？

或許萬象『錯綜複雜』，難得『意義確解』哩！

李白的《靜夜思》在述說些什麼呢？既然已寫了【床前明月光】，作者難道會不知道『月將圓』嗎？為何又寫【疑是地上霜】 的呢！也許人們『知道』一事終將發生，與『感受』那事現實之刻，有巨大的差別，故而『生疑』，總以為時間還很多，事情也還沒發生。所謂『春雪秋霜』，這個 『秋之明月』，又能意指什麼的呢？？在此『中秋賞月』佳節，不是萬戶笙簫？早是團欒樂夥之時？？任誰只消雙手推開，就能見那『窗前月』的了吧！又怎會俳佪 『床前』的呢！！是想『入夢』？又或者『難眠』？？眼巴巴的，竟捱到了『月上中天』，不得不『舉頭望』明月！！或許此刻正『萬籟俱寂 』，夜深人靜，方敢 『思念』，卻不能『想家』，又不忍『思親』 ，祇得說︰【低頭思故鄉】，觸景偶然『思憶』起了『所來之處』 ……

由於一首《静夜思》卻有多個『版本』，據考證，現今『最早』的文本是宋朝郭茂倩《樂府詩集》作：

床前看月光
疑是地上霜
舉頭望山月
低頭思故鄉

又因為這是『詩仙』『李白』流傳最廣泛的一首詩，所以引起學術界許多議論質疑，那位清代文人『蘅塘退士』『孫洙』所編纂的《唐詩三百首》之收錄版本是否是『原作』？如果 Li Bai 不『仙人指路』，那麼我們能夠知道『答案』嗎？或許『流傳的早』就代表『抄寫的對』！！又或許還有尚未發現之『更早版本』？？要是說已知︰李白的《静夜思》創作于唐玄宗開元十四年九月十五日之『揚州旅舍』，時年李白二十六歲。在同時同地所作的還有一首《秋夕旅懷》

涼風度秋海，吹我鄉思飛。
連山去無際，流水何時歸。
目極浮雲色，心斷明月暉。
芳草歇柔豔，白露催寒衣。
夢長銀漢落，覺罷天星稀。
含悲想舊國，泣下誰能揮。

。之後在一個月明星稀的夜晚，詩人抬望天空一輪皓月，思鄉之情油然而生，寫下了這首傳誦古今，中外皆知的名詩。如此我們會怎麼『判斷』這個『版本』的問題呢？駐留於『可能』與『不可能』之間的到底是什麼ㄚ！！

─── 摘自《如何閱讀 W!o+ 之傳真？？》

 

若思假借『幾何』與『代數』談及同一平面之『論述』，應當駕輕就熟的吧？？且讓我們試著使用『比』、『交比』講講所謂『同義反復』也！！

《Cut The Knot》網站上有一篇文章介紹了『交比』︰

Cross-Ratio

The cross-ratio is a surprising and a fundamental concept that plays a key role in projective geometry. In the spirit of duality, cross-ratio is defined for two sets of objects: 4 collinear points and 4 concurrent lines.

The cross-ratio (ABCD) of four collinear points A, B, C, D is defined as the “double ratio”:

(1)

(ABCD) = CA/CB : DA/DB,

where all the segments are thought to be signed. The cross-ratio obviously does not depend on the selected direction of the line ABCD, but does depend on the relative position of the points and the order in which they are listed.

The cross-ratio (abcd) of four (coplanar and) concurrent lines is defined as another double ratio, now of sines:

(abcd) = sin(cMa)/sin(cMb) : sin(dMa)/sin(dMb),

where angles are also considered signed (in a natural way.) If points A, B, C, D are chosen on four lines a, b, c, d concurrent at M, then we often write (abcd) = M(ABCD). The fact that the four points (lines) are grouped into two pairs of points (lines) is reflected in another popular notation: (AB; CD) and (ab; cd).

The relationship between the two definitions is established by the following

Lemma

Remark

When the lines a, b, c, d are defined by the points, as above, it is often convenient to write (abcd) = M(ABCD).

Proof of Lemma

Consider 4 triangles CMA, CMB, DMA, and DMB and represent their areas in two different ways:

	Area(CMA):	h·CA/2	= MC·MA·sin(CMA)/2
	Area(CMB):	h·CB/2	= MC·MB·sin(CMB)/2
	Area(DMA):	h·DA/2	= MD·MA·sin(DMA)/2
	Area(DMB):	h·DB/2	= MD·MB·sin(DMB)/2,

where h is the length of the common altitude of the four triangles from vertex M. The required identity now follows immediately.

The lemma helps explain the significance of the cross-ratio in projective geometry.

Theorem 1

Indeed, from Lemma, (ABCD) = (abcd) = (A’B’C’D’).

It’s worth noting that central projection does not, in general, preserve either the distance or the ratio of two distances.

A permutation of the points may or may not change the cross-ratio. If any two pairs of points are swaped simultaneously, the cross-ratio does not change, e.g., (ABCD) = (BADC) = (DCBA). Wherever it changes, there are only five possible values. If (ABCD) = m, the possible values are 1-m, 1/m, (m-1)/m, 1/(1-m), m/(m-1).

If (ABCD) = 1, then either A = B or C = D. A more important case is where (ABCD) = -1. If (ABCD) = -1 then the points C and D are called harmonic conjugates of each other with respect to the pair A and B. A and B are then also harmonic conjugates with respect to C and D. Each of the pairs is said to divide the other’s segment harmonically. There exists a straight edge only construction of harmonic conjugates. The four lines through an arbitrary point M and four conjugate points are called a harmonic bundle.

One of the four points may lie at infinity. On such occasions, it is useful to consider the limit when a finite point moves to infinity along the common line of the four. The limit is quite simple. For example, if D = , then (ABC) = CA/CB.

The theorem has been established by Pappus in the seventh book of his Mathematical Collections. It was further developed by Desargues starting with 1639 [Wells, p. 41].

或 許可當作探索『幾何意義』的起點。首先為什麼『交比』又稱為『雙比』 double ratio 呢？因為 是『比之比』。這建議了『比』之『定義』可由三『共線點』 給出，因此『交比』 就是 的了。

那 麼這個『比』 有什麼『幾何意義』嗎？事實上它可以用來『確定』直線 上任意點 的『位置』。甚至賦予『座標數值』 ── AC 、BC 同向取正，反向取負 ── 。如果與『共點』 線結合起來︰

可得『邊』、『角』對應關係︰

之面積 。

之面積 。

。

故知對固定之『共點』 與『定點』 而言，『邊角比』

和 點的『位置無關』也。

─── 摘自《GoPiGo 小汽車︰格點圖像算術《投影幾何》【四‧平面國】《壬》》

 

如果說三條『視線』共點 ，假設其為『原點』有何妨乎 ？畢竟『幾何』無此依賴耶！

故而

在 共點也。

今知 『三點共線』，故知恰有一『實數』 ，使得

。所以

。當然

，自然推出

吧！

但是這也等於三角形 與 之面積比，可用

來計算︰

。果然

也。

此時若說有另一點 形成『四點共線』，那麼 當可表示為 吧。這可由 知 而確認哩。所以

。如是『交比』

『邊』、『角』分立左右且『相等』呦◎

縱是『同義反覆』，假使改弦易轍︰

，怕那『交比性質』不易見勒☆

 

 

 

 

 

 

 

 

假設讀者已經知道『複數』

Complex number

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit (which satisfies the equation i² = −1).^[1] In this expression, a is called the real part of the complex number, and b is called the imaginary part. If  , then we write and 

Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the complex numbers are a field extension of the ordinary real numbers, in order to solve problems that cannot be solved with real numbers alone.

As well as their use within mathematics, complex numbers have practical applications in many fields, including physics, chemistry, biology, economics, electrical engineering, and statistics. The Italian mathematician Gerolamo Cardano is the first person known to have introduced complex numbers. He called them “fictitious” during his attempts to find solutions to cubic equations in the 16th century.^[2]

A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. “Re” is the real axis, “Im” is the imaginary axis, and i is the imaginary unit which satisfies i² = −1.

 

是什麼？熟悉它的

運算

通過形式上應用代數的結合律、交換律和分配律，再加上等式i ² = −1，定義複數的加法、減法、乘法和除法:

• 加法： 
• 減法： 
• 乘法：
• 除法：

法則。此處只是簡介如何利用『複數平面』以及『複數運算』表達『平面幾何』的概念。接續文本中祇借 起頭的符號 代表『複數』，其餘符號除非特別指定外，都是『實數』也。

這個『複數』 有兩種表示法

『笛卡爾座標系形式』 ，方便表達

【複數加法‧平移】

。

『極座標形式』 ，容易展現

【複數乘法‧旋轉】

。

若能深入了解

【共軛複數】

，將可用它表達許多概念哩！

‧【實部】

‧【虛部】

‧【長度】

‧……

舉例說 ，那麼

可以看成統和了『內積‧外積』︰

Dot product

In mathematics, the dot product or scalar product^{[note 1]} is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used and often called inner product (or rarely projection product); see also inner product space.

Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry, Euclidean spaces are often defined by using vector spaces. In this case, the dot product is used for defining lengths (the length of a vector is the square root of the dot product of the vector by itself) and angles (the cosine of the angle of two vectors is the quotient of their dot product by the product of their lengths).

The name “dot product” is derived from the centered dot ” · ” that is often used to designate this operation; the alternative name “scalar product” emphasizes that the result is a scalar, rather than a vector, which is the case for the vector product in three-dimensional space.

Scalar projection

Cross product

In mathematics and vector algebra, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space (R³) and is denoted by the symbol ×. Given two linearly independent vectors a and b, the cross product, a × b, is a vector that is perpendicular to both a and b and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with dot product (projection product).

If two vectors have the same direction (or have the exact opposite direction from one another, i.e. are not linearly independent) or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The cross product is anticommutative (i.e., a × b = −(b × a)) and is distributive over addition (i.e., a × (b + c) = a × b + a × c). The space R³ together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.

Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation or “handedness“. The product can be generalized in various ways; it can be made independent of orientation by changing the result to pseudovector, or in arbitrary dimensions the exterior product of vectors can be used with a bivector or two-form result. Also, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can in n dimensions take the product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions.^[1] If one adds the further requirement that the product be uniquely defined, then only the 3-dimensional cross product qualifies. (See § Generalizations, below, for other dimensions.)

The cross-product in respect to a right-handed coordinate system

 

，借之探討幾何上『垂直‧平行』矣。

雖然『SymPy』 的複數運算感覺有點陽春，依舊值得嚐鮮乎！◎

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]: init_printing()

In [3]: z, z1, z2 = symbols('z,z1,z2')

In [4]: x,y,x1,y1,x2,y2,t = symbols('x,y,x1,y1,x2,y2,t', real=True)

In [5]: z = x + y*I

In [6]: z1 = x1 + y1*I

In [7]: z2 = x2 + y2*I

In [8]: z
Out[8]: x + ⅈ⋅y

In [9]: conjugate(z)
Out[9]: x - ⅈ⋅y

In [10]: Abs(z)
Out[10]: 
   _________
  ╱  2    2 
╲╱  x  + y  

In [11]: arg(z)
Out[11]: arg(x + ⅈ⋅y)

In [12]: z1 + z2
Out[12]: x₁ + x₂ + ⅈ⋅y₁ + ⅈ⋅y₂

In [13]: re(z1 + z2)
Out[13]: x₁ + x₂

In [14]: im(z1 + z2)
Out[14]: y₁ + y₂

In [15]: z1 * z2
Out[15]: (x₁ + ⅈ⋅y₁)⋅(x₂ + ⅈ⋅y₂)

In [16]: re(z1 * z2)
Out[16]: x₁⋅x₂ - y₁⋅y₂

In [17]: im(z1 * z2)
Out[17]: x₁⋅y₂ + x₂⋅y₁

In [18]: exp(t*I)
Out[18]: 
 ⅈ⋅t
ℯ   

In [19]: exp(t*I).expand(complex=True)
Out[19]: ⅈ⋅sin(t) + cos(t)

In [20]: re(exp(t*I) *z)
Out[20]: x⋅cos(t) - y⋅sin(t)

In [21]: im(exp(t*I) *z)
Out[21]: x⋅sin(t) + y⋅cos(t)

In [22]: 

 

 

 

 

 

 

 

 

 

《老子》‧第二章

天下皆知美之為美，斯惡已；皆知善之為善，斯不善已。故有無相生，難易相成，長短相形，高下相傾，音聲相和，前後相隨。

是以聖人處無為之事，行不言之教。萬物作焉而不辭，生而不有，為而不恃，功成而弗居。夫唯弗居，是以不去。

何 謂『矛盾』的呢？ 『此』且『非此』，『彼』又『非彼』一類的『概念』。若問『善』與『惡』是『矛盾』的嗎？假使人世間有『不善不惡』之事或『既善且惡』之行，當然『善‧ 惡』就不可說是邏輯的『矛盾』。即使沒有『又善又惡』的事件，那『善‧惡』的概念是否就可窮盡一切現象哩！要是存在『無記』之『非善非惡 』的作為，那『善‧惡』依然不可說是邏輯的『矛盾』！！

 

讀者莫要以為『透視』在『二維』平面上的表達就一定比『三維』空間裡容易？如果說 是一條『投影線』︰

 

代數上講可用 表示也。此時平面上之『任一點』 與這條『投影線』可有多種關係︰

‧I︰ 或 。

‧II︰ 不在『投影線』上，但在 或 『視線』上。

‧III︰ 非 I 且 II ， 在『投影線』上。

‧IV:︰ 非 I、II、III 之『投影線』外一點。

 

此所以前一篇寫來總覺拗口為難矣︰

【相 依三點】 。

如果 ，當下

。

豈能『不相依』耶！◎

 

假設 ，上式不過是說平面裡『任一點』 ，都可以用『任選兩個』 『線性獨立』向量所構成 。若講這 一般也稱作『廣義座標』︰

Generalized coordinates

In analytical mechanics, specifically the study of the rigid body dynamics of multibody systems, the term generalized coordinates refers to the parameters that describe the configuration of the system relative to some reference configuration. These parameters must uniquely define the configuration of the system relative to the reference configuration.^[1] This is done assuming that this can be done with a single chart. The generalized velocities are the time derivatives of the generalized coordinates of the system.

An example of a generalized coordinate is the angle that locates a point moving on a circle. The adjective “generalized” distinguishes these parameters from the traditional use of the term coordinate to refer to Cartesian coordinates: for example, describing the location of the point on the circle using x and y coordinates.

Although there may be many choices for generalized coordinates for a physical system, parameters which are convenient are usually selected for the specification of the configuration of the system and which make the solution of its equations of motion easier. If these parameters are independent of one another, the number of independent generalized coordinates is defined by the number of degrees of freedom of the system.^[2][3]

 

首要『二維』『自由度』不變也。

此時如果

，

怎麼可能在 線上耶？？

然而 『視線』 終有

時乎！！

解析者務請小心也★☆

── 摘自《【Sonic π】電路學之補充《四》無窮小算術‧中下下‧中》

 

 

 

 

 

 

 

 

《LOST 對話錄》

人的存在久遠矣，就像物種的存在一樣，哪有什麼同不同、做不做的事呢？如果存在有道理，那它若不普遍似乎比較神奇的吧！

 誰說人們超越了芝諾，人們果真聽明白了他的話嗎？？有窮與無窮並不是人給的條件，而只是認知之不足的啊。就像諸神的時代已經遠離，人們怎麼還不知道如何過日子哩！！

我親愛的普羅米修斯火種是不夠用的，哪怕你認為把光明帶給世界仍舊徒然？？因為人類根本無法承受那光亮照明的呢！！

䁗奧思，你為什麼這麼說呢？你明知道這火種既不屬於你也不屬於你的孿生兄弟奧德。它從無物反思自身存有迸發出的大霹靂之火而來 。只要還有時間，自然存在機會阿！當虛空歸寂反噬萬有之時，你曾經歷千百萬次，終究無法回想起是吧！！…

 

即使只有『一維視覺』，如果能作『二維運動』，不知『平面國』的科學家會應用什麼樣的數學『描述所見』呢？由於典章早就流失 ，父老傳說幾成神話！與其迷失於想像國度的『奧秘之文』，不如回首人類歷史上的『神奇之數』── 複數 ──

那麽要怎樣理解『複數』 的呢？如果說『複數』起源於『方程式』的『求解』，比方說 ，這定義了『』，但是它的『意義』依然晦澀。即使說從『複數平面』的每一個『點』 都對應著一個『複數』 可能還是不清楚『』的意思到底是什麼？假使再從『複數』的『加法上看』︰

假使 、 ，

那麼 。

這是一種類似『向量』的加法，是否『』的意義就藏在其中的呢？

─── 摘自《【Sonic π】電聲學補充《二》》

 

品嚐一下『代數法』料理『透視』的味道，或可借其『神奇』搭條通往『線性代數』之『橋樑』乎？

此處略講一些『 □ ○ ☆ 術語』作個開始︰

 

【投影線 點對應】 。

也可說『線性相依』︰

。可得

也。

若說兩點間非『線性相依』，則知必是『相異兩點』矣。此時稱作『線性獨立』。

 

【獨立兩點】 。

假設

，那麼

。

這從『聯立方程式』求解可知乎？

。

 

【相依三點】 。

如果 ，當下

。

 

豈能『不相依』耶！◎

 

 

 

 

 

 

 

 

 

無奈這個『一維鏡頭』只有『單點』之象觀，無論『視野』 之『角度』大小︰

Angle of view

In photography, angle of view (AOV)^[1] describes the angular extent of a given scene that is imaged by a camera. It is used interchangeably with the more general term field of view.

A camera’s angle of view can be measured horizontally, vertically, or diagonally.

It is important to distinguish the angle of view from the angle of coverage, which describes the angle range that a lens can image. Typically the image circle produced by a lens is large enough to cover the film or sensor completely, possibly including some vignetting toward the edge. If the angle of coverage of the lens does not fill the sensor, the image circle will be visible, typically with strong vignetting toward the edge, and the effective angle of view will be limited to the angle of coverage.

A camera’s angle of view depends not only on the lens, but also on the sensor. Digital sensors are usually smaller than 35mm film, and this causes the lens to have a narrower angle of view than with 35mm film, by a constant factor for each sensor (called the crop factor). In everyday digital cameras, the crop factor can range from around 1 (professional digital SLRs), to 1.6 (consumer SLR), to 2 (Micro Four Thirds ILC) to 4 (enthusiast compact cameras) to 6 (most compact cameras). So a standard 50mm lens for 35mm photography acts like a 50mm standard “film” lens even on a professional digital SLR, but would act closer to an 80mm lens (1.6 x 50mm) on many mid-market DSLRs, and the 40 degree angle of view of a standard 50mm lens on a film camera is equivalent to a 28 – 35mm lens on many digital SLRs.

In 1916, Northey showed how to calculate the angle of view using ordinary carpenter’s tools.^[2] The angle that he labels as the angle of view is the half-angle or “the angle that a straight line would take from the extreme outside of the field of view to the center of the lens;” he notes that manufacturers of lenses use twice this angle.

In this simulation, adjusting the angle of view and distance of the camera while keeping the object in frame results in vastly differing images. At distances approaching infinity, the light rays are nearly parallel to each other, resulting in a “flattened” image. At low distances and high angles of view objects appear “foreshortened”.

Derivation of the angle-of-view formula

Consider a rectilinear lens in a camera used to photograph an object at a distance  , and forming an image that just barely fits in the dimension,  , of the frame (the film or image sensor). Treat the lens as if it were a pinhole at distance  from the image plane (technically, the center of perspective of a rectilinear lens is at the center of its entrance pupil):^[7]

Now  is the angle between the optical axis of the lens and the ray joining its optical center to the edge of the film. Here  is defined to be the angle-of-view, since it is the angle enclosing the largest object whose image can fit on the film. We want to find the relationship between:

the angle 

the “opposite” side of the right triangle, (half the film-format dimension)

the “adjacent” side,  (distance from the lens to the image plane)

Using basic trigonometry, we find:

 

which we can solve for α, giving:

To project a sharp image of distant objects, needs to be equal to the focal length,  , which is attained by setting the lens for infinity focus. Then the angle of view is given by:

where 

Note that the angle of view varies slightly when the focus is not at infinity (See breathing (lens)), given by  rearranging the lens equation.

Macro photography

For macro photography, we cannot neglect the difference between and  . From the thin lens formula,

.

From the definition of magnification,  , we can substitute  and with some algebra find:

Defining  as the “effective focal length”, we get the formula presented above:

where .

A second effect which comes into play in macro photography is lens asymmetry (an asymmetric lens is a lens where the aperture appears to have different dimensions when viewed from the front and from the back). The lens asymmetry causes an offset between the nodal plane and pupil positions. The effect can be quantified using the ratio (P) between apparent exit pupil diameter and entrance pupil diameter. The full formula for angle of view now becomes:^[5]

─── 摘自《光的世界︰矩陣光學六辛》

 

難免受限於

笛沙格定理

笛沙格（Desargues）定理說明：在射影空間中，有六點A,B,C,a,b,c。Aa,Bb,Cc共點若且唯若AB∩ab,BC∩bc,CA∩ca共線。

在射影幾何的對偶性來看，笛沙格定理是自對偶的。

笛沙格（Desargues）定理說明：在射影空間中，有六點A,B,C,a,b,c。Aa,Bb,Cc共點若且唯若AB∩ab,BC∩bc,CA∩ca共線。

在射影幾何的對偶性來看，笛沙格定理是自對偶的。

 

也。

讀者或可藉著『嘗試』體驗矣！

Desargues’ Theorem

Let A₁B₁C₁ and A₂B₂C₂ be two triangles. Consider two conditions:

1. Lines A₁A₂, B₂B₂, C₁C₂ joining the corresponding vertices are concurrent.
2. Points ab, bc, ca of intersection of the (extended) sides A₁B₁ and A₂B₂, B₁C₁ and B₂C₂, C₁A₁ and C₂A₂, respectively, are collinear.

Desargues’ Theorem claims that 1. implies 2. It’s dual asserts that 1. follows from 2. In particular, the dual to Desragues’ theorem coincides with its converse.

(In the applet below each of the triangles as well as each of the vertices is draggable.)

Created with GeoGebra

Two triangles that satisfy the first condition are said to be perspecitve from a point. Two triangles that satisfy the second condition are said to be perspecitve from a line. Desargues‘ theorem thus claims that two triangles perspective from a point are perspective from a line. Its dual asserts that two triangles perspective from a line are also perspective from a point.

Monge’s theorem can be derived from that of Desargues and in fact is the latter in disguise. The existence of an orthic axis of a triangle is also an immediate consequence of Desargues’ Theorem. In a somewhat disguised form Desargues’ Theorem establishes a relationship between a triangle and a cevian triangle of a point not on a triangle itself.

Curiously, Desragues’ theorem admits an intuitive proof if considered as a statement in the 3-dimensional space, but is not as easy in the 2-dimensional case, where it is often taken as an axiom.

Following is the proof (kindly supplied by Hubert Shutrick) that adopts the 3-dimensional perspective. (The proof fails in some exceptional configurations, e.g., when A₁, A₂, B₁, and B₂ are collinear. These are simple enough to be treated individually and will not be considered here. We simply restrict the theorem to two triangles in general position, i.e., assuming that A₁≠A₂, B₁≠B₂, C₁≠C₂ and that, similarly, no two corresponding side lines coincide.)

 

聽聞此恰是『平面國』之『點投派』將『投影線』『抽象公設化』的時候，所遭遇『四元 □ ○ 完備』困難哩◎

 Complete Quadrangle

If the four points making up a quadrilateral are joined pairwise by six distinct lines, a figure known as a complete quadrangle results. A complete quadrangle is therefore a set of four points, no three collinear, and the six lines which join them. Note that a complete quadrilateral is different from a complete quadrangle.

The midpoints of the sides of any complete quadrangle and the three diagonal points all lie on a conic known as the nine-point conic. If it is an orthocentric quadrilateral, the conic reduces to a circle.

Complete Quadrilateral

The figure determined by four lines, no three of which are concurrent, and their six points of intersection (Johnson 1929, pp. 61-62). Note that this figure is different from a complete quadrangle. A complete quadrilateral has three diagonals (compared to two for an ordinary quadrilateral). The midpoints of the diagonals of a complete quadrilateral are collinear on a line (Johnson 1929, pp. 152-153).

A theorem due to Steiner (Mention 1862ab, Johnson 1929, Steiner 1971) states that in a complete quadrilateral, the bisectors of angles are concurrent at 16 points which are the incenters and excenters of the four triangles. Furthermore, these points are the intersections of two sets of four circles each of which is a member of a conjugate coaxal system. The axes of these systems intersect at the point common to the circumcircles of the quadrilateral.

Newton proved that, if a conic section is inscribed in a complete quadrilateral, then its center lies on (Wells 1991). In addition, the orthocenters of the four triangles formed by a complete quadrilateral lie on a line which is perpendicular to . Plücker proved that the circles having the three diagonals as diameters have two common points which lie on the line joining the four triangles’ orthocenters (Wells 1991).