比較『投影』之『幾何公設化』與『線性代數化』，猶自

Timothy Peil 教授寫之

4.6.2 Fundamental Theorem of Projective Geometry  Printout

All truths are easy to understand once they are discovered; the point is to discover them..
—Galileo Galilei (1564–1643)

 

 

In the introduction to projective geometry, we stated that in a later section we would consider a mapping between two pencils of points. We begin by showing that there exists a projectivity between two pencils of points. (The first video is the lecture and the second video is the construction in Geometer’s Sketchpad.)
Assume A, B, C are elements of a pencil with axis p and A’, B’, C’ are elements of a pencil with axis p’.  Further, assume the points are distinct and the axes p and p’ are distinct. We desire to find a projectivity so that ABC is projectively related to A’B’C’, i.e. . Since a projectivity is a composition of perspectivities, we construct two perspectivities to map ABC to A’B’C’.

To construct the first perspectivity, we define the center of a perspectivity that will map A to A’ and C to itself with the image of B to be found. Let P be a point on AA’ that is distinct from A and A’. (How do we know point P exists?) Let B₁ = BP · A’C. Thus .
For the second perspectivity, we define the center of a perspectivity that maps A’ to itself, B₁ to B’, and C to C’. Let Q = B₁B’ · CC’. Then . Since   and , we have . We have proven the following theorem.

Theorem 4.10. If A, B, C and A’, B’, C’ are distinct elements in pencils of points with distinct axes p and p’, respectively, then there exists a projectivity such that ABC is projectively related to A’B’C’.

 

通往 Christopher Cooper  先生的

 

宛如從 之『比』︰

《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。

此乃『交比不變性』之基礎吧！

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

 

到定義『一維空間』也？！

One-dimensional space

In physics and mathematics, a sequence of n numbers can specify a location in n-dimensional space. When n = 1, the set of all such locations is called a one-dimensional space. An example of a one-dimensional space is the number line, where the position of each point on it can be described by a single number.^[1]

In algebraic geometry there are several structures that are technically one-dimensional spaces but referred to in other terms. A field k is a one-dimensional vector space over itself. Similarly, the projective line over k is a one-dimensional space. In particular, if k = ℂ, the complex numbers, then the complex projective line P¹(ℂ) is one-dimensional with respect to ℂ, even though it is also known as the Riemann sphere.

More generally, a ring is a length-one module over itself. Similarly, the projective line over a ring is a one-dimensional space over the ring. In case the ring is an algebra over a field, these spaces are one-dimensional with respect to the algebra, even if the algebra is of higher dimensionality.

Coordinate systems in one-dimensional space

The most popular coordinate systems are the number line and the angle.

• 

  Number line

• 

  Angle

故此系列文本『不求甚解』，只可說是『交會小品』而已吧◎