射影幾何是一個數學課題。它是研究在射影變換中不變的幾何性質。這意味著，與初等幾何相比，射影幾何有不同的背景、射影空間和一組選擇性的基本幾何概念。
基本的直覺是，對於給定的維數，射影空間比歐幾裡得空間有更多的點，並且允許幾何變換將額外的點(稱為「無窮遠處的點」)轉換成歐幾裡得點，反之亦然。
Projective geometry is a topic of mathematics. It is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts.
The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice versa.
對射影幾何有意義的性質被這種新的變換思想所尊重，它的效果比用變換矩陣和變換(仿射變換)所能表達的更為激進。對於幾何學家來說，第一個問題是什麼樣的幾何結構適合於一種新的情況。
在射影幾何中，不可能像在歐幾裡得幾何中那樣引用角度，因為角度是一個概念的例子，就像在透視圖中看到的那樣，它在射影變換中不是不變的。射影幾何的一個來源就是透視理論。
與初等幾何的另一個不同之處在於，平行線在無窮遠處相交，一旦這個概念被轉換成射影幾何的術語。同樣，這個概念有一個直觀的基礎，比如在透視圖中，鐵軌在地平線處相交。有關二維射影幾何的基礎知識，請參閱射影平面。
Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than expressible by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation.
It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant with respect to projective transformations, as is seen in perspective drawing. One source for projective geometry was indeed the theory of perspective.
Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of projective geometry in two dimensions.
雖然這些思想在早期就有了，但射影幾何主要是19世紀的發展。這包括復射影空間的理論，所使用的坐標(齊次坐標)是複數。
一些主要類型的更抽象的數學(包括不變量理論、義大利代數幾何學派和Felix Klein的Erlangen程序導致了對經典群的研究)是基於射影幾何的。
它本身也是一門有大量實踐者的學科，如合成幾何學。從射影幾何的公理研究中發展起來的另一個課題是有限幾何。
While the ideas were available earlier, projective geometry was mainly a development of the 19th century. This included the theory of complex projective space, the coordinates used (homogeneous coordinates) being complex numbers.
Several major types of more abstract mathematics (including invariant theory, the Italian school of algebraic geometry, and Felix Klein's Erlangen programme resulting in the study of the classical groups) were based on projective geometry.
It was also a subject with a large number of practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry.
射影幾何的主題本身現在被分成許多研究的子主題，其中兩個例子是射影代數幾何(射影變種的研究)和射影微分幾何(射影變換的微分不變量的研究)。
The topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry (the study of projective varieties) and projective differential geometry (the study of differential invariants of the projective transformations).
01概觀 Overview
射影幾何是幾何的基本非格律形式，這意味著它不是基於距離的概念。在二維中，它首先研究點和線的構型。在這個稀疏的背景中確實存在一些幾何興趣，這被認為是由desargument和其他人在探索透視藝術的原理中發展出來的射影幾何。
在高維空間中有超平面(總是滿足的)和其他線性子空間，它們表現出對偶性的原理。對偶性最簡單的例子是在射影平面上，「兩個不同的點決定了一條唯一的線」(即通過它們的線)和「兩個不同的線決定了一個唯一的點」(即它們的交點)的表述與命題具有相同的結構。
射影幾何也可以被看作是只有直邊的結構幾何。由於射影幾何不包括羅盤結構，所以沒有圓，沒有角度，沒有測量，沒有平行線，也沒有中間的概念。人們意識到，那些確實適用於射影幾何的定理是一些更簡單的表述。例如，在復射影幾何中，不同的圓錐截面都是等價的，一些關於圓的定理可以看作是這些一般定理的特例。
Projective geometry is an elementary non-metrical form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines.
That there is indeed some geometric interest in this sparse setting was seen as projective geometry was developed by Desargues and others in their exploration of the principles of perspective art.
In higher dimensional spaces there are considered hyperplanes (that always meet), and other linear subspaces, which exhibit the principle of duality.
The simplest illustration of duality is in the projective plane, where the statements "two distinct points determine a unique line" (i.e. the line through them) and "two distinct lines determine a unique point" (i.e. their point of intersection) show the same structure as propositions.
Projective geometry can also be seen as a geometry of constructions with a straight-edge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels, and no concept of intermediacy.
It was realised that the theorems that do apply to projective geometry are simpler statements. For example, the different conic sections are all equivalent in (complex) projective geometry, and some theorems about circles can be considered as special cases of these general theorems.
在19世紀早期的工作Jean-Victor彭色列，來到卡諾和其他人建立了射影幾何作為一個獨立的數學領域。其嚴格的基礎是由卡爾·馮·Staudt解決和完善由義大利人朱塞佩·皮亞諾，馬裡奧地區、亞歷桑德羅·Padoa和基諾法諾在19世紀晚期。像仿射幾何和歐幾裡德幾何一樣，射影幾何也可以由Felix Klein的Erlangen程序發展而來；射影幾何的特徵是射影群變換下的不變量。
During the early 19th century the work of Jean-Victor Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics. Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa and Gino Fano during the late 19th century.
Projective geometry, like affine and Euclidean geometry, can also be developed from the Erlangen program of Felix Klein; projective geometry is characterized by invariants under transformations of the projective group.
因此，在對這門學科中大量的定理做了大量的工作之後，人們開始理解射影幾何的基礎。關聯結構和交比是射影變換下的基本不變量。射影幾何可以由仿射平面(或仿射空間)加上一條線(超平面)來建模。「在無限遠處」，然後把這條線(或超平面)當作「普通」。用齊次坐標給出了解析幾何中射影幾何的代數模型。另一方面，公理研究揭示了非desarguesian平面的存在，並舉例說明了關聯公理可以(僅在二維中)由無法通過齊次坐標系進行推理的結構來建模。
After much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The incidence structure and the cross-ratio are fundamental invariants under projective transformations.
Projective geometry can be modeled by the affine plane (or affine space) plus a line (hyperplane) "at infinity" and then treating that line (or hyperplane) as "ordinary". An algebraic model for doing projective geometry in the style of analytic geometry is given by homogeneous coordinates.
On the other hand, axiomatic studies revealed the existence of non-Desarguesian planes, examples to show that the axioms of incidence can be modelled (in two dimensions only) by structures not accessible to reasoning through homogeneous coordinate systems.
從基礎意義上講，射影幾何和有序幾何是基本的，因為它們涉及到最小的公理，並且它們都可以用作仿射和歐幾裡得幾何的基礎。射影幾何不是「有序的」，所以它是一個獨特的幾何基礎。
In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine and Euclidean geometry.Projective geometry is not "ordered" and so it is a distinct foundation for geometry.
02歷史 History
第一個射影性質的幾何性質是在3世紀由亞歷山大的帕普斯發現的。Filippo Brunelleschi(1404 1472)在1425年開始研究透視法的幾何學(見透視法的歷史，更深入地討論美術作品，激發了射影幾何學的發展)。
約翰內斯·克卜勒(1571 1630)和傑拉德·德斯格爾(1591 1661)獨立地提出了「無限點」的概念。
desargument開發了一種構建透視圖的替代方法，通過推廣消失點的使用來包括無限遠處的情況。他把歐幾裡得幾何——平行線實際上是平行的——變成了一個包含一切的幾何系統的特例。德斯哈格對圓錐曲線的研究引起了16歲的布萊斯·帕斯卡的注意，並幫助他闡明了帕斯卡定理。
加斯帕德·蒙格在18世紀末19世紀初的作品對後來射影幾何的發展起了重要作用。直到1845年，米歇爾·蔡斯偶然發現了一份手稿，德薩格的作品才被人們所重視。
同時，Jean-Victor Poncelet在1822年發表了關於射影幾何的基礎性論文。
Poncelet將物體的射影屬性分離到單獨的類中，建立了度量和射影屬性之間的關係。此後不久發現的非歐幾裡得幾何最終被證明具有與射影幾何有關的模型，如雙曲空間的克萊因模型。
The first geometrical properties of a projective nature were discovered during the 3rd century by Pappus of Alexandria.Filippo Brunelleschi (1404–1472) started investigating the geometry of perspective during 1425 (see the history of perspective for a more thorough discussion of the work in the fine arts that motivated much of the development of projective geometry).
Johannes Kepler (1571–1630) and Gérard Desargues (1591–1661) independently developed the concept of the "point at infinity".
Desargues developed an alternative way of constructing perspective drawings by generalizing the use of vanishing points to include the case when these are infinitely far away. He made Euclidean geometry, where parallel lines are truly parallel, into a special case of an all-encompassing geometric system.
Desargues's study on conic sections drew the attention of 16-year-old Blaise Pascal and helped him formulate Pascal's theorem. The works of Gaspard Monge at the end of 18th and beginning of 19th century were important for the subsequent development of projective geometry. The work of Desargues was ignored until Michel Chasles chanced upon a handwritten copy during 1845.
Meanwhile, Jean-Victor Poncelet had published the foundational treatise on projective geometry during 1822.
Poncelet separated the projective properties of objects in individual class and establishing a relationship between metric and projective properties.
The non-Euclidean geometries discovered soon thereafter were eventually demonstrated to have models, such as the Klein model of hyperbolic space, relating to projective geometry.
這種19世紀早期的射影幾何是解析幾何到代數幾何的過渡。當用齊次坐標處理時，射影幾何看起來像是用坐標把幾何問題簡化成代數的一種擴展或技術改進，一種減少特殊情況數量的擴展。朱利葉斯普拉克對二次曲面和「線幾何」的詳細研究仍然為研究更一般概念的幾何學家提供了豐富的例子。
This early 19th century projective geometry was intermediate from analytic geometry to algebraic geometry. When treated in terms of homogeneous coordinates, projective geometry seems like an extension or technical improvement of the use of coordinates to reduce geometric problems to algebra, an extension reducing the number of special cases.
The detailed study of quadrics and the "line geometry" of Julius Plücker still form a rich set of examples for geometers working with more general concepts.
Poncelet, Jakob Steiner和其他人的工作並不是為了擴展解析幾何。技術應該是綜合的：實際上，現在所理解的射影空間是按公理引入的。因此，在射影幾何中重新規劃早期的工作以使其滿足當前的嚴格標準可能有些困難。即使在射影平面的情況下，公理方法也會導致模型無法通過線性代數來描述。
The work of Poncelet, Jakob Steiner and others was not intended to extend analytic geometry. Techniques were supposed to be synthetic: in effect projective space as now understood was to be introduced axiomatically.
As a result, reformulating early work in projective geometry so that it satisfies current standards of rigor can be somewhat difficult. Even in the case of the projective plane alone, the axiomatic approach can result in models not describable via linear algebra.
這一時期的幾何學被Clebsch、Riemann、Max Noether等人對一般代數曲線的研究所取代，這些研究擴展了現有的技術，然後又被不變理論所取代。在本世紀末，義大利的代數幾何學派(Enriques, Segre, Severi)突破了傳統的主題，進入了一個需要更深層次技術的領域。
This period in geometry was overtaken by research on the general algebraic curve by Clebsch, Riemann, Max Noether and others, which stretched existing techniques, and then by invariant theory. Towards the end of the century, the Italian school of algebraic geometry (Enriques, Segre, Severi) broke out of the traditional subject matter into an area demanding deeper techniques.
在19世紀後期，射影幾何的詳細研究變得不那麼流行了，儘管文獻是大量的。特別是舒伯特在枚舉幾何方面做了一些重要的工作，這些工作現在被認為是預見了Chern類的理論，被認為是代表了格拉斯曼代數拓撲。
During the later part of the 19th century, the detailed study of projective geometry became less fashionable, although the literature is voluminous. Some important work was done in enumerative geometry in particular, by Schubert, that is now considered as anticipating the theory of Chern classes, taken as representing the algebraic topology of Grassmannians.
保羅·狄拉克(Paul Dirac)研究了射影幾何，並將其作為發展量子力學概念的基礎，儘管他發表的結果總是以代數形式出現。參見一篇關於這個主題的文章和一本書的博客文章，以及狄拉克1972年在波士頓對普通觀眾所做的關於射影幾何的演講，但沒有詳細說明射影幾何在他的物理學中的應用。
Paul Dirac studied projective geometry and used it as a basis for developing his concepts of Quantum Mechanics, although his published results were always in algebraic form. See a blog article referring to an article and a book on this subject, also to a talk Dirac gave to a general audience during 1972 in Boston about projective geometry, without specifics as to its application in his physics.
03描述 Description
射影幾何比歐幾裡得幾何和仿射幾何的限制要少。它本質上是一種非格律幾何，其事實獨立於任何格律結構。在射影變換下，保留了入射結構和射影調和共軛關係。射影範圍是一維的基礎。
射影幾何形成了透視藝術的一個核心原則:平行線在無限遠處相交，因此被畫成那樣。從本質上講，射影幾何可以看作是歐幾裡得幾何的延伸，其中每條線的「方向」都包含在線內作為一個額外的「點」，而與共面線相對應的方向的「視界」則被看作是一條「線」。因此，兩條平行線在水平線上相交，因為它們具有相同的方向。
Projective geometry is less restrictive than either Euclidean geometry or affine geometry. It is an intrinsically non-metrical geometry, whose facts are independent of any metric structure. Under the projective transformations, the incidence structure and the relation of projective harmonic conjugates are preserved. A projective range is the one-dimensional foundation.
Projective geometry formalizes one of the central principles of perspective art: that parallel lines meet at infinity, and therefore are drawn that way.
In essence, a projective geometry may be thought of as an extension of Euclidean geometry in which the "direction" of each line is subsumed within the line as an extra "point", and in which a "horizon" of directions corresponding to coplanar lines is regarded as a "line". Thus, two parallel lines meet on a horizon line in virtue of their possessing the same direction.
理想化的方向被稱為無限遠處的點，而理想化的地平線被稱為無限遠處的線。反過來，所有這些線都在平面上的無窮遠處。然而，無窮大是一個度規概念，所以一個純粹的射影幾何不會在這方面挑出任何點、線或平面，那些在無窮大的就像其他的一樣。
Idealized directions are referred to as points at infinity, while idealized horizons are referred to as lines at infinity. In turn, all these lines lie in the plane at infinity. However, infinity is a metric concept, so a purely projective geometry does not single out any points, lines or plane in this regard—those at infinity are treated just like any others.
由於歐幾裡得幾何包含在射影幾何中，射影幾何的基礎比較簡單，所以歐幾裡得幾何的一般結果可以用一種更透明的方式推導出來，歐幾裡得幾何中分離但相似的定理可以在射影幾何的框架內共同處理。例如，平行線和非平行線不需要作為單獨的情況處理，我們挑出任意的射影平面作為理想平面，用齊次坐標在無窮遠處定位它。
Because a Euclidean geometry is contained within a projective geometry, with projective geometry having a simpler foundation, general results in Euclidean geometry may be derived in a more transparent manner, where separate but similar theorems of Euclidean geometry may be handled collectively within the framework of projective geometry.
For example, parallel and nonparallel lines need not be treated as separate cases – we single out some arbitrary projective plane as the ideal plane and locate it "at infinity" using homogeneous coordinates.
其他重要的基本性質包括德斯爾格定理和帕普斯定理。在三維或三維以上的射影空間中，有一種結構可以用來證明德斯爾格定理。但是對於第2維，它必須是單獨假設的。
Additional properties of fundamental importance include Desargues' Theorem and the Theorem of Pappus. In projective spaces of dimension 3 or greater there is a construction that allows one to prove Desargues' Theorem. But for dimension 2, it must be separately postulated.
利用德斯格定理，結合其它公理，可以從幾何上定義算術的基本運算。得到的運算滿足場的公理——除了乘法的可交換性需要帕普斯的六邊形定理。結果,每一行的點與給定的欄位一一對應,F,加上一個額外的元素,∞,這樣r∞=∞,∞=∞, r +∞=∞, r / 0 =∞, r /∞= 0,∞r =∞=∞。然而,0/0,∞/∞∞+∞∞∞,0∞∞,0仍未定義。
Using Desargues' Theorem, combined with the other axioms, it is possible to define the basic operations of arithmetic, geometrically. The resulting operations satisfy the axioms of a field — except that the commutativity of multiplication requires Pappus's hexagon theorem.
As a result, the points of each line are in one-to-one correspondence with a given field, F, supplemented by an additional element, ∞, such that r ∞ = ∞, ∞ = ∞, r + ∞ = ∞, r / 0 = ∞, r / ∞ = 0, ∞ r = r ∞ = ∞. However, 0 / 0, ∞ / ∞, ∞ + ∞, ∞ ∞, 0 ∞ and ∞ 0 remain undefined.
射影幾何還包括一個完整的圓錐截面理論，一個在歐幾裡得幾何學已經發展得很好的主題。把雙曲線和橢圓區分開來是有好處的因為雙曲線在無窮遠處與直線相交;拋物線的區別在於它與同一條直線相切。整個圓族可以被認為是在無窮遠處通過直線上兩個給定點的二次曲線，代價是需要複雜的坐標。由於坐標不是「合成的」，我們可以用固定一條線和兩個點來代替坐標，並把通過這些點的所有二次曲線的線性系統作為基本的研究對象。這種方法對有才能的幾何學家很有吸引力，並對這一課題進行了深入的研究。這種方法的一個例子是h·f·貝克的多卷專著。
Projective geometry also includes a full theory of conic sections, a subject already very well developed in Euclidean geometry.
There are advantages in being able to think of a hyperbola and an ellipse as distinguished only by the way the hyperbola lies across the line at infinity; and that a parabola is distinguished only by being tangent to the same line. The whole family of circles can be considered as conics passing through two given points on the line at infinity — at the cost of requiring complex coordinates.
Since coordinates are not "synthetic", one replaces them by fixing a line and two points on it, and considering the linear system of all conics passing through those points as the basic object of study. This method proved very attractive to talented geometers, and the topic was studied thoroughly. An example of this method is the multi-volume treatise by H. F. Baker.
射影幾何有很多，可以分為離散和連續兩種:離散幾何由一組點組成，這些點的數量可能是有限的，也可能不是有限的，而連續幾何有無限多個點，中間沒有間隙。
There are many projective geometries, which may be divided into discrete and continuous: a discrete geometry comprises a set of points, which may or may not be finite in number, while a continuous geometry has infinitely many points with no gaps in between.
維數為0的唯一射影幾何是一個單點。維1的射影幾何由一條包含至少3個點的直線組成。算術運算的幾何結構在這兩種情況下都不能執行。在第2維中，由於缺少德斯格爾定理，有一個豐富的結構。
The only projective geometry of dimension 0 is a single point. A projective geometry of dimension 1 consists of a single line containing at least 3 points. The geometric construction of arithmetic operations cannot be performed in either of these cases. For dimension 2, there is a rich structure in virtue of the absence of Desargues' Theorem.
根據Greenberg(1999)等人的研究，最簡單的二維射影幾何是Fano平面，它每條直線上有3個點，共7個點，共7條線，共線性如下:
According to Greenberg (1999) and others, the simplest 2-dimensional projective geometry is the Fano plane, which has 3 points on every line, with 7 points and 7 lines in all, having the following collinearities:
與齊次坐標= (0,0,1),B = (0, 1, 1), C = (0,1,0), D = (1,0, - 1), E = (1,0,0), F = (1, 1, 1), G =(1 1 0),或者,在仿射坐標,A = (0, 0), B = (0, 1), C = (), D = (1,0), E = (0), F =(1,1)和G = (1) Desarguesian中的仿射坐標平面的點指定的點在無窮遠處(在這個例子中,C, E, G)可以被定義在其他幾個方面。
with homogeneous coordinates A = (0,0,1), B = (0,1,1), C = (0,1,0), D = (1,0,1), E = (1,0,0), F = (1,1,1), G = (1,1,0), or, in affine coordinates, A = (0,0), B = (0,1), C = (∞), D = (1,0), E = (0), F = (1,1)and G = (1). The affine coordinates in a Desarguesian plane for the points designated to be the points at infinity (in this example: C, E and G) can be defined in several other ways.
在標準表示法中，有限射影幾何被寫成PG(a, b)其中:
a是射影(或幾何)維數，
b比直線上的點數(稱為幾何級數)少1。
因此，只有7個點的例子被寫成PG(2,2)。
In standard notation, a finite projective geometry is written PG(a, b) where:
a is the projective (or geometric) dimension, and
b is one less than the number of points on a line (called the order of the geometry).
Thus, the example having only 7 points is written PG(2, 2).
「射影幾何」這個術語有時用來表示背後的普遍抽象幾何,有時表示一個特定的幾何廣泛的興趣,如平面空間的度量幾何分析通過使用齊次坐標,和歐幾裡德幾何學的嵌入式(因此得名,擴展歐幾裡德平面)。
The term "projective geometry" is used sometimes to indicate the generalised underlying abstract geometry, and sometimes to indicate a particular geometry of wide interest, such as the metric geometry of flat space which we analyse through the use of homogeneous coordinates, and in which Euclidean geometry may be embedded (hence its name, Extended Euclidean plane).
挑出所有射影幾何圖形的基本性質是橢圓發病率屬性,任何兩個不同的線L和M在射影平面相交於一個點P .解析幾何的特例平行線包含在平滑的線在無窮遠處,P的謊言。因此，無窮遠處的直線與理論中的任何其它直線一樣:它既不特殊，也不特殊。(在Erlangen程序的後期精神中，人們可以指出轉換組可以將任何直線移到無窮遠處的直線上的方式)。
The fundamental property that singles out all projective geometries is the elliptic incidence property that any two distinct lines L and M in the projective plane intersect at exactly one point P.
The special case in analytic geometry of parallel lines is subsumed in the smoother form of a line at infinity on which P lies. The line at infinity is thus a line like any other in the theory: it is in no way special or distinguished. (In the later spirit of the Erlangen programme one could point to the way the group of transformations can move any line to the line at infinity).
橢圓幾何、歐幾裡德幾何和雙曲幾何的平行性質對比如下:
已知直線l和點P不在直線上，
橢圓: 通過P的直線沒有不滿足l的
歐幾裡得: P中只有一條線不與l相交
雙曲: 通過P的不止一條線不滿足l
橢圓幾何的平行性質是導致射影對偶原理的關鍵思想，這可能是所有射影幾何最重要的共同性質。
The parallel properties of elliptic, Euclidean and hyperbolic geometries contrast as follows:
Given a line l and a point P not on the line,
Elliptic : there exists no line through P that does not meet l
Euclidean: there exists exactly one line through P that does not meet l
Hyperbolic: there exists more than one line through P that does not meet l
The parallel property of elliptic geometry is the key idea that leads to the principle of projective duality, possibly the most important property that all projective geometries have in common.
04二元性 Duality
1825年,約瑟夫Gergonne指出射影平面幾何的對偶原理描述:給出任何定理或幾何的定義,用點線,躺在穿過,並發共線,交叉連接,反之亦然,結果在另一個定理或有效的定義,「雙」。類似地，在三維空間中，點和面之間的對偶關係也成立，任何定理都可以通過交換點和面來轉換。更一般地，對於維N的射影空間，維R的子空間和維N r1之間存在對偶性。對於N = 2，它專門針對點和線之間最常見的對偶形式。對偶原理也是由Jean-Victor Poncelet獨立發現的。
In 1825, Joseph Gergonne noted the principle of duality characterizing projective plane geometry: given any theorem or definition of that geometry, substituting point for line, lie on for pass through, collinear for concurrent, intersection for join, or vice versa, results in another theorem or valid definition, the "dual" of the first.
Similarly in 3 dimensions, the duality relation holds between points and planes, allowing any theorem to be transformed by swapping point and plane, is contained by and contains.
More generally, for projective spaces of dimension N, there is a duality between the subspaces of dimension R and dimension NR1. For N = 2, this specializes to the most commonly known form of duality—that between points and lines. The duality principle was also discovered independently by Jean-Victor Poncelet.
要建立對偶性，只需要建立有關維的公理的對偶版本的定理。因此,對於三維空間,需要顯示(1*)每一點在於3不同的飛機,(2*)每兩平面相交於一個獨特的線和雙版本(3*)的效應:如果飛機P和Q的交集是共面與平面R和S的十字路口,那麼各自的十字路口飛機P, R, Q和S(假設飛機P和S是有別於Q和R)。
To establish duality only requires establishing theorems which are the dual versions of the axioms for the dimension in question.
Thus, for 3-dimensional spaces, one needs to show that (1*) every point lies in 3 distinct planes, (2*) every two planes intersect in a unique line and a dual version of (3*) to the effect: if the intersection of plane P and Q is coplanar with the intersection of plane R and S, then so are the respective intersections of planes P and R, Q and S (assuming planes P and S are distinct from Q and R).
在實踐中，對偶原理允許我們在兩個幾何結構之間建立對偶對應關係。其中最著名的是二次曲線(二維)或二次曲面(三維)中兩個圖形的極性或相互作用。一個常見的例子是對稱多面體在同心球面上的往復運動，從而得到對偶多面體。
In practice, the principle of duality allows us to set up a dual correspondence between two geometric constructions. The most famous of these is the polarity or reciprocity of two figures in a conic curve (in 2 dimensions) or a quadric surface (in 3 dimensions). A commonplace example is found in the reciprocation of a symmetrical polyhedron in a concentric sphere to obtain the dual polyhedron.
另一個例子是Brianchon定理，對偶的帕斯卡定理，其證明之一就是將對偶原理應用於帕斯卡定理。下面是這兩個定理的比較陳述(在射影平面的框架內):
Another example is Brianchon's theorem, the dual of the already mentioned Pascal's theorem, and one of whose proofs simply consists of applying the principle of duality to Pascal's. Here are comparative statements of these two theorems (in both cases within the framework of the projective plane):
帕斯卡:如果一個六邊形的所有六個頂點都在一個圓錐曲線上，那麼它的相對邊的交點(被認為是整條線，因為在射影平面上沒有所謂的「線段」)就是三個共線點。連接它們的線稱為六邊形的帕斯卡線。Pascal: If all six vertices of a hexagon lie on a conic, then the intersections of its opposite sides (regarded as full lines, since in the projective plane there is no such thing as a "line segment") are three collinear points. The line joining them is then called the Pascal line of the hexagon.Brianchon:如果一個六邊形的六個邊都與一個圓錐相切，那麼它的對角線(即連接相對頂點的線)就是三條並行的線。它們的交點稱為六邊形的Brianchon點。Brianchon: If all six sides of a hexagon are tangent to a conic, then its diagonals (i.e. the lines joining opposite vertices) are three concurrent lines. Their point of intersection is then called the Brianchon point of the hexagon.(如果圓錐曲線退化成兩條直線，帕斯卡定理就變成了帕普斯定理，而帕普斯 定理沒有有趣的對偶性，因為Brianchon點平凡地變成了兩條直線的交點。)(If the conic degenerates into two straight lines, Pascal's becomes Pappus's theorem, which has no interesting dual, since the Brianchon point trivially becomes the two lines' intersection point.)
05射影幾何的公理 Axioms of projective geometry
任何給定的幾何都可以從一組適當的公理推導出來。射影幾何的特徵是「橢圓平行」公理，即任意兩個平面總是在一條直線上相交，或在平面上任意兩條直線總是在一點上相交。換句話說，在射影幾何中不存在平行線或平面這樣的東西。許多射影幾何公理的替代集合已經被提出(例如參見Coxeter 2003, Hilbert &科恩-沃森1999，格林伯格1980)。
Any given geometry may be deduced from an appropriate set of axioms. Projective geometries are characterised by the "elliptic parallel" axiom, that any two planes always meet in just one line, or in the plane, any two lines always meet in just one point. In other words, there are no such things as parallel lines or planes in projective geometry.
Many alternative sets of axioms for projective geometry have been proposed (see for example Coxeter 2003, Hilbert & Cohn-Vossen 1999, Greenberg 1980).
Whitehead公理 Whitehead's axioms
這些公理是基於Whitehead的「射影幾何公理」。有兩種類型，點與線，點與線之間有一種「關聯」關係。這三個公理是:
These axioms are based on Whitehead, "The Axioms of Projective Geometry". There are two types, points and lines, and one "incidence" relation between points and lines. The three axioms are:
G1:每行至少包含3個點Every line contains at least 3 pointsG2: 每兩個點，A和B，在一條唯一的直線AB上。Every two points, A and B, lie on a unique line, AB.G3: 如果AB線和CD線相交，那麼AC線和BD線也相交(假設A線和D線與B線和C線不同)。If lines AB and CD intersect, then so do lines AC and BD (where it is assumed that A and D are distinct from B and C).假設每一行至少包含3個點的原因是為了消除一些退化的情況。滿足這三個公理的空間要麼至多只有一條直線，要麼是除法環上的某個維的射影空間，要麼是非德賽斯平面。
Thereasoneachline is assumed to contain at least 3 points is to eliminatesomedegeneratecases.
Thespacessatisfyingthesethreeaxiomseitherhave at mostoneline, or areprojectivespaces of somedimensionover a division ring, or arenon-Desarguesianplanes.
可以添加限制維度或坐標環的更多公理。例如，Coxeter的射影幾何在上面的三個公理中引用了Veblen，另外還有5個公理使維3和坐標環成為一個特徵不是2的交換域。
Onecanaddfurtheraxiomsrestrictingthedimension or thecoordinatering.
Forexample,Coxeter'sProjectiveGeometry, referencesVeblen in thethreeaxiomsabove,togetherwith a further 5 axiomsthatmakethedimension 3 andthecoordinatering a commutativefield of characteristicnot 2.
06使用三元關係的公理 Axioms using a ternary relation
可以通過假設一個三元關係來追求公理化，[ABC]表示當三個點(不一定完全不同)共線時。公理化也可以用這種關係來表示:
Onecanpursueaxiomatization by postulating a ternaryrelation,[ABC] to denotewhenthreepoints(notallnecessarilydistinct)arecollinear. An axiomatizationmay be writtendown in terms of thisrelation as well:
C0:[ABA]C1: 如果A和B是兩個點，使得[ABC]和[ABD]，則[BDC]If A and B aretwopointssuchthat[ABC]and[ABD]then[BDC]C2: 如果A和B是兩個點，那麼還有第三個點C，使得[ABC]If A and B aretwopointsthenthere is a thirdpoint C suchthat[ABC]C3: 如果A和C是兩個點，B和D也是[BCE]， [ADE]而不是[ABE]，那麼F點就是 [ACF]和[BDF]。If A and C are two points, B and D also, with [BCE], [ADE] but not [ABE] then there is a point F such that [ACF] and [BDF].對於兩個不同的點A和B，直線AB被定義為由[ABC]對應的所有點C組成。公理C0和C1提供了G2的形式化形式;C2代表G1, C3代表G3。
Fortwodifferentpoints, A and B, theline AB is defined as consisting of allpoints C forwhich[ABC].Theaxioms C0 and C1 thenprovide a formalization of G2; C2 for G1 and C3 forG3.
線的概念可推廣到平面和高維子空間。因此，可以用子空間AB…X來遞歸地定義一個子空間AB…X，即包含所有直線YZ上的所有點，即Z在AB…X上取值。共線性進而推廣到「獨立性」關係。如果{A, B，…，Z}是子空間AB…Z的最小生成子集，則A集合{A, B，…，Z}是獨立的，[AB，…，Z]。
The concept of line generalizes to planes and higher-dimensional subspaces. A subspace, AB…XY may thus be recursively defined in terms of the subspace AB…X as that containing all the points of all lines YZ, as Z ranges over AB…X.
Collinearity then generalizes to the relation of "independence". A set {A, B, …, Z} of points is independent, [AB…Z] if {A, B, …, Z} is a minimal generating subset for the subspace AB…Z.
射影公理可以通過進一步的空間尺度限制公理加以補充。最小尺寸由所需尺寸的獨立集的存在性決定。對於最低維度，相關條件可以用等價形式表述如下。射影空間為:
Theprojectiveaxiomsmay be supplemented by furtheraxiomspostulatinglimits on thedimension of thespace.Theminimumdimension is determined by theexistence of an independentset of therequiredsize.
Forthelowestdimensions,therelevantconditionsmay be stated in equivalentform as follows. A projectivespace is of:
(L1) 如果它至少有1個點，那麼至少是0維at least dimension 0 if it has at least 1 point,(L2) 至少維度1，如果它至少有兩個不同的點(因此是一條線)at least dimension 1 if it has at least 2 distinct points (and therefore a line),(L3) 至少維度2，如果它有至少3個非共線點(或兩條線，或一條線和不在直線上的一個點)at least dimension 2 if it has at least 3 non-collinear points (or two lines, or a line and a point not on the line),(L4) 如果至少有4個非共面點，則至少為3維。at least dimension 3 if it has at least 4 non-coplanar points.最大尺寸也可以以類似的方式確定。對於最低維度，它們具有以下形式。射影空間為:
Themaximumdimensionmayalso be determined in a similarfashion.Forthelowestdimensions,theytake on thefollowingforms. A projectivespace is of:
(M1) 最多是0，如果它不超過1個點，at most dimension 0 if it has no more than 1 point,(M2) 最多1維，如果它不超過1行，at most dimension 1 if it has no more than 1 line,(M3) 最多2維，如果它不超過1個平面，at most dimension 2 if it has no more than 1 plane,等等。這是一個普遍的定理(公理(3)的結果)，所有共面線相交的原則，射影幾何本來打算體現。因此，性質(M3)可以等價地表述為所有的線相互相交。
and so on. It is a general theorem (a consequence of axiom (3)) that all coplanar lines intersect—the very principle Projective Geometry was originally intended to embody. Therefore, property (M3) may be equivalently stated that all lines intersect one another.
一般認為射影空間至少是二維的。在某些情況下，如果焦點在射影平面上，可以假設M3的變體。例如，(Eves 1997: 111)的公理包括(1)、(2)、(L3)和(M3)。Axiom(3)在(M3)下變得空洞而真實，因此在這個上下文中不需要它。
It is generally assumed that projective spaces are of at least dimension 2. In some cases, if the focus is on projective planes, a variant of M3 may be postulated. The axioms of (Eves 1997: 111), for instance, include (1), (2), (L3) and (M3). Axiom (3) becomes vacuously true under (M3) and is therefore not needed in this context.
07射影平面的公理 Axioms for projective planes
在入射幾何中，多數作者將Fano平面PG(2,2)作為最小有限射影平面來處理。實現這一目標的公理系統如下：
In incidence geometry, most authors give a treatment that embraces the Fano plane PG(2, 2) as the minimal finite projective plane. An axiom system that achieves this is as follows:
(P1) 任何兩個不同的點都位於一條唯一的直線上。Any two distinct points lie on a unique line.(P2) 任何兩條不同的線在一個唯一的點上相交。Any two distinct lines meet in a unique point.(P3) 至少存在四個點，其中沒有三個點是共線的。There exist at least four points of which no three are collinear.Coxeter幾何的介紹給了五個公理的列表更嚴格的射影平面的概念歸因於巴赫曼,增加冠毛定理上面列出的公理(消除non-Desarguesian飛機)和不含射影平面/領域的特點2(那些不滿足範諾公理)。以這種方式給出的約束平面更接近真實射影平面。
Coxeter's Introduction to Geometry gives a list of five axioms for a more restrictive concept of a projective plane attributed to Bachmann, adding Pappus's theorem to the list of axioms above (which eliminates non-Desarguesian planes) and excluding projective planes over fields of characteristic 2 (those that don't satisfy Fano's axiom).
The restricted planes given in this manner more closely resemble the real projective plane.