Projective plane

Japanese: 射影平面 - しゃえいへいめん

A mathematical plane with a line at infinity added to it. A projective plane can be considered from both topological and analytical aspects. If you think of a line segment as a string, roll it up and overlap the two ends, it will close like a circle. In mathematical terms, if you identify the two end points of a line segment, it will be topologically isomorphic to a circle. Similarly, if you identify sides AB and DC of a quadrangle like (1) in the figure , it will be topologically isomorphic to the cylindrical surface of (2) in the figure . In mathematics, a new set is often created by identifying several points in a set of points according to a certain rule. Two points that are the end points of a diameter on a sphere are called the diameter pairs. If you consider the set of points created by all the points on a sphere and identify the diameter pairs, a new set is created ((3) in the figure ). This set is called a projective plane or two-dimensional projective space. In this case, the components of the projective plane (called points) are the two points that are identified (in the normal sense).

Next, if we cut off the sphere south of the equator, the remaining set is a hemisphere where the diametric pairs are identified only on the equator, such as Q and Q', and R and R' ( Figure (4)). Such a set is also a representation of a projective plane. In this case, the point near Q in the northern hemisphere is naturally close to Q'. This projective plane is further deformed. This hemisphere is turned down onto a plane and orthogonally projected onto the plane from directly above. The set that can be created on the plane with this one-to-one correspondence is a disk (the inside, including the circumference of the circle), and the set of all the points on the disk is also a representation of a projective plane, under the condition that the diametric pairs are identified on the circumference of the edge. In Figure (5), the strip surrounded by Q, R, Q', and R' is homeomorphic to a Möbius strip, since QR is identified to Q'R' inversely. As such, the projective plane contains a Möbius strip as a part of it, and so it is an unsignifiable (unorientable surface). We present an analytical construction method for the projective plane starting from the xy plane. For a point (x, y),

Let us make the numbers X, Y, and Z correspond such that (X, Y, Z) are called the homogeneous coordinates of the point (x, y). (X', Y', Z') where X:Y:Z = X':Y':Z' is also a homogeneous coordinate of the same point, so homogeneous coordinates can be determined excluding the factor of proportionality. Now, let us consider the symbols (X,Y,0) (X and Y are not simultaneously zero) and (X',Y',0) to be the same when X:Y = X':Y', and put these into the family of points, calling them points at infinity. Of course, these new points do not exist on the xy plane. Linear equation of X, Y, Z (1) aX + bY + cZ = 0
The set of all points that satisfy this is called a projected line. The solution of (1) where Z ≠ 0 is (2) ax + by + c = 0
Since it is the set of points that satisfy the above equation, it is a straight line on the xy plane. The solution to (1) where Z = 0 is the point at infinity, (X, Y, Z) = (b, -a, 0). Therefore, a projective line is a line on the xy plane with a point at infinity added to it. Since the line parallel to (2) is of the form ax + by + c' = 0, it can be seen that these two parallel lines intersect at the point at infinity (b, -a, 0). The set of points at infinity forms a single projective line. This is called the line at infinity. If the set of points obtained by adding the line at infinity to the xy plane is transformed topologically, ignoring the concept of distance, it becomes the projective plane mentioned earlier. In general, for points (x ₁ , x ₂ , ... , x _n ) in an n-dimensional space,

If we call (X ₁ , X ₂ , ……, X _n , X ₀ ) homogeneous coordinates, then an n-dimensional projective space can be constructed in the same way as in the case of n = 2.

[Tachibana Shunichi]

Projection plane diagram (figure)
©Shogakukan ">

Projection plane diagram (figure)

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

無限遠直線を付加した数学的平面をいう。射影平面は、位相的側面と、解析的側面から考察することができる。一つの線分を紐(ひも)と考え、丸めて両端を重ねれば円のように閉じる。数学のことばでは、線分の両端点を同一視すれば円と位相同形になるという。同様に図の(1)のような四角形は辺ABとDCを同一視すれば図の(2)の円柱面と位相同形になる。数学ではこのように、一つの点集合のなかのいくつかの点どうしをある法則によって同一視して新しい集合をつくることが多い。球面上で一つの直径の両端点となっている2点を互いの直径対点という。球面上の点全体のつくる点集合を考え、直径対点どうしを同一視すれば新しい集合ができる（図の(3)）。この集合を射影平面または二次元射影空間という。この場合、射影平面の構成要素（これを点とよぶ）は同一視された（普通の意味の）2点である。

　次に、球面の赤道から南を切り捨ててしまうと、残った集合は赤道上だけQとQ′、RとR′のように直径対点が同一視されている半球面である（図の(4)）。このような集合も射影平面（の一つの表現）である。この場合、北半球でQの近くの点は当然Q′に近い。この射影平面をさらに変形する。この半球を平面上に伏せて真上から平面上に正射影する。この一対一対応で平面上にできる集合は円盤（円の周も含めた内部）で、円盤上の点全体のつくる集合は、その縁(ふち)の円周上では直径対点を同一視するという条件のもとで、また射影平面の一つの表現である。図の(5)でQ、R、Q′、R′の囲む帯状の部分はQRがQ′R′に逆さに同一視されているのでメビウスの帯と位相同形である。このように射影平面はメビウスの帯を一部分として含むことから不可符号（向きづけ不可能な曲面）であることがわかる。xy平面から出発した射影平面の解析的構成法を述べる。点（x, y）に対して

なる数X、Y、Zを対応させ、（X, Y, Z）を点（x, y）の斉次(せいじ)座標とよぶ。X：Y：Z＝X′：Y′：Z′なる（X′, Y′, Z′）も同1点の斉次座標となるから、斉次座標は比例因子を除いて定まる。いま記号（X, Y, 0）（X、Yは同時にはゼロでないとする）と記号（X′, Y′, 0）はX：Y＝X′：Y′なるとき同一視し、これらを点の仲間に入れて無限遠点とよぶことにする。もちろんこれらの新しい点はxy平面上には存在しない。X、Y、Zの一次方程式
　　(1)　　aX＋bY＋cZ＝0
を満たす点全体の集合を射影直線とよぶ。Z≠0なる(1)の解は
　　(2)　　ax＋by＋c＝0
を満たす点の全体であるから、xy平面上の直線である。Z＝0なる(1)の解は（X, Y, Z）＝（b, －a, 0）なる無限遠点である。したがって、射影直線とはxy平面上の直線に一つの無限遠点を付加したものである。(2)に平行な直線はax＋by＋c′＝0の形なので、これら平行2直線は無限遠点（b, －a, 0）で交わることがわかる。無限遠点の全体は一つの射影直線をつくる。これを無限遠直線とよぶ。xy平面に無限遠直線を付加した点集合は、距離の概念を無視して位相的に変形すれば、先に述べた射影平面になる。一般にn次元数空間の点（x₁, x₂,……, x_n）に対して

と置いて（X₁, X₂,……, X_n, X₀）を斉次座標とよべば、n＝2の場合と同様にしてn次元射影空間が構成される。