射影几何里线段的长度和简单比值在中心投影下都会改变，为什么由四个共线点构造的交比却保持不变？“交比相等”又能告诉我们什么？
Lengths and simple ratios of segments change under central projection, so why does the cross-ratio of four collinear points remain invariant? And what does equality of cross-ratios tell us?
任一射影映射限制在直线上是线性分式变换 x ↦ (ax+b)/(cx+d)；代入交比 (AC·BD)/(AD·BC) 后各坐标差的公共因子互相约去，数值不变。反之，交比相等的两组共线四点可经唯一的射影变换互化，一个数便刻画了四点的射影等价类；取值为 -1 时得调和点列，是完全四角形构形的核心。
Restricted to a line, every projective map acts as a fractional linear transformation x ↦ (ax+b)/(cx+d); substituting this into the cross-ratio (AC·BD)/(AD·BC) makes the common factors of the coordinate differences cancel, so the value survives projection. Conversely, two collinear quadruples with equal cross-ratio are interchanged by exactly one projectivity, so a single number classifies four points up to projective equivalence. The special value -1 yields the harmonic range, central to the complete quadrangle construction.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
H. S. M. Coxeter《Projective Geometry》（第2版），Springer，1987
H. S. M. Coxeter, Projective Geometry, 2nd ed., Springer, 1987
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780387965321 (1987)
#射影几何 #交比 #射影不变量 #调和点列
透视投影为什么让平行线‘交于一点’？
Why does perspective projection make parallel lines meet at a point?
人眼和相机把三维点沿视线投影到像平面，平行但不在视线方向的线会在无限远处汇聚到‘灭点’。这是射影几何的核心现象。
Eyes and cameras project 3D points along sight lines onto an image plane; parallels not along the sight direction converge to a 'vanishing point' at infinity. This is central to projective geometry.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#投影 #灭点
射影几何为什么要给每族平行线添上一个“无穷远点”？这样做的好处是什么？
Why does projective geometry attach a “point at infinity” to each family of parallels, and what is gained by doing so?
在普通平面上，“两条直线要么相交、要么平行”迫使一切涉及直线的命题分类讨论。射影平面为每个方向补上一个无穷远点，让平行线在那里相遇，于是任意两条不同直线恰有唯一交点，命题无条件成立。例外消失后，中心投影不再制造新类型的特殊情形，相交关系在投影下保持不变。绘画中的灭点正是它的直观形象：画面上的铁轨汇聚于一点。
On the ordinary plane, “two lines either meet or are parallel” forces every line-related statement to split into cases. The projective plane appends one point at infinity for each direction, where the parallels meet, so any two distinct lines now share exactly one point, unconditionally. With exceptions gone, central projection creates no new kinds of special cases and incidence relations survive projection unchanged. The vanishing point in perspective painting is its everyday image: rails in the picture converge to a single spot.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
H. S. M. Coxeter《Projective Geometry》，第 2 版，Springer，1987
H. S. M. Coxeter, Projective Geometry, 2nd ed., Springer, 1987
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780387965321 (1987)
#射影几何 #无穷远点 #中心投影