Numero 给了欧几里得几何和双曲几何的平行线定义。欧几里得几何是处处曲率为1 的特别特殊的几何,双曲几何是无数非欧几何中一种特别的几何,罗巴切夫斯基重点研究了双曲几何,黎曼重点研究了椭圆几何。这些都是一些特例,随便说一句,双曲几何可以引无数条平行线。你给的两个特例定义,完全不能否定从平行线相交的质疑。而且椭圆几何的平行线,确实是相交的,如地球经线相交在南极北极。
hyperbolic geometry, also called Lobachevskian Geometry, a non-Euclidean geometry that rejects the validity of Euclid’s fifth, the “parallel,” postulate. Simply stated, this Euclidean postulate is: through a point not on a given line there is exactly one line parallel to the given line. In hyperbolic geometry, through a point not on a given line there are at least two lines parallel to the given line. The tenets of hyperbolic geometry, however, admit the other four Euclidean postulates.
Although many of the theorems of hyperbolic geometry are identical to those of Euclidean, others differ. In Euclidean geometry, for example, two parallel lines are taken to be everywhere equidistant. In hyperbolic geometry, two parallel lines are taken to converge in one direction and diverge in the other. In Euclidean, the sum of the angles in a triangle is equal to two right angles; in hyperbolic, the sum is less than two right angles. In Euclidean, polygons of differing areas can be similar; and in hyperbolic, similar polygons of differing areas do not exist.
The first published works expounding the existence of hyperbolic and other non-Euclidean geometries are those of a Russian mathematician, Nikolay Ivanovich Lobachevsky, who wrote on the subject in 1829, and, independently, the Hungarian mathematicians Farkas and János Bolyai, father and son, in 1831.
欧氏几何起源于Euclid的Elements,主要围绕卷1-4 和6的plane geometry, 其中包括5条公理。争议点在于第五公理,也就是平行公理,是独立存在还是可以从其他公理证明。平行公理的内容是: For any line l and any point P not on l, there exists a unique line m through P and parallel to l.
我说的一个故事
Lobachevsky上中学的时候,发现可以让平行线相交 建立新的几何学。被老师斥责无可救药。他苦恼至死,死后12年公认 - 兄贵
我写了这么一句后,被两个网友质疑:
• 更正一下:是线外过一点可以有不止一条平行线。任何几何学中“平行线”的定义都是不相交的线。 - trivial -
你那个Lobachevsky的典故理解错了,给你批判一下 LOL 来源: Numero 于 2023-03-20 13:23:04 [档案] [博客] [旧帖] [给我悄悄话] 本文已被阅读: 44 次 (469 bytes) 本文内容已被 [ Numero ] 在 2023-03-20 13:42:51 编辑过。如有问题,请报告版主或论坛管理删除.任何几何学中平行线的定义是:如果两条线没交点则称为平行。
欧氏几何中是这样:“过线外一点有且只有一条线和已知线平行”
而非欧氏几何(Lobachevskian geometry),也就是双曲几何,平行线多了而已
是这样的: “过线外一点至少有两条不同的线和已知线平行“
其实可以有无数多条。。。
这两位应该是大学数学教授,所以我觉得应该回答一下。
首先我说的是 一个众所周知的事实,不知道他们反对什么,网上可以查,类似的很多:
https://k.sina.com.cn/article_6501934712_1838ba67800100rviq.html?from=science
首先我说的没有错。再来看他们的论点:
trivial说的,任何几何学中“平行线”的定义都是不相交的线 反而是错的,诺巴切夫斯基和黎曼都是从质疑平行线是否相交开创的非欧几何的研究,而且平行线相交也是一种流行说法,见:
Numero 给了欧几里得几何和双曲几何的平行线定义。欧几里得几何是处处曲率为1 的特别特殊的几何,双曲几何是无数非欧几何中一种特别的几何,罗巴切夫斯基重点研究了双曲几何,黎曼重点研究了椭圆几何。这些都是一些特例,随便说一句,双曲几何可以引无数条平行线。你给的两个特例定义,完全不能否定从平行线相交的质疑。而且椭圆几何的平行线,确实是相交的,如地球经线相交在南极北极。
在非欧几何中,不存在直线的概念,所以,我从来不用平行线这个词。非欧几何中最短的是测地线,距离的系数是度规,从度规可以计算曲率,各个地方的曲率可以完全不同,甚至难以计算。欧几里得几何的平行线,在微分几何中,平行线相当于是曲面或者流形上的于测地线成90度角的系列线,因为测地线是曲率最小的曲线,沿着测地线平移的向量的大小和方向不会改变,对于给定的度规张量,这些“平行线”可以通过求测地线的微分方程来求解。
所以,Numero 的两个定义是可取的,但是不全面,更没有动摇我的说法。而trivial是完全不靠谱的。
我是再也弄不明白了。。这些网站也不给个明确的回答。。。
而且是挑战数学教授,只凭这一点就够牛!
根据之前一贯的推理,兄贵不是学计算机出身,不是学金融出身,肯定也不是学数学出身。闯入别人领域指点江山,一剑孤胆如入无人之境,这份潇洒,舍兄贵其谁?
hyperbolic geometry, also called Lobachevskian Geometry, a non-Euclidean geometry that rejects the validity of Euclid’s fifth, the “parallel,” postulate. Simply stated, this Euclidean postulate is: through a point not on a given line there is exactly one line parallel to the given line. In hyperbolic geometry, through a point not on a given line there are at least two lines parallel to the given line. The tenets of hyperbolic geometry, however, admit the other four Euclidean postulates.
Although many of the theorems of hyperbolic geometry are identical to those of Euclidean, others differ. In Euclidean geometry, for example, two parallel lines are taken to be everywhere equidistant. In hyperbolic geometry, two parallel lines are taken to converge in one direction and diverge in the other. In Euclidean, the sum of the angles in a triangle is equal to two right angles; in hyperbolic, the sum is less than two right angles. In Euclidean, polygons of differing areas can be similar; and in hyperbolic, similar polygons of differing areas do not exist.
The first published works expounding the existence of hyperbolic and other non-Euclidean geometries are those of a Russian mathematician, Nikolay Ivanovich Lobachevsky, who wrote on the subject in 1829, and, independently, the Hungarian mathematicians Farkas and János Bolyai, father and son, in 1831.
In hyperbolic geometry, two parallel lines are taken to converge in one direction and diverge in the other.
the given line 平行
我没时间给你科普太多, 就基本几条。
平行是几何概念,既不依赖于坐标系,也不依赖于任何测度。平行线的定义就是线不相交。(这里也没有所谓直线曲线之分)。
欧氏几何起源于Euclid的Elements,主要围绕卷1-4 和6的plane geometry, 其中包括5条公理。争议点在于第五公理,也就是平行公理,是独立存在还是可以从其他公理证明。平行公理的内容是: For any line l and any point P not on l, there exists a unique line m through P and parallel to l.
近两千年来数学家的重点都集中在试图证明平行公理可以证明上。所说的non-Euclidean plane geometry, 是指承认Euclid 前四条公理的情况下,平行公理可否不成立。
Elliptic geometry 中任两条线相交,也就没有平行线。但是它不全满足前面四条欧式公理,所以不需考虑。
前面四条欧式公理可以推出平行线的存在性,所以问题在于唯一性。
Hyperbolic parallel property is: for any line l and any point P not on l, there are more than one line through P and parallel to l.
Hyperbolic geometry = a geometry with Euclidean's first four postulates and hyperbolic parallal property.
Lobachevsky 的贡献在于他论证hyperbolic geometry 可以存在。 他不是唯一的一个,与他思路类似的还有几乎同期的匈牙利数学家Janos Bolyai。 他们的工作当时没有被迅速接受,因为他们都没给出实例。 第一个例子是1868年Eugenio Beltrami 构造的。
(至于你说什么拿垂直测地线定义平行, 了解一下Lambert quadrilateral, 这个四边形有三个内角是直角,另一个不是。 按你的定义,对边是平行还是不平行啊?
https://en.wikipedia.org/wiki/Lambert_quadrilateral )