动直线产生的曲面方程

tommywong

求与三直线：
$\displaystyle \frac{x}{-2}=\frac{y-1}{0}=\frac{z}{1}$
$\displaystyle \frac{x-2}{0}=\frac{y}{1}=\frac{z}{1}$
$\displaystyle \frac{x}{2}=\frac{y+1}{0}=\frac{z}{1}$
相交的动直线产生的曲面方程

青青子衿

回复 1# tommywong

求与三直线：
$\displaystyle \frac{x}{-2}=\frac{y-1}{0}=\frac{z}{1}$
$\displaystyle \frac{x-2}{0}=\frac{y}{1}=\frac{z}{1}$
$\displaystyle \frac{x}{2}=\frac{y+1}{0}=\frac{z}{1}$ ...
tommywong 发表于 2014-7-8 08:21

2008年辽宁高考理数第11题
1997年全国数学联赛
(记\(l_1,l_2,l_3\))连接\(l_1\)上\(P\)与\(l_2\)构成平面与\(l_3\)相交，这是射影空间内的透视映射，因而是射影变换空间异面直线上点列（成射影变换的）连线构成二次曲面，特别地，若三直线为实直线，一般为单叶双曲面，若一直线为无穷远直线，曲面为双曲抛物面
（修改）
tieba.baidu.com/p/1377931815

hbghlyj

Line intersecting 3 other lines
Three non-coplanar lines in the 3D-space always have a fourth one that intersect them all?
Line intersecting three (or four) given lines
Find a ruled quadratic surface, given 3 of its lines.

hbghlyj

There is a general result by Hilbert [that can be found in his book with Cohn-Vossen : "Mathematics and the imagination" 《直观几何 》] saying that, being given 3 "skew" lines in general position in $\Bbb R^3$, there is a unique (ruled) quadric that contains these lines ; it is in general an hyperboloid with one sheet ; exceptionally (as is the case here), it is a hyperbolic paraboloid. See (Equation of a regulus) and as well (Hyperboloids of one sheet, hyperbolic paraboloids, and Hilbert's famous "three skew lines").

青青子衿 发表于 2014-7-8 03:54
...（成射影变换的）连线构成二次曲面

Steiner's method of projective generation of conics

Coxeter "Introduction to Geometry." 2nd ed. pp. 253.

hbghlyj

青青子衿 发表于 2014-7-8 03:54
特别地，若三直线为实直线，一般为单叶双曲面，若一直线为无穷远直线，曲面为双曲抛物面

单叶双曲面$\bf\underline∼$双曲抛物面