由三角形引导的一个几何变换——三坐标反演

hujunhua

先占个位置。

非退化的三次曲线与三角形的三边皆有至少一个实交点，故其像曲线通过三个顶点。
若像曲线仍为非退化三次曲线，则源曲线亦通过三个顶点。
故非退化三次曲线在三坐标反演下仍为非退化三次曲线当且仅当曲线通过三个顶点。它们的代数方程中 `x^3, y^3, z^3` 项的系数都是 0:\[
ax^2y+bxy^2+cy^2z+dyz^2+ez^2x+fzx^2+gxyz=0
\]两边除以 `xyz`, 写作成对互为倒数的分式项，方程形式在三坐标反演下的不变性更明显：\[
a\frac xz+b\frac yz+c\frac yx+d\frac zx+e\frac zy+f\frac xy+g=0\]
显然，上述曲线要为不变三次曲线，互倒项的系数应对应相等，即应具有以下形式\[
ax(y^2+z^2)+by(z^2+x^2)+cz(x^2+y^2)=dxyz\\
a(\frac yz+\frac zy)+b(\frac zx+\frac xz)+c(\frac xy+\frac yx)=d
\]要画出一条具体的曲线，又需要作旋转变换了，我可心算不来。

数学星空

借用一下Wayne 的代码哈~

为什么不是\(\frac{x}{y}+\frac{y}{x}+\frac{y}{z}+\frac{z}{y}+\frac{z}{x}+\frac{x}{z}=C\)呢？

zeroieme

数学星空 发表于 2019-3-21 19:04
借用一下Wayne 的代码哈~

为什么不是\(\frac{x}{y}+\frac{y}{x}+\frac{y}{z}+\frac{z}{y}+\frac{z}{x}+\ ...

可以多图Show一起比较下

wayne

\[\left(\frac{x}{y}+\frac{y}{x}\right)+\left(\frac{x}{z}+\frac{z}{x}\right)+\left(\frac{y}{z}+\frac{z}{y}\right)\]
变换成
\[\frac{12 z^3-3 (x+y) \left(x^2-4 x y+y^2\right)}{(x+y-z) \left(x^2-2 z (x+y)-4 x y+y^2-2 z^2\right)}\]
画图如下：


跟前面的六次曲线是一样的形状

hujunhua

上面的曲线可能是对的，真是大跌眼镜！

自觉对三次曲线素有研究，没想过三次曲线还能如此这般。阅后一想，豁然开朗，三次曲线如此呈现实属正常，因为一般情况下三次曲线与无穷远线有3个不同的交点，相应就可有三条渐近线，那不就是这样么。

平常所见的例子，无穷远线总是很特殊，还真没用一般割线做过无穷远线。以至形成了三次曲线最多有两个分支的固定印象。


习惯与偏好造就盲点！

hujunhua

`(x+1)(x^2-3y^2-4x+4)=1`

数学星空

10#命题1：通过两个顶点和两个不动点（无三点共线）的二次曲线为变换下的不变曲线。

     因为两个不动点确定一条直线，而直线通过变换后形成一个双曲线，这个双曲线为命题说的不变曲线吗？

30# 我们已经算出\([x_0,y_0]\)变成\([x,y]\),且满足关系：

    \( -ab^2yy_0+ac^2yy_0+a^2sy+a^2sy_0-2asxy_0-2asx_0y+b^2sy+b^2sy_0-c^2sy-c^2sy_0-4as^2+4s^2x+4s^2x_0=0\)

    \(3a^4yy_0+2a^2b^2yy_0-2a^2c^2yy_0-b^4yy_0+2b^2c^2yy_0-c^4yy_0+4a^2sxy_0+4a^2sx_0y-4b^2sxy_0-4b^2sx_0y+4c^2sxy_0+4c^2sx_0y-16s^2xx_0=0\)

    例：我们取\(a = 3, b = 4, c = 5, s = 6\)


     得到：

\(x=\frac{(10368x_0^2-31104x_0)}{6(576x_0^2-432x_0y_0+324y_0^2-1728x_0)}\)

\(y=\frac{-4(-576x_0^2+432x_0y_0+1728x_0-1296y_0)}{576x_0^2-432x_0y_0+324y_0^2-1728x_0}\)


经过验算\((x_0=1,y_0=1)\longrightarrow (x = \frac{96}{35}, y = \frac{32}{35})\longrightarrow (x_0=1,y_0=1)\longrightarrow (x = \frac{96}{35}, y = \frac{32}{35})....\)

直接将代数式代入迭代表达式也是二次循环的结果，这就是对合变换吗？

老胡得到的不变点集表达式在直角坐标系中应该是什么？不知有没有计算方案

mathe

我们可一看hujunhua的面积坐标系中$A(1,0,0),B(0,1,0),C(0,0,1),O(1,1,1)$
对应正常坐标系下$A({-sqrt(3)}/2,-1/2,1), B({sqrt(3)}/2,-1/2,1),C(0,1,1),O(0,0,1)$
设变换P将面积坐标变换成普通坐标，那么根据A,B,C可以假设
\(P=\begin{pmatrix}-\frac{\sqrt{3}}2u_1&\frac{\sqrt{3}}2u_2&0\\-\frac12u_1&-\frac12u_2&u_3\\u_1&u_2&u_3\end{pmatrix}\)
再根据O就可以计算出$u_1=u_2=u_3$
所以我们有
\(P=\begin{pmatrix}-\frac{\sqrt{3}}2&\frac{\sqrt{3}}2&0\\-\frac12&-\frac12&1\\1&1&1\end{pmatrix}\)
计算可知
\(P^{-1}=\begin{pmatrix}-\frac{\sqrt{3}}3&-\frac13&\frac13\\ \frac{\sqrt{3}}3&-\frac13&\frac13\\0&\frac23&\frac13\end{pmatrix}\)
于是对于面积坐标中任意一个二次曲线方程\(X'AX=0\),
可以转变为普通射影坐标下\(X^tP^tAPX=0\)
比如过B,C点不变曲线的面积坐标下方程为\(2dx^2+2dyz+2ezx+2exy=0\)
对应矩阵\(A=\begin{pmatrix}2d&e&e\\e&0&d\\e&d&0\end{pmatrix}\)
所以普通射影坐标下对应方程系数为\(P'AP=\begin{pmatrix}\frac d2-\frac{\sqrt{3}e}2&-\frac{5d}{2}&\frac d2-\sqrt{3}e\\-\frac{5d}{2}&\frac d2+\frac{\sqrt{3}e}2&\frac d2+\sqrt{3}e\\\frac d2-\sqrt{3}e&\frac d2+\sqrt{3}e&2d\end{pmatrix}\)
特别的，如果这条曲线还过不动点O,那么$d+e=0$,得出普通射影平面下方程系数矩阵为
\(P^tAP=\begin{pmatrix}\frac 12+\frac{\sqrt{3}}2&-\frac{5}{2}&\frac 12+\sqrt{3}\\-\frac{5}{2}&\frac 12-\frac{\sqrt{3}}2&\frac 12-\sqrt{3}\\\frac 12+\sqrt{3}&\frac 12-\sqrt{3}&2\end{pmatrix}\)
这应该是双曲线而不是椭圆。
而且上面满足过O(1,1,1)的曲线$X'AX=0$显然不过点(-1,1,1), 不知道哪里有问题。
那个曲线应该经过三个不动点而不是两个，很奇怪，不好理解

数学星空

我把mathe 提出的另一种三角形内点关于三边作对称点，然后取三个对称点的圆心变换（下面叫对称变换）也做了计算，结果如下：

\(a^5y0-2a^4x_{00}y0-2a^3b^2y0-2a^3c^2y0+2a^2b^2x_0y0+2a^2b^2x_{00}y0+2a^2c^2x_0y0+2a^2c^2x_{00}y0+ab^4y0-2ab^2c^2y0+ac^4y0-2b^4x_0y0+4b^2c^2x_0y0-2c^4x_0y0-4a^3sx_0+4a^2sx_0^2+8a^2sx_0x_{00}+4a^2sy0^2+4ab^2sx_0-4ac^2sx_0-8asx_0^2x_{00}-8asx_{00}y0^2-4b^2sx_0^2+4b^2sy0^2+4c^2sx_0^2-4c^2sy0^2=0\)

\(a^5x_0-a^4x_0^2-a^4y0^2-2a^4y0y_{00}-2a^3b^2x_0-2a^3c^2x_0+2a^2b^2x_0^2+2a^2b^2y0y_{00}+2a^2c^2x_0^2+2a^2c^2y0y_{00}+ab^4x_0-2ab^2c^2x_0+ac^4x_0-b^4x_0^2+b^4y0^2+2b^2c^2x_0^2-2b^2c^2y0^2-c^4x_0^2+c^4y0^2+4a^3sy0+8a^2sx_0y_{00}-4ab^2sy0+4ac^2sy0-8asx_0^2y_{00}-8asy0^2y_{00}+8b^2sx_0y0-8c^2sx_0y0=0\)

\(a^9k^2x_{00}-a^8k^2x_{00}^2-a^8k^2y_{00}^2-4a^7b^2k^2x_{00}+4a^6b^2k^2x_{00}^2+2a^6b^2k^2y_{00}^2+2a^6c^2k^2y_{00}^2+6a^5b^4k^2x_{00}-4a^5b^2c^2k^2x_{00}-2a^5c^4k^2x_{00}-6a^4b^4k^2x_{00}^2+4a^4b^2c^2k^2x_{00}^2+2a^4c^4k^2x_{00}^2-4a^3b^6k^2x_{00}+8a^3b^4c^2k^2x_{00}-4a^3b^2c^4k^2x_{00}+4a^2b^6k^2x_{00}^2-2a^2b^6k^2y_{00}^2-8a^2b^4c^2k^2x_{00}^2+2a^2b^4c^2k^2y_{00}^2+4a^2b^2c^4k^2x_{00}^2+2a^2b^2c^4k^2y_{00}^2-2a^2c^6k^2y_{00}^2+ab^8k^2x_{00}-4ab^6c^2k^2x_{00}+6ab^4c^4k^2x_{00}-4ab^2c^6k^2x_{00}+ac^8k^2x_{00}-b^8k^2x_{00}^2+b^8k^2y_{00}^2+4b^6c^2k^2x_{00}^2-4b^6c^2k^2y_{00}^2-6b^4c^4k^2x_{00}^2+6b^4c^4k^2y_{00}^2+4b^2c^6k^2x_{00}^2-4b^2c^6k^2y_{00}^2-c^8k^2x_{00}^2+c^8k^2y_{00}^2+2a^9ky_{00}+4a^8ktx_{00}-8a^7b^2ky_{00}-4a^7c^2ky_{00}+4a^7k^2sy_{00}-4a^7ktx_{00}^2-4a^7kty_{00}^2-12a^6b^2ktx_{00}+4a^6b^2kx_{00}y_{00}-4a^6c^2ktx_{00}+12a^5b^4ky_{00}-12a^5b^2k^2sy_{00}+12a^5b^2ktx_{00}^2+8a^5b^2kty_{00}^2+4a^5c^4ky_{00}-4a^5c^2k^2sy_{00}+4a^5c^2ktx_{00}^2+8a^5c^2kty_{00}^2+12a^4b^4ktx_{00}-12a^4b^4kx_{00}y_{00}-8a^4b^2c^2ktx_{00}+8a^4b^2k^2sx_{00}y_{00}-4a^4c^4ktx_{00}-4a^4c^4kx_{00}y_{00}+8a^4c^2k^2sx_{00}y_{00}-8a^3b^6ky_{00}+12a^3b^4c^2ky_{00}+12a^3b^4k^2sy_{00}-12a^3b^4ktx_{00}^2-4a^3b^4kty_{00}^2-8a^3b^2c^2k^2sy_{00}+8a^3b^2c^2ktx_{00}^2+8a^3b^2c^2kty_{00}^2-4a^3c^6ky_{00}-4a^3c^4k^2sy_{00}+4a^3c^4ktx_{00}^2-4a^3c^4kty_{00}^2-4a^2b^6ktx_{00}+12a^2b^6kx_{00}y_{00}+12a^2b^4c^2ktx_{00}-16a^2b^4c^2kx_{00}y_{00}-16a^2b^4k^2sx_{00}y_{00}-12a^2b^2c^4ktx_{00}-4a^2b^2c^4kx_{00}y_{00}+16a^2b^2c^2k^2sx_{00}y_{00}+4a^2c^6ktx_{00}+8a^2c^6kx_{00}y_{00}+2ab^8ky_{00}-8ab^6c^2ky_{00}-4ab^6k^2sy_{00}+4ab^6ktx_{00}^2+12ab^4c^4ky_{00}+12ab^4c^2k^2sy_{00}-12ab^4c^2ktx_{00}^2-8ab^2c^6ky_{00}-12ab^2c^4k^2sy_{00}+12ab^2c^4ktx_{00}^2+2ac^8ky_{00}+4ac^6k^2sy_{00}-4ac^6ktx_{00}^2-4b^8kx_{00}y_{00}+16b^6c^2kx_{00}y_{00}+8b^6k^2sx_{00}y_{00}-24b^4c^4kx_{00}y_{00}-24b^4c^2k^2sx_{00}y_{00}+16b^2c^6kx_{00}y_{00}+24b^2c^4k^2sx_{00}y_{00}-4c^8kx_{00}y_{00}-8c^6k^2sx_{00}y_{00}-a^9x_{00}+4a^8ty_{00}+a^8x_{00}^2+a^8y_{00}^2+4a^7b^2x_{00}+4a^7c^2x_{00}-8a^7ksx_{00}+4a^7t^2x_{00}-12a^6b^2ty_{00}-4a^6b^2x_{00}^2-2a^6b^2y_{00}^2-8a^6c^2ty_{00}-4a^6c^2x_{00}^2-2a^6c^2y_{00}^2+16a^6ksty_{00}+8a^6ksx_{00}^2+8a^6ksy_{00}^2-4a^6t^2x_{00}^2-4a^6t^2y_{00}^2-6a^5b^4x_{00}-4a^5b^2c^2x_{00}+24a^5b^2ksx_{00}-8a^5b^2t^2x_{00}+4a^5b^2tx_{00}y_{00}-6a^5c^4x_{00}+8a^5c^2ksx_{00}-8a^5c^2t^2x_{00}-4a^5c^2tx_{00}y_{00}+12a^4b^4ty_{00}+6a^4b^4x_{00}^2+4a^4b^2c^2x_{00}^2-32a^4b^2ksty_{00}-24a^4b^2ksx_{00}^2-8a^4b^2ksy_{00}^2+8a^4b^2t^2x_{00}^2+8a^4b^2t^2y_{00}^2+4a^4c^4ty_{00}+6a^4c^4x_{00}^2-16a^4c^2ksty_{00}-8a^4c^2ksx_{00}^2-8a^4c^2ksy_{00}^2+8a^4c^2t^2x_{00}^2+8a^4c^2t^2y_{00}^2+4a^3b^6x_{00}-4a^3b^4c^2x_{00}-24a^3b^4ksx_{00}+4a^3b^4t^2x_{00}-8a^3b^4tx_{00}y_{00}-4a^3b^2c^4x_{00}+16a^3b^2c^2ksx_{00}-8a^3b^2c^2t^2x_{00}+16a^3b^2kstx_{00}y_{00}+4a^3c^6x_{00}+8a^3c^4ksx_{00}+4a^3c^4t^2x_{00}+8a^3c^4tx_{00}y_{00}+16a^3c^2kstx_{00}y_{00}-4a^2b^6ty_{00}-4a^2b^6x_{00}^2+2a^2b^6y_{00}^2+8a^2b^4c^2ty_{00}+4a^2b^4c^2x_{00}^2-2a^2b^4c^2y_{00}^2+16a^2b^4ksty_{00}+24a^2b^4ksx_{00}^2-8a^2b^4ksy_{00}^2-4a^2b^4t^2x_{00}^2-4a^2b^4t^2y_{00}^2-4a^2b^2c^4ty_{00}+4a^2b^2c^4x_{00}^2-2a^2b^2c^4y_{00}^2-16a^2b^2c^2ksty_{00}-16a^2b^2c^2ksx_{00}^2+8a^2b^2c^2t^2x_{00}^2+8a^2b^2c^2t^2y_{00}^2-4a^2c^6x_{00}^2+2a^2c^6y_{00}^2-8a^2c^4ksx_{00}^2+8a^2c^4ksy_{00}^2-4a^2c^4t^2x_{00}^2-4a^2c^4t^2y_{00}^2-ab^8x_{00}+4ab^6c^2x_{00}+8ab^6ksx_{00}+4ab^6tx_{00}y_{00}-6ab^4c^4x_{00}-24ab^4c^2ksx_{00}-12ab^4c^2tx_{00}y_{00}-16ab^4kstx_{00}y_{00}+4ab^2c^6x_{00}+24ab^2c^4ksx_{00}+12ab^2c^4tx_{00}y_{00}+32ab^2c^2kstx_{00}y_{00}-ac^8x_{00}-8ac^6ksx_{00}-4ac^6tx_{00}y_{00}-16ac^4kstx_{00}y_{00}+b^8x_{00}^2-b^8y_{00}^2-4b^6c^2x_{00}^2+4b^6c^2y_{00}^2-8b^6ksx_{00}^2+8b^6ksy_{00}^2+6b^4c^4x_{00}^2-6b^4c^4y_{00}^2+24b^4c^2ksx_{00}^2-24b^4c^2ksy_{00}^2-4b^2c^6x_{00}^2+4b^2c^6y_{00}^2-24b^2c^4ksx_{00}^2+24b^2c^4ksy_{00}^2+c^8x_{00}^2-c^8y_{00}^2+8c^6ksx_{00}^2-8c^6ksy_{00}^2-4a^7sy_{00}-16a^6stx_{00}+12a^5b^2sy_{00}+4a^5c^2sy_{00}+16a^5st^2y_{00}+16a^5stx_{00}^2+16a^5sty_{00}^2+32a^4b^2stx_{00}-8a^4b^2sx_{00}y_{00}+32a^4c^2stx_{00}+8a^4c^2sx_{00}y_{00}-12a^3b^4sy_{00}+8a^3b^2c^2sy_{00}-16a^3b^2st^2y_{00}-32a^3b^2stx_{00}^2-16a^3b^2sty_{00}^2+4a^3c^4sy_{00}-16a^3c^2st^2y_{00}-32a^3c^2stx_{00}^2-16a^3c^2sty_{00}^2-16a^2b^4stx_{00}+16a^2b^4sx_{00}y_{00}+32a^2b^2c^2stx_{00}-16a^2c^4stx_{00}-16a^2c^4sx_{00}y_{00}+4ab^6sy_{00}-12ab^4c^2sy_{00}+16ab^4stx_{00}^2+12ab^2c^4sy_{00}-32ab^2c^2stx_{00}^2-4ac^6sy_{00}+16ac^4stx_{00}^2-8b^6sx_{00}y_{00}+24b^4c^2sx_{00}y_{00}-24b^2c^4sx_{00}y_{00}+8c^6sx_{00}y_{00}=0\)

取\(a = 3, b = 4, c = 5, s = 6\)

分别得到两种变换：

1.内点与各边中点的对称点构成的变换（称交点变换）

\(x= \frac{10368x_0^2-31104x_0}{6(576x_0^2-432x_0y0+324y0^2-1728x_0)}, y = -\frac{4(-576x_0^2+432x_0y0+1728x_0-1296y0)}{576x_0^2-432x_0y0+324y0^2-1728x_0}\)

2.内点与各边对称点构成圆心的变换（称对称变换）

\(x_{00} = \frac{3x_0^2+4x_0y0-9x_0-12y0}{x_0^2+y0^2-3x_0-4y0}, y_{00} = \frac{4x_0^2-3x_0y0-12x_0+9y0}{x_0^2+y0^2-3x_0-4y0)}\)

若内点沿直线\(y=kx+t\)运动，则

1.交点变换的曲线方程

\(-248832kx^2-82944tx^2+62208txy-46656ty^2+746496kx+248832tx+331776x^2-248832xy-995328x+746496y=0\)

2.对称变换的曲线方程

\(3kx_{00}^2+4kx_{00}y_{00}+tx_{00}^2+ty_{00}^2-9kx_{00}-12ky_{00}-3tx_{00}-4ty_{00}-4x_{00}^2+3x_{00}y_{00}+12x_{00}-9y_{00}=0\)

进一步取\(k=\frac{1}{2},t=\frac{1}{2}\)

\(-4x_{00}^2+10x_{00}y_{00}+y_{00}^2+12x_{00}-34y_{00}=0\)

\(165888x^2-217728xy-23328y^2-497664x+746496y=0\)

画图得到：



红色为交点变换曲线，蓝色为对称变换曲线

hujunhua

三次不变曲线还有如下一种类型\[
ax(y^2-z^2)+by(z^2-x^2)+cz(x^2-y^2)=0\\
a(\frac yz-\frac zy)+b(\frac zx-\frac xz)+c(\frac xy-\frac yx)=0
\]这一类型的曲线除了过3个顶点，还通过 4 个不动点。
比如取`a=b=c`的不变曲线\[
x(y^2-z^2)+y(z^2-x^2)+z(x^2-y^2)=0
\]左边可分解为\[
(y-z)(z-x)(x-y)=0
\]这是三条不变直线，不动点(1,1,1)是它的奇点（三直线的交点）。
再如取`a=1,b=2,c=3`时，曲线在XOY平面上的投影如图，是非退化的。