某有理参数旋转曲面的隐式方程

青青子衿

\begin{cases}  
\begin{split}      
x&=&-&\dfrac{(2s-1)(2t^3-3t^2+4)}{2(2 s^2-2s+1)}\\      
y&=&-&\dfrac{s(s-1)(2t^3-3t^2+4)}{2 s^2 - 2 s + 1}\\   
z&=&-&\dfrac{3(t^3-9t^2+18t-12)}{40}   
\end{split}  
\end{cases}
\begin{gather*}
324\bigg(x^2+y^2\bigg)\bigg(1600z^2+6900z+3x^2+3y^2+1197\bigg)^2\\
=\bigg(128000z^3-306000z^2+2160x^2z+2160y^2z-120960z-11745x^2-11745y^2-11124\bigg)^2
\end{gather*}
参见《计算机代数（第二版）》王东明;夏壁灿;李子明
[M]清华大学出版社 2007 P186  Grobner基的应用

orzweb111

回复 1# 青青子衿

What is this about?

hbghlyj

回复 2# orzweb111
应该是将参数$s,t$消去的算法

hbghlyj

Asymptote: HTML format
import palette;
import graph3;

triple f(pair st) {
  real s = st.x;
  real t = st.y;
  return (
    -(2s-1)*(2t^3-3t^2+4)/(2(2s^2-2s+1)),
    -s*(s-1)*(2t^3-3t^2+4)/(2s^2-2s+1),
    -3*(t^3-9t^2+18t-12)/40
  );
}

size(8cm);
currentprojection = perspective(6,3,6);

real xmin = -1, xmax = 1, ymin = -0.5, ymax = 0.5, zmin = -1, zmax = 1;
surface mySurface = surface(f, (xmin,ymin), (xmax,ymax),30,10);
mySurface.colors(palette(mySurface.map(zpart),Rainbow()));
draw(mySurface,render(merge=true));