一类三次旋转曲面的参数化

青青子衿

\begin{align*}
\color{black}{
\left\{
\begin{split}
x&=\dfrac{u^2}{3}+\dfrac{1}{3u}\cdot\dfrac{v^2-3}{v^2+3}+\dfrac{1}{u}\cdot\dfrac{2v}{v^2+3}\\
y&=\dfrac{u^2}{3}+\dfrac{1}{3u}\cdot\dfrac{v^2-3}{v^2+3}-\dfrac{1}{u}\cdot\dfrac{2v}{v^2+3}\\
z&=\dfrac{u^2}{3}-\dfrac{2}{3u}\cdot\dfrac{v^2-3}{v^2+3}
\end{split}\right.}
\end{align*}
\[\large\color{black}{x^3+y^3+z^3-3xyz=1}\]
.

1. X = u^2/3 + 1/(3 u)*(v^2 - 3)/(v^2 + 3) + 1/u*2 v/(v^2 + 3)
2. Y = u^2/3 + 1/(3 u)*(v^2 - 3)/(v^2 + 3) - 1/u*2 v/(v^2 + 3)
3. Z = u^2/3 - 2/(3 u)*(v^2 - 3)/(v^2 + 3)
4. X^3 + Y^3 + Z^3 - 3 X*Y*Z // FullSimplify

复制代码

.
A Cubic Surface of Revolution      Mark B. Villarino
Article (PDF Available) in The Mathematical Gazette
98(542) · January 2013 with 56 Reads
DOI: 10.1017/S0025557200001327 · Source: arXiv
researchgate.net/publication/234018008_A_Cubic_Surface_of_Revolution

青青子衿

回复 1# 青青子衿
还有一条曲线在该曲面上
\[ \left(\sum\limits_{k=0}^{+\infty}\dfrac{u^{3k}}{\left(3k\right)!}\right)^3+\left(\sum\limits_{k=0}^{+\infty}\dfrac{u^{3k+1}}{\left(3k+1\right)!}\right)^3+\left(\sum\limits_{k=0}^{+\infty}\dfrac{u^{3k+2}}{\left(3k+2\right)!}\right)^3-3\left(\sum\limits_{k=0}^{+\infty}\dfrac{u^{3k}}{\left(3k\right)!}\right)\left(\sum\limits_{k=0}^{+\infty}\dfrac{u^{3k+1}}{\left(3k+1\right)!}\right)\left(\sum\limits_{k=0}^{+\infty}\dfrac{u^{3k+2}}{\left(3k+2\right)!}\right)=1 \]

hbghlyj

Asymptote HTML format(关于surfacepen参考TeX.SE)
import graph3;
import palette;

size(8cm);

currentprojection = perspective(3,6,2);

triple parametricSurface(pair p) {
  real u=p.x, v=p.y;
  real X = (u^(2/3)) + ((v^2-3)/(3*u*(v^2+3))) + ((2*v)/(u*(v^2+3)));
  real Y = (u^(2/3)) + ((v^2-3)/(3*u*(v^2+3))) - ((2*v)/(u*(v^2+3)));
  real Z = (u^(2/3)) - ((2*(v^2-3))/(3*u*(v^2+3)));
  return (X,Y,Z);
}
currentlight = light(white,(5,5,5));
surface s = surface(parametricSurface, (1,-5), (4,5), nu=15, nv=15, Spline);

s.colors(palette(s.map(zpart),Rainbow()));

draw(s);

xaxis3(Label("$x$"),red,Arrow3);
yaxis3(Label("$y$"),red,Arrow3);
zaxis3(Label("$z$"),red,Arrow3);

hbghlyj

hbghlyj 发表于 2023-2-18 22:32
Asymptote HTML format(关于surfacepen参考TeX.SE)

Asymptote 为什么x轴 缺少了一段?

青青子衿

青青子衿 发表于 2019-8-14 10:21
回复 1# 青青子衿
还有一条曲线在该曲面上

虚数解を持たない３次方程式とオイラーの公式の２変数化
高村　明¤
Cubic equations without imaginary solutions and Euler's formula with two variables
Akira TAKAMURA¤
jstage.jst.go.jp/article/toyotakosenkiyo/55/0/55_55-10/_pdf/-char/ja