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青青子衿 发表于 2019-7-29 15:55

一类三次旋转曲面的参数化

[i=s] 本帖最后由 青青子衿 于 2019-7-29 16:22 编辑 [/i]

\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*}
\[\large\color{black}{x^3+y^3+z^3-3xyz=1}\]
.[code]X = u^2/3 + 1/(3 u)*(v^2 - 3)/(v^2 + 3) + 1/u*2 v/(v^2 + 3)
Y = u^2/3 + 1/(3 u)*(v^2 - 3)/(v^2 + 3) - 1/u*2 v/(v^2 + 3)
Z = u^2/3 - 2/(3 u)*(v^2 - 3)/(v^2 + 3)
X^3 + Y^3 + Z^3 - 3 X*Y*Z // FullSimplify[/code].
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
[url]https://www.researchgate.net/publication/234018008_A_Cubic_Surface_of_Revolution[/url]

青青子衿 发表于 2019-8-14 10:21

[i=s] 本帖最后由 青青子衿 于 2019-10-21 22:49 编辑 [/i]

[b]回复 [url=http://kuing.orzweb.net/redirect.php?goto=findpost&pid=32888&ptid=6387]1#[/url] [i]青青子衿[/i] [/b]
还有一条曲线在该曲面上
\[ \left(\sum\limits_{k=0}^{+\infty}\dfrac{u^{3k}}{\left(3k\right)!}\right)^3+\left(\sum\limits_{k=0}^{+\infty}\dfrac{u^{3k+1}}{\left(3k+1\right)!}\right)^3+\left(\sum\limits_{k=0}^{+\infty}\dfrac{u^{3k+2}}{\left(3k+2\right)!}\right)^3-3\left(\sum\limits_{k=0}^{+\infty}\dfrac{u^{3k}}{\left(3k\right)!}\right)\left(\sum\limits_{k=0}^{+\infty}\dfrac{u^{3k+1}}{\left(3k+1\right)!}\right)\left(\sum\limits_{k=0}^{+\infty}\dfrac{u^{3k+2}}{\left(3k+2\right)!}\right)=1 \]

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