一类以封闭代数曲面为界区域的体积

青青子衿

\begin{align*}
\operatorname{Volume}\left[\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}\leqslant\dfrac{2x}{A}+\dfrac{2y}{B}+\dfrac{2z}{C}\right]
&=\dfrac{4\pi\,abc}{3}\sqrt{\left(\dfrac{a^2}{A^2}+\dfrac{b^2}{B^2}+\dfrac{c^2}{C^2}\right)^3}\\
\\
\operatorname{Volume}\left[\left(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}\right)^2\leqslant\sqrt{\dfrac{a^2}{A^2}+\dfrac{b^2}{B^2}+\dfrac{c^2}{C^2}}\,\bigg(\dfrac{z}{c}\bigg)\right]
&=\dfrac{\pi\,abc}{3}\sqrt{\dfrac{a^2}{A^2}+\dfrac{b^2}{B^2}+\dfrac{c^2}{C^2}}\\
\\
\operatorname{Volume}\left[\left(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}\right)^2\leqslant\dfrac{x}{A}+\dfrac{y}{B}+\dfrac{z}{C}\right]
&=\dfrac{\pi\,abc}{3}\sqrt{\dfrac{a^2}{A^2}+\dfrac{b^2}{B^2}+\dfrac{c^2}{C^2}}\\
\end{align*}

青青子衿

\begin{gather*}
\left\{
\begin{split}
\left(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}\right)^2
&\leqslant\dfrac{x}{A}+\dfrac{y}{B}+\dfrac{z}{C}\\
x&\geqslant0\\
y&\geqslant0\\
z&\geqslant0
\end{split}\right.\\
\\
V=\dfrac{abc}{60}\left(\left(\dfrac{a}{A}+\dfrac{b}{B}\right)\left(\dfrac{a^2}{A^2}+\dfrac{b^2}{B^2}\right)+\left(\dfrac{a^2}{A^2}+\dfrac{ab}{AB}+\dfrac{b^2}{B^2}\right)\dfrac{c}{C}+\left(\dfrac{a}{A}+\dfrac{b}{B}\right)\dfrac{c^2}{C^2}+\dfrac{c^3}{C^3}\right)\\
\\
V=\dfrac{abc}{60}\bigg(\left(\dfrac{a}{A}+\dfrac{b}{B}+\dfrac{c}{C}\right)^3-2\left(\dfrac{a}{A}+\dfrac{b}{B}+\dfrac{c}{C}\right)\left(\dfrac{a\,b}{AB}+\dfrac{b\,c}{BC}+\dfrac{a\,c}{AC}\right)+\dfrac{a\,b\,c}{ABC}\bigg)
\end{gather*}

hbghlyj

青青子衿 发表于 2019-6-9 06:21
\begin{gather*}
\left\{
\begin{split}
\left(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}\right)^2
&\leqslant\dfrac{x}{A}+\dfrac{y}{B}+\dfrac{z}{C}\\
x&\geqslant0\\
y&\geqslant0\\
z&\geqslant0
\end{split}\right.
\end{gather*}

Asymptote HTML format
import graph3;
real a=2,b=2,c=2.1,A=3,B=2,C=1.2;
currentprojection=orthographic(-1.7,1.2,0.5);
triple f(pair p) {
  real x=p.x, u=p.y,
  y = (b*B*(a*A*u*(c-C*u)-A*c*x+a*C*x))/(a*A*(B*c-b*C)),
  z = (c*C*(A*b*x-a*(A*u*(b-B*u)+B*x)))/(a*A*(B*c-b*C));
  return (x,y,z);
}

size3(8cm);

draw(surface(f, (0,-.7), (2,2)), paleyellow);
xaxis3("$x$");
yaxis3("$y$");
zaxis3("$z$");
其中, 曲面的参数是$(u,x)$
由$\cases{u=\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}\\u^2=\dfrac{x}{A}+\dfrac{y}{B}+\dfrac{z}{C}}$
解得

y = (b*B*(a*A*u*(c-C*u)-A*c*x+a*C*x))/(a*A*(B*c-b*C)),

z = (c*C*(A*b*x-a*(A*u*(b-B*u)+B*x)))/(a*A*(B*c-b*C))