理工系の数学 B の授業で，楕円の周長や面積，回転楕円体の表面積や体積を求めているので。
塗りつぶした楕円
Maxima で長半径 $a$，単半径 $b$ の楕円を描く例。ellipse() を使ってみる。
? ellipse;
-- Graphic object: ellipse (<xc>, <yc>, <a>, <b>, <ang1>, <ang2>)
Draws ellipses and circles in 2D.
2D
'ellipse (<xc>, <yc>, <a>, <b>, <ang1>, <ang2>)' plots an ellipse
centered at '[<xc>, <yc>]' with horizontal and vertical semi axis
<a> and <b>, respectively, starting at angle <ang1> with an
amplitude equal to angle <ang2>.In [1]:
a: 2$
b: 1$
xlim: a+0.5$
ylim: b+0.5$
draw2d(
font_size = 14, dimensions=[640, 400],
/* 縦横比 */
proportional_axes = xy,
/* 表示範囲 */
xrange = [-xlim, xlim], yrange = [-ylim, ylim],
/* x 軸 y 軸の表示 */
xaxis = true, yaxis = true,
xtics = {["-a", -a], ["a", a]},
ytics = {["-b", -b], ["b", b]},
user_preamble = "set grid front;",
transparent = false, fill_color = yellow,
line_width = 3,
color = dark-blue,
ellipse(0, 0, a, b, 0, 360)
)$
塗りつぶし無しの楕円
In [2]:
a: 2$
b: 1$
draw2d(
font_size = 14, dimensions=[640, 400],
/* 縦横比 */
proportional_axes = xy,
/* 表示範囲 */
xrange = [-xlim, xlim], yrange = [-ylim, ylim],
/* x 軸 y 軸の表示 */
xaxis = true, yaxis = true,
xtics = {["-a", -a], ["a", a]},
ytics = {["-b", -b], ["b", b]},
user_preamble = "set grid front;",
transparent = true,
line_width = 3,
color = dark-blue,
ellipse(0, 0, a, b, 0, 360)
)$
長軸の周りに回転した回転楕円体
$\displaystyle y = b \sqrt{1-\frac{x^2}{a^2}} $ を $x$ の周りに回転してできる回転楕円体の表面。
In [3]:
draw2d(
font_size = 14, dimensions=[640, 400],
/* 縦横比 */
proportional_axes = xy,
/* 表示範囲 */
xrange = [-xlim, xlim], yrange = [-ylim, ylim],
/* x 軸 y 軸の表示 */
xaxis = true, yaxis = true,
xtics = {["-a", -a], ["a", a]},
ytics = {["-b", -b], ["b", b]}, user_preamble = "set grid front;",
transparent = false,
fill_color = light-blue,
line_width = 3,
color = dark-blue,
/* 上半分の楕円を描く */
ellipse(0, 0, a, b, 0, 180),
color = blue, line_width = 2,
explicit(0, x, -xlim, xlim)
)$
In [4]:
f(x):= b*sqrt(1-x**2/a**2)$
a: 2$
b: 1$
xlim: a+0.5$
ylim: b+0.5$
draw3d(
dimensions=[640, 640], proportional_axes = xyz,
xrange = [-xlim, xlim],
yrange = [-xlim, xlim],
zrange = [-xlim, xlim],
/* 回転体の表面 */
color = blue,
line_width=0.5, xu_grid = 40, yv_grid = 30,
parametric_surface(x, f(x)*cos(t), f(x)*sin(t),
x, -a, a, t, 0, 2*%pi),
surface_hide =true, view=[60, 10],
axis_3d = false,
/* x 軸 */
color=blue, line_width = 2,
parametric(x, 0, 0, x, -2.2, 2.2),
/* いわゆる「y = y(x)」，実は z = z(x) */
nticks=100, color=dark-blue, line_width = 3,
parametric(a*cos(t), 0, b*sin(t), t, 0, %pi)
)$
短軸の周りに回転した回転楕円体
$\displaystyle x = a \sqrt{1-\frac{y^2}{b^2}} $ を $y$ の周りに回転してできる回転楕円体の表面。
In [5]:
draw2d(
font_size = 14, dimensions=[640, 400],
/* 縦横比 */
proportional_axes = xy,
/* 表示範囲 */
xrange = [-xlim, xlim], yrange = [-ylim, ylim],
/* x 軸 y 軸の表示 */
xaxis = true, yaxis = true,
xtics = {["-a", -a], ["a", a]},
ytics = {["-b", -b], ["b", b]},
user_preamble = "set grid front;",
transparent = false,
fill_color = light-pink,
line_width = 3,
color = dark-red,
/* 右半分の楕円を描く */
ellipse(0, 0, a, b, 90, -180),
color = red, line_width = 2,
parametric(0, y, y, -ylim, ylim)
)$
In [6]:
g(z):= a*sqrt(1-z**2/b**2)$
draw3d(
dimensions=[640, 640], proportional_axes = xyz,
xrange = [-a, a],
yrange = [-a, a],
zrange = [-a, a],
/* 回転体の表面 */
color = red,
line_width=0.5, xu_grid = 40, yv_grid = 30,
parametric_surface(g(z)*cos(t), g(z)*sin(t), z,
z, -b, b, t, 0, 2*%pi),
surface_hide=true, view=[70, 20],
axis_3d = false,
/* いわゆる「y 軸」，実は z 軸 */
color=red, line_width = 2,
parametric(0, 0, z, z, -xlim, xlim),
/* いわゆる「x = x(y)」，実は x = x(z) */
nticks=100, color=dark-red, line_width = 3,
parametric(a*cos(t), 0, b*sin(t), t, -%pi/2, %pi/2)
)$
