绘图

Sage 可以生成二维和三维图形。

二维图形

在二维中,Sage 可以绘制圆、线和多边形;在直角坐标系中绘制函数图形; 还可以绘制极坐标图、轮廓图和矢量场图。本文档展示了若干这些图形的例子。 有关使用 Sage 绘图的更多例子,请参见 求解微分方程Maxima, 以及 Sage Constructions 文档。

该命令生成一个位于原点的半径为 1 的黄色圆:

sage: circle((0,0), 1, rgbcolor=(1,1,0))
Graphics object consisting of 1 graphics primitive
>>> from sage.all import *
>>> circle((Integer(0),Integer(0)), Integer(1), rgbcolor=(Integer(1),Integer(1),Integer(0)))
Graphics object consisting of 1 graphics primitive

你还可以生成一个填充的圆:

sage: circle((0,0), 1, rgbcolor=(1,1,0), fill=True)
Graphics object consisting of 1 graphics primitive
>>> from sage.all import *
>>> circle((Integer(0),Integer(0)), Integer(1), rgbcolor=(Integer(1),Integer(1),Integer(0)), fill=True)
Graphics object consisting of 1 graphics primitive

你还可以通过将圆赋值给变量来创建圆;这样做不会将圆绘制出来:

sage: c = circle((0,0), 1, rgbcolor=(1,1,0))
>>> from sage.all import *
>>> c = circle((Integer(0),Integer(0)), Integer(1), rgbcolor=(Integer(1),Integer(1),Integer(0)))

要想绘制它,可以使用 c.show()show(c),如下所示:

sage: c.show()
>>> from sage.all import *
>>> c.show()

或者,使用 c.save('filename.png') 将绘图保存到给定文件。

现在,这些“圆”看起来更像椭圆,因为坐标轴的比例不同。你可以这样修复这个问题:

sage: c.show(aspect_ratio=1)
>>> from sage.all import *
>>> c.show(aspect_ratio=Integer(1))

命令 show(c, aspect_ratio=1) 可以完成同样的事情, 或者你可以使用 c.save('filename.png', aspect_ratio=1) 保存图片。

绘制基本函数很容易:

sage: plot(cos, (-5,5))
Graphics object consisting of 1 graphics primitive
>>> from sage.all import *
>>> plot(cos, (-Integer(5),Integer(5)))
Graphics object consisting of 1 graphics primitive

一旦你指定了变量名称,你还可以创建参数化图形:

sage: x = var('x')
sage: parametric_plot((cos(x),sin(x)^3),(x,0,2*pi),rgbcolor=hue(0.6))
Graphics object consisting of 1 graphics primitive
>>> from sage.all import *
>>> x = var('x')
>>> parametric_plot((cos(x),sin(x)**Integer(3)),(x,Integer(0),Integer(2)*pi),rgbcolor=hue(RealNumber('0.6')))
Graphics object consisting of 1 graphics primitive

一定要注意,只有当原点在图形的视图范围内时,图形的轴才会相交,并且对于非常大的数值可能会使用科学计数法:

sage: plot(x^2,(x,300,500))
Graphics object consisting of 1 graphics primitive
>>> from sage.all import *
>>> plot(x**Integer(2),(x,Integer(300),Integer(500)))
Graphics object consisting of 1 graphics primitive

你可以通过将多个图形相加来将他们组合在一起:

sage: x = var('x')
sage: p1 = parametric_plot((cos(x),sin(x)),(x,0,2*pi),rgbcolor=hue(0.2))
sage: p2 = parametric_plot((cos(x),sin(x)^2),(x,0,2*pi),rgbcolor=hue(0.4))
sage: p3 = parametric_plot((cos(x),sin(x)^3),(x,0,2*pi),rgbcolor=hue(0.6))
sage: show(p1+p2+p3, axes=false)
>>> from sage.all import *
>>> x = var('x')
>>> p1 = parametric_plot((cos(x),sin(x)),(x,Integer(0),Integer(2)*pi),rgbcolor=hue(RealNumber('0.2')))
>>> p2 = parametric_plot((cos(x),sin(x)**Integer(2)),(x,Integer(0),Integer(2)*pi),rgbcolor=hue(RealNumber('0.4')))
>>> p3 = parametric_plot((cos(x),sin(x)**Integer(3)),(x,Integer(0),Integer(2)*pi),rgbcolor=hue(RealNumber('0.6')))
>>> show(p1+p2+p3, axes=false)

生成填充形状的一个好方法是生成点列表(示例中的 L), 然后使用 polygon 命令绘制由这些点构成边界的形状。 例如,下面是一个绿色的三角形:

sage: L = [[-1+cos(pi*i/100)*(1+cos(pi*i/100)),
....:     2*sin(pi*i/100)*(1-cos(pi*i/100))] for i in range(200)]
sage: p = polygon(L, rgbcolor=(1/8,3/4,1/2))
sage: p
Graphics object consisting of 1 graphics primitive
>>> from sage.all import *
>>> L = [[-Integer(1)+cos(pi*i/Integer(100))*(Integer(1)+cos(pi*i/Integer(100))),
...     Integer(2)*sin(pi*i/Integer(100))*(Integer(1)-cos(pi*i/Integer(100)))] for i in range(Integer(200))]
>>> p = polygon(L, rgbcolor=(Integer(1)/Integer(8),Integer(3)/Integer(4),Integer(1)/Integer(2)))
>>> p
Graphics object consisting of 1 graphics primitive

输入 show(p, axes=false) 来查看没有任何坐标轴的图形。

你可以向图形中添加文本:

sage: L = [[6*cos(pi*i/100)+5*cos((6/2)*pi*i/100),
....:     6*sin(pi*i/100)-5*sin((6/2)*pi*i/100)] for i in range(200)]
sage: p = polygon(L, rgbcolor=(1/8,1/4,1/2))
sage: t = text("hypotrochoid", (5,4), rgbcolor=(1,0,0))
sage: show(p+t)
>>> from sage.all import *
>>> L = [[Integer(6)*cos(pi*i/Integer(100))+Integer(5)*cos((Integer(6)/Integer(2))*pi*i/Integer(100)),
...     Integer(6)*sin(pi*i/Integer(100))-Integer(5)*sin((Integer(6)/Integer(2))*pi*i/Integer(100))] for i in range(Integer(200))]
>>> p = polygon(L, rgbcolor=(Integer(1)/Integer(8),Integer(1)/Integer(4),Integer(1)/Integer(2)))
>>> t = text("hypotrochoid", (Integer(5),Integer(4)), rgbcolor=(Integer(1),Integer(0),Integer(0)))
>>> show(p+t)

微积分老师经常在黑板上绘制以下图形: arcsin 的多个周期: 即 \(y=\sin(x)\) 对于 \(x\)\(-2\pi\)\(2\pi\) 区间的图像, 围绕 45 度线翻转。以下 Sage 命令构造此图形:

sage: v = [(sin(x),x) for x in srange(-2*float(pi),2*float(pi),0.1)]
sage: line(v)
Graphics object consisting of 1 graphics primitive
>>> from sage.all import *
>>> v = [(sin(x),x) for x in srange(-Integer(2)*float(pi),Integer(2)*float(pi),RealNumber('0.1'))]
>>> line(v)
Graphics object consisting of 1 graphics primitive

由于正切函数的值域比正弦函数大得多,如果你使用相同技巧绘制反正切的图像,你应该更改 x 轴的最大和最小坐标:

sage: v = [(tan(x),x) for x in srange(-2*float(pi),2*float(pi),0.01)]
sage: show(line(v), xmin=-20, xmax=20)
>>> from sage.all import *
>>> v = [(tan(x),x) for x in srange(-Integer(2)*float(pi),Integer(2)*float(pi),RealNumber('0.01'))]
>>> show(line(v), xmin=-Integer(20), xmax=Integer(20))

Sage 还能计算极坐标图、轮廓图和矢量场图(针对特殊类型的函数)。这里是一个轮廓图的例子:

sage: f = lambda x,y: cos(x*y)
sage: contour_plot(f, (-4, 4), (-4, 4))
Graphics object consisting of 1 graphics primitive
>>> from sage.all import *
>>> f = lambda x,y: cos(x*y)
>>> contour_plot(f, (-Integer(4), Integer(4)), (-Integer(4), Integer(4)))
Graphics object consisting of 1 graphics primitive

三维图形

Sage 还可以用于创建三维图形。 在 notebook 和 REPL 中,这些图形将默认使用开源软件包 [ThreeJS] 显示, 该软件包支持使用鼠标交互式旋转和缩放图形。

使用 plot3d 绘制形如 \(f(x, y) = z\) 的函数图像:

sage: x, y = var('x,y')
sage: plot3d(x^2 + y^2, (x,-2,2), (y,-2,2))
Graphics3d Object
>>> from sage.all import *
>>> x, y = var('x,y')
>>> plot3d(x**Integer(2) + y**Integer(2), (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)))
Graphics3d Object

或者,你可以使用 parametric_plot3d 绘制参数曲面, 其中每个 \(x, y, z\) 由一个或两个变量(通常是 \(u\)\(v\))的函数确定。 前面的图形可以参数化地表达如下:

sage: u, v = var('u, v')
sage: f_x(u, v) = u
sage: f_y(u, v) = v
sage: f_z(u, v) = u^2 + v^2
sage: parametric_plot3d([f_x, f_y, f_z], (u, -2, 2), (v, -2, 2))
Graphics3d Object
>>> from sage.all import *
>>> u, v = var('u, v')
>>> __tmp__=var("u,v"); f_x = symbolic_expression(u).function(u,v)
>>> __tmp__=var("u,v"); f_y = symbolic_expression(v).function(u,v)
>>> __tmp__=var("u,v"); f_z = symbolic_expression(u**Integer(2) + v**Integer(2)).function(u,v)
>>> parametric_plot3d([f_x, f_y, f_z], (u, -Integer(2), Integer(2)), (v, -Integer(2), Integer(2)))
Graphics3d Object

在 Sage 中绘制 3D 曲面的第三种方法是 \(implicit_plot3d`\), 它绘制形如 \(f(x, y, z) = 0\) 的函数的轮廓(这定义了一组点)。 我们使用经典公式绘制一个球体:

sage: x, y, z = var('x, y, z')
sage: implicit_plot3d(x^2 + y^2 + z^2 - 4, (x,-2, 2), (y,-2, 2), (z,-2, 2))
Graphics3d Object
>>> from sage.all import *
>>> x, y, z = var('x, y, z')
>>> implicit_plot3d(x**Integer(2) + y**Integer(2) + z**Integer(2) - Integer(4), (x,-Integer(2), Integer(2)), (y,-Integer(2), Integer(2)), (z,-Integer(2), Integer(2)))
Graphics3d Object

下面是更多的例子:

Yellow Whitney's umbrella:

sage: u, v = var('u,v')
sage: fx = u*v
sage: fy = u
sage: fz = v^2
sage: parametric_plot3d([fx, fy, fz], (u, -1, 1), (v, -1, 1),
....:   frame=False, color="yellow")
Graphics3d Object
>>> from sage.all import *
>>> u, v = var('u,v')
>>> fx = u*v
>>> fy = u
>>> fz = v**Integer(2)
>>> parametric_plot3d([fx, fy, fz], (u, -Integer(1), Integer(1)), (v, -Integer(1), Integer(1)),
...   frame=False, color="yellow")
Graphics3d Object

Cross cap:

sage: u, v = var('u,v')
sage: fx = (1+cos(v))*cos(u)
sage: fy = (1+cos(v))*sin(u)
sage: fz = -tanh((2/3)*(u-pi))*sin(v)
sage: parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi),
....:   frame=False, color="red")
Graphics3d Object
>>> from sage.all import *
>>> u, v = var('u,v')
>>> fx = (Integer(1)+cos(v))*cos(u)
>>> fy = (Integer(1)+cos(v))*sin(u)
>>> fz = -tanh((Integer(2)/Integer(3))*(u-pi))*sin(v)
>>> parametric_plot3d([fx, fy, fz], (u, Integer(0), Integer(2)*pi), (v, Integer(0), Integer(2)*pi),
...   frame=False, color="red")
Graphics3d Object

挠环面:

sage: u, v = var('u,v')
sage: fx = (3+sin(v)+cos(u))*cos(2*v)
sage: fy = (3+sin(v)+cos(u))*sin(2*v)
sage: fz = sin(u)+2*cos(v)
sage: parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi),
....:   frame=False, color="red")
Graphics3d Object
>>> from sage.all import *
>>> u, v = var('u,v')
>>> fx = (Integer(3)+sin(v)+cos(u))*cos(Integer(2)*v)
>>> fy = (Integer(3)+sin(v)+cos(u))*sin(Integer(2)*v)
>>> fz = sin(u)+Integer(2)*cos(v)
>>> parametric_plot3d([fx, fy, fz], (u, Integer(0), Integer(2)*pi), (v, Integer(0), Integer(2)*pi),
...   frame=False, color="red")
Graphics3d Object

双纽线:

sage: x, y, z = var('x,y,z')
sage: f(x, y, z) = 4*x^2 * (x^2 + y^2 + z^2 + z) + y^2 * (y^2 + z^2 - 1)
sage: implicit_plot3d(f, (x, -0.5, 0.5), (y, -1, 1), (z, -1, 1))
Graphics3d Object
>>> from sage.all import *
>>> x, y, z = var('x,y,z')
>>> __tmp__=var("x,y,z"); f = symbolic_expression(Integer(4)*x**Integer(2) * (x**Integer(2) + y**Integer(2) + z**Integer(2) + z) + y**Integer(2) * (y**Integer(2) + z**Integer(2) - Integer(1))).function(x,y,z)
>>> implicit_plot3d(f, (x, -RealNumber('0.5'), RealNumber('0.5')), (y, -Integer(1), Integer(1)), (z, -Integer(1), Integer(1)))
Graphics3d Object