Bijection of interval and $\mathbb R$

In [4]:
x=var('x')
plot(x/sqrt(1-abs(x)^2), (-1,1),  ymin=-2, ymax=2)
Out[4]:

Parametrized function

In [11]:
f = (lambda u,v: v*cos(u), lambda u,v: v*sin(u), lambda u,v: u)
parametric_plot3d(f, (0,2*pi), (-4,4))
Out[11]:
In [32]:
from sage.plot.plot3d.shapes import *
f = (lambda u,v: v*cos(u), lambda u,v: v*sin(u), lambda u,v: u)
g = (lambda u,v: 1+u, lambda u,v: 0+v, lambda u,v: v) 
h = (lambda u,v: -1+u, lambda u,v: v, lambda u,v: pi-v) 

parametric_plot3d(f, (-pi,2*pi), (-2,2)) + parametric_plot3d(g, (-1,1), (-1,1), color='green', opacity=0.7,) +Sphere(.05, color='red').translate(1,0,0)+ parametric_plot3d(h, (-1,1), (-1,1), color='green', opacity=0.7,) +Sphere(.05, color='red').translate(-1,0,pi)
Out[32]: