some 3d space curves in SAGE

[David Joyner]

Hi:
Here are a few cool space curves.
I'd like to make more from
http://www.math.umd.edu/research/bianchi/Gifspacecurves/spcu.html
and
http://virtualmathmuseum.org/SpaceCurves/
If anyone knows of some parametric equations for them, I'd appreciate it.

#trefoil knot
#http://en.wikipedia.org/wiki/Trefoil_knot
sage: fx = 0.01*(41*cos(t) - 18*sin(t) - 83*cos(2*t)-83*sin(2*t) -
11*cos(3*t) + 27*sin(3*t))
sage: fy = 0.01*(36*cos(t) + 27*sin(t) - 113*cos(2*t)+30*sin(2*t) +
11*cos(3*t) - 27*sin(3*t))
sage: fz = 0.01*(45*sin(t) - 30*cos(2*t) + 113*sin(2*t)-11*cos(3*t) +
27*sin(3*t))
sage: parametric_plot3d( (fx, fy, fz), (t, 0, 6*pi), plot_points =
500, frame=False)

#figure-eight knot
#http://en.wikipedia.org/wiki/Figure-eight_knot_%28mathematics%29
sage: fx = (2 + cos(2*t))*cos(3*t)
sage: fy = (2 + cos(2*t))*sin(3*t)
sage: fz = sin(4*t)
sage: parametric_plot3d( (fx, fy, fz), (t, 0, 6*pi), plot_points =
500, frame=False)

#Cinquefoil Knot (a=2 is this one):
#http://en.wikipedia.org/wiki/Skein_relation
#http://virtualmathmuseum.org/SpaceCurves/index.html#Classic
fx = cos(t)*(2 - cos(2a*t/(2*a + 1)))
fy = sin(t)*(2 - cos(2*t/(2*a + 1)))
fz = - sin(2*t/(2*a + 1))
sage: parametric_plot3d( (fx, fy, fz), (t, 0, 20*pi), plot_points =
500, frame=False)

#3-leafed space curve
sage: a = 1; b = 1; c = 1; d = 3; e = 2
sage: fx = (a + b*cos(d*t))*cos(e*t)
sage: fy = (a + b*cos(d*t))*sin(e*t)
sage: fz = c*sin(d*t)
sage: parametric_plot3d( (fx, fy, fz), (t, 0, 4*pi), plot_points =
100, frame=False)

# The Viviani Curve
#http://en.wikipedia.org/wiki/Viviani's_curve
sage: a = 1
sage: fx = a*2*sin(t/2)
sage: fy = a*sin(t)
sage: fz = a*(1 + cos(t))
sage: parametric_plot3d( (fx, fy, fz), (t, 0, 4*pi), plot_points =
100, frame=False)

#loxodrome
#http://en.wikipedia.org/wiki/Loxodrome
sage: fx = sin(t)/sqrt(1+t^2)
sage: fy = cos(t)/sqrt(1+t^2)
sage: fz = t/sqrt(1+t^2)
sage: parametric_plot3d( (fx, fy, fz), (t, -4*pi, 4*pi), plot_points =
100, frame=False)