This notebook contains the space curves that we plotted in class. I made these using SageMath (http://sagemath.org) in a web-based Jupyter notebook (http://jupyter.org).
In [1]:
version()
Out[1]:
In [2]:
var('t');
Toroidal Spiral¶
Parametric Equations: $$ \begin{cases} x(t) = \big(4 + \sin(20 t)\big)\cos t,\\[1 ex] y(t) = \big(4 + \sin(20 t)\big)\sin t,\\[1 ex] z(t) = \cos(20 t) \end{cases}$$
In [3]:
f = [(4 + sin(20*t))*cos(t),(4+sin(20*t))*sin(t),cos(20*t)];
parametric_plot3d(f,(0,2*pi),plot_points=1000,aspect_ratio=1,viewer='tachyon')
Out[3]:
Trefoil Knot¶
Parametric Equations: $$\begin{cases} x(t) = \big(2 + \cos(\tfrac{3}{2} t)\big)\cos t,\\[1 ex] y(t) = \big(2 + \cos(\tfrac{3}{2} t)\big)\sin t,\\[1 ex] z(t) = \sin(\tfrac{3}{2} t) \end{cases}$$
In [4]:
g = [(2 + cos(3/2*t))*cos(t),(2+cos(3/2*t))*sin(t),sin(3/2*t)];
parametric_plot3d(g,(0,4*pi),plot_points=1000,aspect_ratio=1,viewer='tachyon')
Out[4]:
Twisted Cubic¶
Parametric Equations: $$\begin{cases} x(t) = t, \\[1ex] y(t) = t^2, \\[1ex] z(t) = t^3 \end{cases}$$
In [5]:
h = [t,t**2,t**3];
parametric_plot3d(h,(-2,2),plot_points=1000,viewer='tachyon')
Out[5]:
