Frenet, kappa, and tau.sagews

#initializations
from sage.plot.plot3d.shapes import Box
t = var('t')

#length function for 3-dimensional vectors
def length(v):
    return sqrt(v[0]^2+v[1]^2+v[2]^2)

#components of the path gamma
x(t) = (cos(3*t)+2)*cos(2*t)
y(t) = (cos(3*t)+2)*sin(2*t)
z(t) = -sin(3*t)

#the path gamma
gamma = vector([x,y,z])

#start and end points a,b
a = 0
b = 2*pi

#path plot
P = parametric_plot3d(gamma(t),(t,a,b),frame=False)

#auxiliary transparent boundary box for better-looking animations
#set box dimensions so that each frame lies inside it
boundary = Box([3.1,3.1,2.1],opacity=0,frame=False)

#Frenet frame
T = gamma.diff(t)/length(gamma.diff(t))
N = T.diff(t)/length(T.diff(t))
B = T.cross_product(N)

#graphical representations of the Frenet frame
def Tarrow(t):
    return arrow(gamma(t),gamma(t)+T(t),color='green',frame=False)
def Narrow(t):
    return arrow(gamma(t),gamma(t)+N(t),color='red',frame=False)
def Barrow(t):
    return arrow(gamma(t),gamma(t)+B(t),color='purple',frame=False)

#all of the elements we want to plot simultaneously at time t
def frenet_plot(t):
    return P + boundary + Tarrow(t) + Narrow(t) + Barrow(t)

#curvature and torsion
#note the novel extrinsic expression for tau
kappa = length(gamma.diff(t).cross_product(gamma.diff(t).diff(t)))/length(gamma.diff(t))^3
tau = (gamma.diff(t).cross_product(gamma.diff(t).diff(t))).dot_product(gamma.diff(t).diff(t).diff(t))/length(gamma.diff(t).cross_product(gamma.diff(t).diff(t)))^2

#simultaneous plot of curvature and torsion
ktplot = Graphics()
ktplot += plot(kappa,(t,a,b),color='green')
ktplot += plot(tau,(t,a,b),color='red')

#add labels to the curvature/torsion plot
ktplot += text("curvature",(b,1.5),color='green')
ktplot += text("torsion",(b,1),color='red')

#set the number of frames in the movie
#test your setup with low num_frames (say 5 or 10) and then boost num_frames to 60 or 100 (or more?) for your final animation
num_frames = 100

#animate the frames of the movie
frenet_movie = animate([frenet_plot(t) for t in srange(a,b,(b-a)/num_frames)])

#display the animation and the curvature/torsion plot
frenet_movie.show()
ktplot.show()

#save the animation and plot
frenet_movie.save('frenet_movie.gif')
ktplot.save('ktplot.png')