3334.sagews

var('x,y,z,t,r')
r=vector([sin(t),cos(t),2*t]) 
print"Problem 1: "
print"1. + 2. = graph"

def tangentVector(r,t):
 return(diff(r,t)/norm(diff(r,t)))

def normalVector(r,t):
 return(diff(tangentVector(r,t),t)/(norm(diff(tangentVector(r,t),t))))

def binormalVector(r,t):
 return(tangentVector(r,t).cross_product(normalVector(r,t)))

def Plottangent(r,t,to,**kwarg):
 var('u,to')
 r0=r(t=to)
 Ta=tangentVector(r,t)(t=to)
 return parametric_plot3d(list(r0+Ta*u),(u,0,1),**kwarg)

def Plotnormal(r,t,to,**kwarg):
 var('u,to')
 r0=r(t=to)
 No=normalVector(r,t)(t=to)
 return parametric_plot3d(list(r0+No*u),(u,0,1),**kwarg)

def Plotbinormal(r,t,to,**kwarg):
 var('u,to')
 r0=r(t=to)
 Bi=binormalVector(r,t)(t=to)
 return parametric_plot3d(list(r0+Bi*u),(u,0,1),**kwarg)

A = parametric_plot3d(r(t),(t,-10,10), thickness = 5)
B = Plottangent(r(t),t,-2,thickness=5,color='red')
C = Plotnormal(r(t),t,-2,thickness=5,color='green')
D = Plotbinormal(r(t),t,-2,thickness=5,color='orange')
E = implicit_plot3d(x==-19.99,(x,-20,20),(y,-20,20),(z,-20,20),color='white')

F = numerical_approx(integral(norm(diff(r,t)),t,0,-2))
print"3. length of the curve from t=0 to t=-2: ",F

r1 = diff(r,t)
r2 = diff(r1,t)
curvature = (norm(r1.cross_product(r2))/((norm(r1))^3))
print"4. curvature at t=-2: ",numerical_approx(curvature(t=2))

A+B+C+D+E

Problem 1: 1. + 2. = graph 3. length of the curve from t=0 to t=-2: -4.47213595499958 4. curvature at t=-2: 0.200000000000000

asdf = vector([(e^(t))*sin(t),t,(e^(t))*cos(t)])
G = parametric_plot3d(asdf,(t,-10,10),thickness=5)
print"5. + 6. = graph"
H = Plottangent(asdf,t,-2,thickness=5,color='red')
I = Plotnormal(asdf,t,-2,thickness=5,color='green')
J = Plotbinormal(asdf,t,-2,thickness=5,color='orange')

K = implicit_plot3d(x==-19999.99,(x,-20000,20000),(y,-20000,20000),(z,-20000,20000),color='white')

L = numerical_approx(integral(norm(diff(asdf,t)),t,0,-2))
print"7. length of the curve from t=0 to t=-2: ",L

asdf1=diff(asdf,t)
asdf2=diff(asdf1,t)
M = (norm(asdf1.cross_product(asdf2))/((norm(asdf1))^3))
print"8. curvature at t=-2: ",numerical_approx(M(t=-2))

G+H+I+J+K

5. + 6. = graph 7. length of the curve from t=0 to t=-2: -2.41102905936836 8. curvature at t=-2: 0.258788978266708