3337.sagews

var('x,y,z,t,r')
r=vector([sin(t),cos(t),2*t]) 
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,**a):
 var('b,to')
 r1=r(t=to)
 TX=tangentVector(r,t)(t=to)
 return parametric_plot3d(list(r1+TX*b),(b,0,1),**a)

def Plotnormal(r,t,to,**a):
 var('b,to')
 r1=r(t=to)
 N=normalVector(r,t)(t=to)
 return parametric_plot3d(list(r1+N*b),(b,0,1),**a)

def Plotbinormal(r,t,to,**a):
 var('b,to')
 r1=r(t=to)
 BN=binormalVector(r,t)(t=to)
 return parametric_plot3d(list(r1+BN*b),(b,0,1),**a)

A = parametric_plot3d(r(t),(t,-10,10), thickness = 10)
B = Plottangent(r(t),t,-2,thickness=10,color='green')
C = Plotnormal(r(t),t,-2,thickness=10,color='orange')
D = Plotbinormal(r(t),t,-2,thickness=10,color='pink')
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 curve 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

1&2=graph 3.length of curve t=0 to t=-2: -4.47213595499958 4.curvature at t=-2: 0.200000000000000

a = vector([(e^(t))*sin(t),t,(e^(t))*cos(t)])
G = parametric_plot3d(a,(t,-10,10),thickness=20)
print"5&6=graph"
H = Plottangent(a,t,-2,thickness=20,color='green')
I = Plotnormal(a,t,-2,thickness=20,color='orange')
J = Plotbinormal(a,t,-2,thickness=20,color='pink')

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

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

a1=diff(a,t)
a2=diff(a1,t)
M =(norm(a1.cross_product(a2))/((norm(a1))^3))
print"8.curvature at t=-2: ",numerical_approx(M(t=-2))

G+H+I+J+K

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