SageMath syntax for Section 3.3

SageMath is a free open-source mathematics software system. You can use a SageMath cell online by going here.

• Numerical integration for arc length: $\quad \displaystyle \int_0^1 \|\vec{r}'(t)\| \, dt$

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var("t")
r=vector([t,t^2,t^3])
numerical_integral( diff(r,t).norm(), 0, 1 )

• Curvature for $\vec{r}(t)$

x
var("t")
r=vector([sin(3*t),-cos(t),t])
rp=diff(r,t)
rpp=diff(rp,t)
cross=rp.cross_product(rpp)
kappa=cross.norm()/(rp.norm())^3
kappa(t=0)

• Plot a 2d-curve, a principal unit tangent vector, and a principal unit normal vector.

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var("t")
r=vector([sin(2*t),sin(t)*cos(2*t)])
rp=diff(r,t)
T=rp/rp.norm()
Tp=diff(T,t)
N=Tp/Tp.norm()
p1=parametric_plot(r,(t,0,pi))
p2=plot(T(t=pi/3), start=r(t=pi/3), color="green")
p3=plot(N(t=pi/3), start=r(t=pi/3), color="red")
show(p1+p2+p3)