Problem 2. Show that the evolute of an ellipse is an asteroid. with(LinearAlgebra): alpha:=<a*cos(t),b*sin(t)>;

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpNz1DOg==

alphap:=map(diff,alpha,t);

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpSzxDOg==

nu:=Norm(alphap,2,conjugate=false);

KiQsJiomSSJhRzYiIiIjLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNJInRHRiZGJyIiIiomSSJiR0YmRictSSRjb3NHRipGLUYnRi8jRi9GJw==

T:=alphap/nu;

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpczpDOg==

Tprime:=simplify(map(diff,T,t));

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpSzhDOg==

R:=< <0|-1>, <1| 0> >; N:=R.T;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKD9ZVCk=

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpNzVDOg==

kappa:=simplify(DotProduct(Tprime,N,conjugate=false)/nu);

KigsKComSSJiRzYiIiIjLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNJInRHRiZGJyIiIiokSSJhR0YmRidGLyomRjFGJ0YoRichIiIjISIkRidGMUYvRiVGLw==

epsilon:=simplify(alpha+1/kappa*N);

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpX0MwOQ==

plot(subs({a=1,b=2/3}, [[alpha[1],alpha[2],t=0..2*Pi],[epsilon[1],epsilon[2],t=0..2*Pi]]),color=[red,blue],scaling=constrained);

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

Problem 4 frenet:=proc(alpha,t) local alphap, alphapp, alphappp, T, N, B, nu, kappa, tau, apcrossapp, n_apcrossapp, norm; alphap:=map(diff,alpha,t); alphapp:=map(diff,alphap,t); alphappp:=map(diff,alphapp,t); norm:=x->LinearAlgebra[Norm](x,2,conjugate=false); nu:=simplify(norm(alphap)); apcrossapp:=LinearAlgebra[CrossProduct](alphap,alphapp); n_apcrossapp:=simplify(norm(apcrossapp)); kappa:=simplify(n_apcrossapp/nu^3); T:=simplify(alphap/nu); B:=simplify(apcrossapp/n_apcrossapp); N:=simplify(LinearAlgebra[CrossProduct](B,T)); tau:= simplify(LinearAlgebra[DotProduct](apcrossapp,alphappp,conjugate=false)/n_apcrossapp^2); return [ nu, kappa, tau, T, N, B]; end;

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

alpha:=<a*cos(t), a*sin(t), b*t>;

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpIT16OiI=

frenet(alpha,t) assuming a>0, b>0;

NygqJCwmKiRJImJHNiIiIiMiIiIqJEkiYUdGJ0YoRikjRilGKComRitGKUYkISIiKiZGJEYuRiZGKS0mSSdWZWN0b3JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRic2I0knY29sdW1uR0YnNiMvSSQlaWRHRiciKVNfKD0iLUYxNiMvRjoiKWtIXzgtRjE2Iy9GOiIpc3psOA==

frenet_plane:=proc(alpha,t) local alphap, alphapp, T, Ns, nu, kappas, norm; alphap:=map(diff,alpha,t); alphapp:=map(diff,alphap,t); norm:=x->LinearAlgebra[Norm](x,2,conjugate=false); nu:=simplify(norm(alphap)); T:=simplify(alphap/nu); Ns:=<-T[2],T[1]>; kappas:=simplify(LinearAlgebra[DotProduct](alphapp,Ns,conjugate=false)/nu^2); return [ nu, kappas, T, Ns ]; end;

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

frenet_plane(<a*cos(omega*t),a*sin(omega*t)>,t) assuming a>0, omega>0;

NyYqJkkiYUc2IiIiIkkmb21lZ2FHRiVGJiokRiQhIiItJkknVmVjdG9yRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNJJ2NvbHVtbkdGJTYjL0kkJWlkR0YlIilDZ045LUYrNiMvRjQiKWtnTjk=

Problem 5. beta:=<(1+s)^(3/2)/3,(1-s)^(3/2)/3,s/sqrt(2)>;

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpVyg+QSI=

fbeta:=frenet(beta,s) assuming s::RealRange(-1,1);

NygiIiIsJComIiIjI0YjRiYsJkYjRiMqJEkic0c2IkYmISIiI0YsRiYjRiMiIiUsJCooRiZGJywmRiNGI0YqRiNGLSwmRiNGI0YqRixGLUYuLSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKzYjSSdjb2x1bW5HRis2Iy9JJCVpZEdGKyIpX2hpOS1GNTYjL0Y+IikjPkVZIi1GNTYjL0Y+IilLaWk5

simplify(fbeta[3]/fbeta[2]) assuming s::RealRange(-1,1);

IiIi

Since the ratio of torsion to curvature is constant, this curve must be a cylindrical helix. We can say more. It makes a 45 degree angle with respect to the axis of the cylinder. arccos(LinearAlgebra[DotProduct](fbeta[4],<0,0,1>));

LCRJI1BpRyUqcHJvdGVjdGVkRyMiIiIiIiU=

Also, we can say something about its projection onto the plane. simplify(beta[1]^(2/3)+beta[2]^(2/3)) assuming s::RealRange(-1,1);

LCQqJCIiJCMiIiJGJCMiIiNGJA==

So the projection of the curve is an arc of an asteroid. beta:=<t+sqrt(3)*sin(t),2*cos(t),sqrt(3)*t-sin(t)>;

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpT0c7OQ==

fbeta:=frenet(beta,t);

NygsJCokIiIjIyIiIkYlRiUjRiciIiUjISIiRiktJkknVmVjdG9yRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJ2NvbHVtbkdGMjYjL0kkJWlkR0YyIilfKUcyIi1GLTYjL0Y3Iikzb1w3LUYtNiMvRjciKGNdRSk=

Since curvature and torsion are both constant, the curve must be a circular helix. TTdSMApJNVJUQUJMRV9TQVZFLzE1MjQxODEyWColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiIyIjKiYlImFHIiIiLSUkY29zRzYjJSJ0R0YpKiYlImJHRiktJSRzaW5HRixGKUYmTTdSMApJNVJUQUJMRV9TQVZFLzE1MjQxNzMyWColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiIyIjLCQqJiUiYUciIiItJSRzaW5HNiMlInRHRiohIiIqJiUiYkdGKi0lJGNvc0dGLUYqRiY=TTdSMApJNVJUQUJMRV9TQVZFLzE1MjQxNTcyWColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiIyIjLCQqKCwmKiYlImFHIiIjLSUkc2luRzYjJSJ0R0YsIiIiKiYlImJHRiwtJSRjb3NHRi9GLEYxIyEiIkYsRitGMUYtRjFGNyooRilGNkYzRjFGNEYxRiY=TTdSMApJNVJUQUJMRV9TQVZFLzE1MjQxMzMyWColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiIyIjLCQqKiUiYUciIiItJSRjb3NHNiMlInRHRiolImJHIiIjLCgqJkYvRjBGK0YwRioqJEYpRjBGKiomRilGMEYrRjAhIiIjISIkRjBGNSwkKipGL0YqLSUkc2luR0YtRipGKUYwRjFGNkY1RiY=TTdSMApJNFJUQUJMRV9TQVZFLzg0MTQ2MjBYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISMlIiMiIyIiISIiIiEiIkYnRiY=TTdSMApJNVJUQUJMRV9TQVZFLzE1MjQxMDEyWColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiIyIjLCQqKCwmKiYlImFHIiIjLSUkc2luRzYjJSJ0R0YsIiIiKiYlImJHRiwtJSRjb3NHRi9GLEYxIyEiIkYsRjNGMUY0RjFGNywkKihGKUY2RitGMUYtRjFGN0YmTTdSMApJNVJUQUJMRV9TQVZFLzE0MDUyNDUyWColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiIyIjKigsJiokJSJiRyIiIyEiIiokJSJhR0YrIiIiRi9GLkYsLSUkY29zRzYjJSJ0RyIiJCwkKihGKEYvLSUkc2luR0YyRjRGKkYsRixGJg==TTdSMApJNVJUQUJMRV9TQVZFLzExNTc5MTgwWColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiJCIkKiYlImFHIiIiLSUkY29zRzYjJSJ0R0YpKiZGKEYpLSUkc2luR0YsRikqJiUiYkdGKUYtRilGJg==TTdSMApJNVJUQUJMRV9TQVZFLzExODc1MjQwWColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiJCIkLCQqKCwmKiQlImJHIiIjIiIiKiQlImFHRixGLSMhIiJGLEYvRi0tJSRzaW5HNiMlInRHRi1GMSooRilGMEYvRi0tJSRjb3NHRjRGLSomRilGMEYrRi1GJg==TTdSMApJNVJUQUJMRV9TQVZFLzEzNTIyOTY0WColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiJCIkLCQtJSRjb3NHNiMlInRHISIiLCQtJSRzaW5HRipGLCIiIUYmTTdSMApJNVJUQUJMRV9TQVZFLzEzNjU3OTcyWColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiJCIkKigsJiokJSJiRyIiIyIiIiokJSJhR0YrRiwjISIiRitGKkYsLSUkc2luRzYjJSJ0R0YsLCQqKEYoRi9GKkYsLSUkY29zR0YzRixGMComRihGL0YuRixGJg==TTdSMApJNVJUQUJMRV9TQVZFLzE0MzU2MDI0WColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiIyIjLCQtJSRzaW5HNiMqJiUmb21lZ2FHIiIiJSJ0R0YtISIiLSUkY29zR0YqRiY=TTdSMApJNVJUQUJMRV9TQVZFLzE0MzU2MDY0WColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiIyIjLCQtJSRjb3NHNiMqJiUmb21lZ2FHIiIiJSJ0R0YtISIiLCQtJSRzaW5HRipGL0YmTTdSMApJNVJUQUJMRV9TQVZFLzEyMjE5NzQ0WColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiJCIkLCQqJCwmIiIiRiolInNHRiojIiIkIiIjI0YqRi0sJCokLCZGKkYqRishIiJGLEYvLCQqJkYrRipGLiNGKkYuRjZGJg==TTdSMApJNVJUQUJMRV9TQVZFLzE0NjI2MTUyWColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiJCIkLCQqJCwmIiIiRiolInNHRiojRioiIiNGLCwkKiQsJkYqRipGKyEiIkYsI0YxRi0sJCokRi1GLEYsRiY=TTdSMApJNVJUQUJMRV9TQVZFLzE0NjI2MTkyWColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiJCIkLCQqKCIiIyMiIiJGKSwmKiYsJkYrRislInNHRitGKiwmRitGK0YvISIiRipGKyokLCZGK0YrKiRGL0YpRjFGKkYrRitGLiNGMUYpI0YrIiIlLCQqKEYpRipGLEYrRjBGNUY2IiIhRiY=TTdSMApJNVJUQUJMRV9TQVZFLzE0NjI2MjMyWColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiJCIkLCQqJiwmIiIiRioqJCUic0ciIiMhIiIjRipGLSwmRipGKkYsRi4jRi5GLUYxLCQqJkYpRi8sJkYqRipGLEYqRjFGLywkKiRGLUYvRi9GJg==TTdSMApJNVJUQUJMRV9TQVZFLzE0MTYyODM2WColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiJCIkLCYlInRHIiIiKiYiIiQjRikiIiMtJSRzaW5HNiNGKEYpRiksJC0lJGNvc0dGMEYtLCYqJkYrRixGKEYpRilGLiEiIkYmTTdSMApJNVJUQUJMRV9TQVZFLzEwNzI4ODUyWColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiJCIkLCQqJiIiIyMiIiJGKSwmRitGKyomIiIkRiotJSRjb3NHNiMlInRHRitGK0YrI0YrIiIlLCQqJkYpRiotJSRzaW5HRjFGKyMhIiJGKSwkKiZGKUYqLCYqJEYuRipGK0YvRjpGK0YzRiY=TTdSMApJNVJUQUJMRV9TQVZFLzEyNDk2ODA4WColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiJCIkLCQqJiIiJCMiIiIiIiMtJSRzaW5HNiMlInRHRisjISIiRiwsJC0lJGNvc0dGL0YyLCRGLUYqRiY=TTdSMApJNFJUQUJMRV9TQVZFLzgyNjUwNTZYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISIkIiQsJComIiIjIyIiIkYpLCYqJiIiJEYqLSUkY29zRzYjJSJ0R0YrRishIiJGK0YrI0YrIiIlLCQqJkYpRiotJSRzaW5HRjFGKyNGM0YpLCQqJkYpRiosJkYvRisqJEYuRipGK0YrI0YzRjVGJg==