UVxvL0xpYnJhcnkvRnJhbWV3b3Jrcy9NYXBsZS5mcmFtZXdvcmsvVmVyc2lvbnMvMTEvYmluLkFQUExFX1BQQ19PU1g2Ig==

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;

Zio2JEkmYWxwaGFHNiJJInRHRiU2LkknYWxwaGFwR0YlSShhbHBoYXBwR0YlSSlhbHBoYXBwcEdGJUkiVEdGJUkiTkdGJUkiQkdGJUkjbnVHRiVJJmthcHBhR0YlSSR0YXVHRiVJK2FwY3Jvc3NhcHBHRiVJLW5fYXBjcm9zc2FwcEdGJUklbm9ybUdGJUYlRiVDLz44JC1JJG1hcEclKnByb3RlY3RlZEc2JUklZGlmZkdGOTkkOSU+OCUtRjg2JUY7RjZGPT44Ji1GODYlRjtGP0Y9PjgvZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLSZJLkxpbmVhckFsZ2VicmFHRiU2I0klTm9ybUdGJTYlRjwiIiMvSSpjb25qdWdhdGVHRjlJJmZhbHNlR0Y5RiVGJUYlPjgqLUkpc2ltcGxpZnlHRiU2Iy1GRzYjRjY+OC0tJkZQNiNJLUNyb3NzUHJvZHVjdEdGJTYkRjZGPz44Li1GZW42Iy1GRzYjRmpuPjgrLUZlbjYjKiZGYW8iIiJGWSEiJD44Jy1GZW42IyomRjZGW3BGWSEiIj44KS1GZW42IyomRmpuRltwRmFvRmJwPjgoLUZlbjYjLUZcbzYkRmRwRl5wPjgsLUZlbjYjKiYtJkZQNiNJK0RvdFByb2R1Y3RHRiU2JUZqbkZDRlVGW3BGYW8hIiNPNyhGWUZnb0ZfcUZecEZpcEZkcEYlRiVGJQ==

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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

Problem 1. alpha:=<(1+2*cos(t))*cos(t), (1+2*cos(t))*sin(t)>;

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIoL2oqKSk=

plot([alpha[1],alpha[2],t=0..2*Pi]);

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

alpha_f:=frenet_plane(alpha,t);

NyYqJCwmLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJInRHRioiIiUiIiYiIiIjRi8iIiMsJComLCZGJUYxIiIkRi9GL0YkIyEiJEYxRjUtJkknVmVjdG9yR0YnNiNJJ2NvbHVtbkdGKjYjL0kkJWlkR0YqIihbR0ApLUY5NiMvRj8iKCd6KD4q

kappa:=alpha_f[2];

LCQqJiwmLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJInRHRioiIiMiIiQiIiJGLywmRiUiIiUiIiZGLyMhIiRGLUYu

If kappa has a minimum or maximum, its derivative must be zero there. kappaprime:=simplify(diff(kappa,t));

LCQqKC1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ0R0YpIiIiLCYtSSRjb3NHRiZGKkYsIiIjRixGLCwmRi4iIiUiIiZGLCMhIiZGMCIjNw==

If the derivative is zero, the numerator must be zero. zero:=numer(kappaprime);

LCQqJi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ0R0YpIiIiLCYtSSRjb3NHRiZGKkYsIiIjRixGLCIjNw==

This expression is zero if one of the two factors is zero. But cos(t)+2 is never zero. So kappaprime is zero if and only if sin(t) is zero, which happens exactly twice on [0,2*pi]. plot(zero,t=0..2*Pi);

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

Problem 5. x:=<a*v*cos(u),a*v*sin(u),b*u>;

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIoZ0VoKg==

xu:=map(diff,x,u); xv:=map(diff,x,v);

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpayRRNCI=

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpZygqUTY=

simplify(LinearAlgebra[Norm](LinearAlgebra[CrossProduct](xu,xv),2,conjugate=false));

KiQqJkkiYUc2IiIiIywmKiRJImJHRiVGJiIiIiomRiRGJkkidkdGJUYmRipGKiNGKkYm

Since a>0 and b>0, the above expression is at least as large as ab, and therefore is non-zero. So x is regular. Problem 6. x:=u*v;

KiZJInVHNiIiIiJJInZHRiRGJQ==

Note that the coordinate curves are themselves lines. When we plot in maple, we get to see the coordinate curves. So the following plot shows the ruilings on this doubly ruled surface. plot3d(x,u=-2..2,v=-2..2);

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