Tube Plots To get started, execute the worksheet, by clicking on Edit above, go down to Execute and select Worksheet. restart; with(plots): Warning, the name changecoords has been redefined #Define a few useful functions: dprod := (a,b) -> sum(a[jj]*b[jj],jj=1..nops(a)): #Dot product cprod := (a,b) -> [ a[2]*b[3] - b[2]*a[3], - (a[1]*b[3] - a[3]*b[1]), a[1]*b[2] - a[2]*b[1]]: #Crossproduct len := a -> sqrt(dprod(a,a)): #Length of a vector #The next functions makes a tubular surface around the space curve gamma of radius rr, given that the axes BB and NN are the unit binormal and unit normal to the curve: tubeit := (gamma,BB,NN,rr) ->expand(gamma+ rr*cos(theta)*BB + rr*sin(theta)*NN): You can change the entries in the function on the next line as long as you keep it in terms of t. After you make your changes, re-execute the work sheet as described above. f := [t,t^2,t^3]: fp := diff(f,t): fpp := diff(fp,t): fppp := diff(fpp,t): T := expand(fp/len(fp)): #Unit Tangent B1 := cprod(fp,fpp): B := expand(B1/len(B1)): #Unit Binormal N := simplify(cprod(B,T)): # Unit Normal kappa := sqrt(dotprod(B1,B1))/sqrt(dotprod(fp,fp))^3: #Formula from the book for kappa (not used, but might be interesting) tau := simplify(dotprod(fppp,B1)/dotprod(B1,B1)): #Formula for torsion (not used, but might be interesting) plot3d(tubeit(f,B,N,.03),t=0..1,theta=0..2*Pi,color=magenta,grid=[30,10],axes=boxed); -%'PLOT3DG6$-%%MESHG6$X.6"F)I)anythingGI*protectedGF+[gl'!%"!!$_cn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