Gauss and Mean Curvature of Surfaces The initial formulas: Many of the formulas for curvature require the use of the dot product, cross product and Eudlidean norm. Procedures are written for each of these. dp := proc(X,Y) X*Y+X*Y+X*Y; end: xp := proc(X,Y) local a,b,c; a := X*Y-X*Y; b := X*Y-X*Y; c := X*Y-X*Y; [a,b,c]; end: nrm := proc(X) sqrt(dp(X,X)); end: The following procedure gives the unit normal to an input surface X, where X is in paramaterized form and in square brackets. UN := proc(X) local Xu,Xv,Z,s; Xu := [diff(X,u),diff(X,u),diff(X,u)]; Xv := [diff(X,v),diff(X,v),diff(X,v)]; Z := xp(Xu,Xv); s := nrm(Z); simplify([Z/s,Z/s,Z/s],sqrt,symbolic); end: The next procedure creates the metric E=x_u dot x_u, F=x_u dot x_v and G=x_v dot x_v. This metric will be used later in the formulas. EFG := proc(X) local Xu,Xv,E,F,G; Xu := [diff(X,u),diff(X,u),diff(X,u)]; Xv := [diff(X,v),diff(X,v),diff(X,v)]; E := dp(Xu,Xu); F := dp(Xu,Xv); G := dp(Xv,Xv); simplify([E,F,G]); end: The formula for Guass Curvature requires certain partial derivatives being dotted with the unit normal vector. lmn := proc(X) local Xu,Xv,Xuu,Xuv,Xvv,U,l,m,n; Xu := [diff(X,u),diff(X,u),diff(X,u)]; Xv := [diff(X,v),diff(X,v),diff(X,v)]; Xuu := [diff(Xu,u),diff(Xu,u),diff(Xu,u)]; Xuv := [diff(Xu,v),diff(Xu,v),diff(Xu,v)]; Xvv := [diff(Xv,v),diff(Xv,v),diff(Xv,v)]; U := UN(X); l := dp(U,Xuu); m := dp(U,Xuv); n := dp(U,Xvv); simplify([l,m,n]); end: The following computes the Gauss Curvature, K. GK := proc(X) local E,F,G,l,m,n,S,T; S := EFG(X); T := lmn(X); E := S; F := S; G := S; l := T; m := T; n := T; simplify((l*n-m^2)/(E*G-F^2)); end: Mean Curvature, H, is given now. MK := proc(X) local E,F,G,l,m,n,S,T; S := EFG(X); T := lmn(X); E := S; F := S; G := S; l := T; m := T; n := T; simplify((G*l+E*n-2*F*m)/(2*E*G-2*F^2)); end:
<Text-field style="Heading 1" layout="Heading 1">Some Examples</Text-field> Rsphere:=[R*cos(u)*cos(v),R*sin(u)*cos(v),R*sin(v)]; GK(Rsphere); The following is the catenoid, which results when the catenary is revolved around the x-axis. Note that the catenoid is a minimal surface. catenoid:=[cosh(u)*cos(v),cosh(u)*sin(v),u]; EFG(catenoid);GK(catenoid); MK(catenoid); plot3d(catenoid, u=-2..2, v=0..2*Pi,axes=box); The following surface is the helicoid obtained by revolving a moving line about the z-axis. The helicoid is a ruled surface. It is also a minimal surface. helicoid:=[sinh(u)*cos(v),sinh(u)*sin(v),v]; EFG(helicoid);GK(helicoid); MK(helicoid); plot3d(helicoid, u=-2..2, v=0..2*Pi,axes=box); Discuss intrinsic Gauss curvature arguments for each of the above surfaces.