<Text-field style="Heading 1" layout="Heading 1">Example 2.2.13: A sphere</Text-field>

The shape operator - some examples

L.K. Norris

October, 2007

I will use standard spherical coordinates rather than Oprea's geometrical coordinates.

f:=<R*sin(v)*cos(u)|R*sin(v)*sin(u)|R*cos(v)>;

with(LinearAlgebra):

x[u]:= map(diff,f,u);

x[v]:= map(diff,f,v);

TheCrossProduct:= simplify(CrossProduct(x[u],x[v]));

TheDenom:= simplify(VectorNorm(TheCrossProduct,2,conjugate=false),symbolic);

UnitNormalVector:= evalm(TheCrossProduct/TheDenom);

To understand the shape operator it is enough to know what the shape operator does to the basis {x_sub_u, x_sub_v}.

S[p](x[u]) = -map(diff,UnitNormalVector,u);

S[p](x[v]) = -map(diff,UnitNormalVector,v);

Note the following:

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 and 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

This means that the shape operator multiplies each basis vector by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJm1mcmFjR0YkNigtRiM2JC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGOi1GIzYkLUYsNiVRIlJGJy8lJ2l0YWxpY0dRJXRydWVGJy9GO1EnaXRhbGljRidGOi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGTC8lKWJldmVsbGVkR1EmZmFsc2VGJ0Y6RitGOg==. Since the shape operator is linear this means the for any tangent vector v we have

Hence the matrix of the shape operator (in this coordinate system) is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJm1mcmFjR0YkNigtRiM2JC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGOi1GIzYkLUYsNiVRIlJGJy8lJ2l0YWxpY0dRJXRydWVGJy9GO1EnaXRhbGljRidGOi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGTC8lKWJldmVsbGVkR1EmZmFsc2VGJ0Y6RitGOg== times the Identity matrix.

NOTE: Opera uses a different coordinate system that results in a negative sign in the above.

<Text-field style="Heading 1" layout="Heading 1">Example 2.2.14: A right circular cylinder</Text-field>

restart;

f:=<R*cos(u)|R*sin(u)|v>;

with(LinearAlgebra):

x[u]:= map(diff,f,u);

x[v]:= map(diff,f,v);

TheCrossProduct:= simplify(CrossProduct(x[u],x[v]));

TheDenom:= simplify(VectorNorm(TheCrossProduct,2,conjugate=false),symbolic);

UnitNormalVector:= evalm(TheCrossProduct/TheDenom);

To understand the shape operator it is enough to know what the shape operator does to the basis {x_sub_u, x_sub_v}.

S[p](x[u]) = -map(diff,UnitNormalVector,u);

S[p](x[v]) = -map(diff,UnitNormalVector,v);

Note the following:

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 and 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

This means that the surface looks like a PLANE in the v-direction (no shape!) and it looks like a sphere in the u-direction.

<Text-field style="Heading 1" layout="Heading 1">Example 2.2.15: The Torus</Text-field>

restart;

f:=<(R+r*cos(u))*cos(v)|(R+r*cos(u))*sin(v)|r*sin(u)>;

with(LinearAlgebra):

x[u]:= map(diff,f,u);

x[v]:= map(diff,f,v);

TheCrossProduct:= simplify(CrossProduct(x[u],x[v]));

TheDenom:= simplify(VectorNorm(TheCrossProduct,2,conjugate=false),symbolic);

UnitNormalVector:= evalm(TheCrossProduct/TheDenom);

To understand the shape operator it is enough to know what the shape operator does to the basis {x_sub_u, x_sub_v}.

S[p](x[u]) = simplify( -map(diff,UnitNormalVector,u),symbolic);

S[p](x[v]) = -map(diff,UnitNormalVector,v);

Note the following:

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 and 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

This means that the surface looks like a Sphere in the u-direction .

<Text-field style="Heading 1" layout="Heading 1">Example 2.2.16: The Saddle surface </Text-field>

restart;

f:=<u|v|u*v>;

with(LinearAlgebra):

x[u]:= map(diff,f,u);

x[v]:= map(diff,f,v);

TheCrossProduct:= simplify(CrossProduct(x[u],x[v]));

TheDenom:= simplify(VectorNorm(TheCrossProduct,2,conjugate=false),symbolic);

UnitNormalVector:= evalm(TheCrossProduct/TheDenom);

To understand the shape operator it is enough to know what the shape operator does to the basis {x_sub_u, x_sub_v}.

S[p](x[u]) = simplify( -map(diff,UnitNormalVector,u),symbolic); GG:= evalm( -map(diff,UnitNormalVector,u));

S[p](x[v]) = -map(diff,UnitNormalVector,v); HH:= evalm(-map(diff,UnitNormalVector,v));

W:= a*x[u] + b*x[v];

sols1:= solve({W[1]=GG[1],W[2]=GG[2],W[3]=GG[3]},{a,b});

S_sub_p_on_x_u:= eval(a,sols1)*x_sub_u + eval(b,sols1)*x_sub_v;

WW:= aa*x[u] + bb*x[v];

sols2:= solve({WW[1]=HH[1],WW[2]=HH[2],WW[3]=HH[3]},{aa,bb});

S_sub_p_on_x_v:= eval(aa,sols2)*x_sub_u + eval(bb,sols2)*x_sub_v;