For application of object recognition with 3d sensor

plot3d(subs(c=1, a*b+a*c), a=-1..1, b=-1..1);
plot3d(subs(c=1, a+a*b*c), a=-1..1, b=-1..1);

after draw above two equations, they looked having similar structure

i find a book

Differential Geometry and its application John Oprea, page 98

having some measure to determine the shape, however they seems for

2 variables and need parametric form

how to use them to determine above two equations which have 3 variables

a*b+a*c  and  a+a*b*c

though not know the 3 variables version of below code, 

i think subs(c=1, a*b+a*c), subs(a=1, a*b+a*c), subs(b=1, a*b+a*c)

3 sets of equations after substitute third variables with 1, is it still possible to determine the shape?

EFG := proc(X)
local Xu, Xv, E, F, G;
Xu := <diff(X[1],u), diff(X[2],u), diff(X[3],u)>;
Xv := <diff(X[1],v), diff(X[2],v), diff(X[3],v)>;
E := DotProduct(Xu,Xu,conjugate=false);
F := DotProduct(Xu,Xv,conjugate=false);
G := DotProduct(Xv,Xv,conjugate=
false);
simplify([E,F,G]);
end:
UN := proc(X)
local Xu, Xv, Z, s;
Xu := <diff(X[1],u), diff(X[2],u), diff(X[3],u)>;
Xv := <diff(X[1],v), diff(X[2],v), diff(X[3],v)>;
Z := CrossProduct(Xu,Xv);
s := VectorNorm(Z,Euclidean,conjugate=false);
simplify(<Z[1]/s|Z[2]/s|Z[3]/s>,sqrt,trig,symbolic);
end:
lmn := proc(X)
local Xu,Xv,Xuu,Xuv,Xvv,U,l,m,n;
Xu := <diff(X[1],u), diff(X[2],u), diff(X[3],u)>;
Xv := <diff(X[1],v), diff(X[2],v), diff(X[3],v)>;
Xuu := <diff(Xu[1],u), diff(Xu[2],u), diff(Xu[3],u)>;
Xuv := <diff(Xu[1],v), diff(Xu[2],v), diff(Xu[3],v)>;
Xvv := <diff(Xv[1],v), diff(Xv[2],v), diff(Xv[3],v)>;
U := UN(X);
l := DotProduct(U, Xuu, conjugate=false);
m := DotProduct(U, Xuv, conjugate=false);
n := DotProduct(U, Xvv, conjugate=false);
simplify([l,m,n],sqrt,trig,symbolic);
end:
shape := proc(X)
local Y,Z,a,b,c,d;
Y:=EFG(X);
Z:=lmn(X);
a:=simplify((Z[1]*Y[3]-Z[2]*Y[2])/(Y[1]*Y[3]-Y[2]^2));
b:=simplify((Z[2]*Y[1]-Z[1]*Y[2])/(Y[1]*Y[3]-Y[2]^2));
c:=simplify((Z[2]*Y[3]-Z[3]*Y[2])/(Y[1]*Y[3]-Y[2]^2));
d:=simplify((Z[3]*Y[1]-Z[2]*Y[2])/(Y[1]*Y[3]-Y[2]^2));
[S(x_u)=a*x_u+b*x_v,S(x_v)=c*x_u+d*x_v];
end:
shape_matrix := proc(X)
local Y,Z,a,b,c,d;
Y:=EFG(X);
Z:=lmn(X);
a:=simplify((Z[1]*Y[3]-Z[2]*Y[2])/(Y[1]*Y[3]-Y[2]^2));
b:=simplify((Z[2]*Y[1]-Z[1]*Y[2])/(Y[1]*Y[3]-Y[2]^2));
c:=simplify((Z[2]*Y[3]-Z[3]*Y[2])/(Y[1]*Y[3]-Y[2]^2));
d:=simplify((Z[3]*Y[1]-Z[2]*Y[2])/(Y[1]*Y[3]-Y[2]^2));
Matrix([[a,c],[b,d]]);
end:

how to convert between (system of polynomials or module) and rational function which is a four dimensional space

1. is module in algebraic geometry or group cohomology to classify homology group?

2. is final result like this? for example,

a dictionary  or table store

homology group 1, key=invariant = 1 hole in topological space , value = module 1, module 2 module 3 etc

homology group 2, key=invariant = 2 holes in topological space , value =module 4, module 5, module 6 etc

homology group 3, key=invariant = 3 holes in topological space , value = module 7, module 8.. etc

 ... etc

3. i find betti number can count holes, however how to know the form for

all combination?

i mean if it is module, how to change the module to get the full combination

http://www.singular.uni-kl.de/Manual/html/sing_159.htm

i find betti number's input is just ideal, but it is not module

if ideal is enough, why need module?

how to permuate module? and what do it classify and result in module?

1. is module in algebraic geometry for classification of topological space which a poset is a frame

2. which invariant is for doing this classification of topological space in algebraic geometry or group cohomology?

3. if want to do full combination before classification, which kind of polynomials be a full combination

4. is poset just like function fst and snd function for meet and join in functional programming instead of using "and" and "or" logic? how a matrix group related with topological space which a poset is a frame?

5. is there any invariant function for classification of topological space in maple?

#page 320 and 322 of book Singular introduction to commutative algebra

it return too many recursion 

hilbertseries([a+a*c, a+a*b, a+b+c]);

eq1 := a+a*c;

eq2 := a+a*b;

eq3 := a+b+c;

eq1a := Homogenize(eq1, h);

eq2a := Homogenize(eq2, h);

eq3a := Homogenize(eq3, h);

T3:=lexdeg([a,b,c,h]);

GB := Basis([eq1a,eq2a,eq3a], T3); #a

#MonomialHilbertPoincare(LeadingMonomial(GB[1],T3), LeadingMonomial(GB[2],T3), LeadingMonomial(GB[3],T3));

with(PolynomialIdeals):

MonomialHilbertPoincare := proc (I3)

#I3:=[LeadingMonomial(GB[1],T3), LeadingMonomial(GB[2],T3), LeadingMonomial(GB[3],T3)];

T2:=lexdeg([h,c,b,a]);

varj := [h,c,b,a];

I2 := InterReduce(I3, T2);

s := nops(I2);

if I2[1] = 0 then return 1 end if:

if I2[1] = 1 then return 0 end if:

if degree(I2[s]) = 1 then return (1-varj[1])^s end if:

lt := LeadingTerm(I2[s],T2);

leadexp := [degree(lt[2],h),degree(lt[2],c),degree(lt[2],b),degree(lt[2],a)];

j := 1;

for z from 1 to nops(leadexp) do

                if leadexp[j] = 0 then

                                j := j + 1;

                end if:

od:

finallist := [];

for z from 1 to nops(GB) do

                finallist := [op(finallist), GB[z]+varj[j]];

od:

quotientlist := Generators(Quotient(GB, varj[j]));

finallist2 := [];

for z from 1 to nops(quotientlist) do

                finallist2 := [op(finallist2), op(z,quotientlist)];

od:

return MonomialHilbertPoincare(finallist) + varj[1]*MonomialHilbertPoincare(finallist2);

end proc;

F:=[LeadingMonomial(GB[1],T3), LeadingMonomial(GB[2],T3), LeadingMonomial(GB[3],T3)];

MonomialHilbertPoincare(F);