Experimental format for projective vectors

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Question:Experimental format for projective vectors

Posted: Ronan 1267 Product: Maple 2024

November 20 2024

I am experimenting using the this format of Vector( [Vector] ) to make projective vectors a different data type to Vectors. I don't want to use 1 x 3 or 3 x 1 matrices. The format holds some promise.
I would like to be able to copy the Maple format of Vector or Vector[column] and Vector[row] for my varaition.

ProjVectoC and ProjVectorR so ProjVector or ProjVector[column] and ProjVector[row]
A secondary question is on type checking (see previous question How to setup special type check in a procedure? - MaplePrimes ). Would it be possible to have the type check return ProjVector[column] or ProjVector[row]?
The attached worksheet contains a procedure for factor reducing the vectors to to a minimal format of <x,y,z>. Also Cross product and Dot product procedures to suit.

I am open to any efficiency improvements.

restart

>	interface(rtablesize=50)

(1)

>	with(LinearAlgebra):

FactReduce:=overload([
     proc(v::{list,Vector})
          option overload;
          description " removes linear factor from",
                      " a list, vector, matrix or expression";
          uses LinearAlgebra;
          local i, num,tgdc,dnm, V1;
          num:=`ifelse`(type(v,Vector),numelems(v),nops(v));
          dnm:=frontend(lcm, [seq(denom(v[i]),i=1..num)]);
          V1:=radnormal(v*~dnm);
          tgdc:=V1[1];

          for i from 2 to num do
               tgdc:=frontend(gcd, [tgdc, V1[i]]);
          end do;

          return simplify(V1/~tgdc);
     end proc,

     proc(M::{Matrix})
          option overload;
          uses LinearAlgebra;
          local i, num,r,c, tgdc,dnm, V1, Ml;
          r,c:=Dimension(M);
          num:=r*c;
          V1:=convert(M,list);
          dnm:=frontend(lcm, [seq(denom(V1[i]),i=1..num)]);
          Ml:=radnormal(dnm*~M);
          V1:=convert(Ml,list);#print((dnm,V1));
          tgdc:=V1[1];#print("xx")

          for i from 2 to num do
               tgdc:=frontend(gcd, [tgdc, V1[i]])
          end do;

          return simplify(Ml/~tgdc);
     end proc,

     proc(l::{`+`,`*`,`=`, `symbol`,procedure}, {vars::list:=[:-x,:-y]})
          option overload;
          uses LinearAlgebra;
          local i, num,f1,f1a,lv,lr, tgdc,dnm, V1,Vs;
          f1 := `if`(l::procedure, l(vars[]), l);
               f1a:=`if`(f1::`=`,lhs(f1)-rhs(f1),f1) ; # Remequal(f1);
          lr:=primpart(f1a,vars);
          return lr
end proc

]):

>	ProjVectorC := proc(a, b, c) local cfs, vectr; description " A Projective Column (Line) Vector in Reduced format"; cfs := FactReduce([a, b, c]); vectr := <[<cfs>]>; end proc:

>	ProjVectorR := proc(a, b, c) local cfs, vectr; description " A Projective Row (Point) Vector"; cfs := sign(c)*FactReduce([a, b, c]); vectr := <[<cfs>^%T]>^%T; end proc:

>	`&otimes;` := proc(A, B) local cp; description "Cross Product of Projective Vectors in Reduced format"; cp :=sign(c)* FactReduce(LinearAlgebra:-`&x`(A[1], B[1]))^%T; cp := ifelse(cp[3] <> 0, <[sign(cp[3]) *~ cp]>, cp); #makes sure format is [x,y,z] and not [x,y-z] end proc: