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Question: ReScaliing a projective vector

Question:ReScaliing a projective vector

Ronan 1267 Maple 2024
gcd frontend
May 09 2024
0

I want to rescale a projective vector. Have been using gcd on the numerators and denominators. This works in simple situations. It doesn;t work well here, admitadely the points have been just made up for the question.  Square roots seem to make it mal-preform. I run into a lot of squate roots in symbolic situations. What would be a better way? I have been wondering if frontend would help?

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restart
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Prntmsg::boolean:=true;
Normalise_Projective_Point:=1;
ReScl::boolean:=true;

true

  

1

  

true

 (1)
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ProjLP:=overload([

      proc(A::Vector[row],B::Vector[row],prnt::boolean:=Prntmsg)
      description "2 projective points to create a projective line vector";
      option overload;
      local Vp ,gcdn,gcdd,vp ;
      uses LinearAlgebra;
       
      Vp:=CrossProduct(A,B)^%T;#print("2nd ",Vp);
      if ReScl then
         gcdn := gcd(gcd(numer(Vp[1]),numer(Vp[2])), numer(Vp[3]));
         gcdd := gcd(gcd(denom(Vp[1]),denom(Vp[2])), denom(Vp[3]));
         Vp:=simplify(Vp*gcdd/gcdn);
      end if;
      if Prntmsg then
         print("Line vector from two projective points. " );
      end if;
      return Vp
      end proc,



      proc(A::Vector[column],B::Vector[column],prnt::boolean:=Prntmsg)
      description "2 lines to get intersection projective point";
      option overload;
      uses LinearAlgebra;
      local  Vp;
    
      Vp:=CrossProduct(A,B)^%T;
     
     
      if Vp[3]<>0 and Normalise_Projective_Point<>0 then
           Vp:=Vp/Vp[3];
      end if;
      if Prntmsg then
           print("Meet of two Lines ");
      end if;
      return Vp
   end proc
     
]);
> 

 

proc () option overload; [proc (A::(Vector[row]), B::(Vector[row]), prnt::boolean := Prntmsg) local Vp, gcdn, gcdd, vp; option overload; description "2 projective points to create a projective line vector"; Vp := LinearAlgebra:-CrossProduct(A, B)^%T; if ReScl then gcdn := gcd(gcd(numer(Vp[1]), numer(Vp[2])), numer(Vp[3])); gcdd := gcd(gcd(denom(Vp[1]), denom(Vp[2])), denom(Vp[3])); Vp := simplify(Vp*gcdd/gcdn) end if; if Prntmsg then print("Line vector from two projective points. ") end if; return Vp end proc, proc (A::(Vector[column]), B::(Vector[column]), prnt::boolean := Prntmsg) local Vp; option overload; description "2 lines to get intersection projective point"; Vp := LinearAlgebra:-CrossProduct(A, B)^%T; if Vp[3] <> 0 and Normalise_Projective_Point <> 0 then Vp := Vp/Vp[3] end if; if Prntmsg then print("Meet of two Lines ") end if; return Vp end proc] end proc

 (2)
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#maplemint(ProjLP)
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pt1:=<a|sqrt(b^2+c^2)|1>:
pt2:=<c|sqrt(b^2+a^2)|1>:
pt3:=<f^2/sqrt(a^2+b^2)|f^2/sqrt(c^2+b^2)+sqrt(a^2+b^2)|1>:
pt4:=<b^2/sqrt(a^2+b^2)|f^2/sqrt(c^2+b^2)-sqrt(a^2+b^2)|1>:
> 

 

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l1:=ProjLP(pt1,pt2)

"Line vector from two projective points. "

  

Vector[column](%id = 36893490491002736020)

 (3)
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l2:=ProjLP(pt3,pt4)

"Line vector from two projective points. "

  

Vector[column](%id = 36893490491002712420)

 (4)
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l3:=ProjLP(pt1,pt4)

"Line vector from two projective points. "

  

Vector[column](%id = 36893490491064062908)

 (5)
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l4:=ProjLP(pt2,pt4)

"Line vector from two projective points. "

  

Vector[column](%id = 36893490491064037372)

 (6)
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pl1l2:=simplify(ProjLP(l1,l2))

"Meet of two Lines "

  

Vector[row](%id = 36893490491002741932)

 (7)
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pl2l3:=simplify(ProjLP(l2,l3))

"Meet of two Lines "

  

Vector[row](%id = 36893490491113907252)

 (8)
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(ProjLP(pl1l2,pl2l3));
length(%)

"Line vector from two projective points. "

  

Vector[column](%id = 36893490491113907972)

  

6223

 (9)
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ReScl:=false

false

 (10)
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# doing nothing seems to work better here than rescaling
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(ProjLP(pl1l2,pl2l3));
length(%)

"Line vector from two projective points. "

  

Vector[column](%id = 36893490491125667468)

  

2796

 (11)
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