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Ronan

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14 years, 30 days
East Grinstead, United Kingdom
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Using row/column matrices as projective ...

Ronan 1401 Maple 2024
vector physics dotproduct
August 16 2024
0 1

I need to seperate vectors  from projective geometry vectors in my package. I am experimenting with using row and column matrices for projective points and lines reppresentation. Two approaches are use here. The 1st converts the Matrices to Vectors and back in the procedure. I dont like that way. The 2nd approach, I worked out the equivalent of CrossProduct and DotProduct for the row and column matrices. That I think is better.

1st Question are there any other approaches worth looking at?Is there something in the Physics package that could be of use?

2nd Qusetion could a new data type be made with the properties of a vector that could be displayed differently.?

Say, for a point [ x : y : z ] and a line < x : y : z >, because projecive quantities as basically ratios..


 

> 

restart
> 

with(LinearAlgebra):
> 

 

> 

#
> 

projcross:=overload([

proc(A::Matrix(1,3),B::Matrix(1,3))
option overload;
description "join of points";
local a:=convert(A,Vector[row]),b:=convert(B,Vector[row]);
uses LinearAlgebra;
print("line thro' points");
convert((a &x b)^%T,Matrix);
end proc,

proc(A::Matrix(3,1),B::Matrix(3,1))
option overload;
description "meet of lines";
local a:=convert(A,Vector[column]),b:=convert(B,Vector[column]);
uses LinearAlgebra;
print("intersect point");
convert((a &x b)^%T,Matrix);
end proc,

proc(A::{Matrix(1,3),Matrix(3,1)},B::{Matrix(1,3),Matrix(3,1)})
option overload;
description "incidence of point and line";
uses LinearAlgebra;
local a,b;
if type(A,'Matrix'(1,3))and type(B,'Matrix'(3,1)) or type(A,'Matrix'(3,1))and type(B,'Matrix'(1,3))  then #not working
a:=convert(A,Vector[row]);
b:=convert(B,Vector[row]);
end if;
print("if = 0 coincident");
a.b ;
end proc


]):
> 

projcross(<<1|2|4>>,<<3|-5|7>>)

"line thro' points"

  

Matrix([[34], [5], [-11]])

 (1)
> 

projcross(<<1,2,4>>,<<3,-5,7>>)

"intersect point"

  

Matrix([[34, 5, -11]])

 (2)
> 

projcross(<<1|2|4>>,<<34,5,-11>>)

"if = 0 coincident"

  

0

 (3)
> 

projcross(<<34,5,-11>> ,<<3|-5|7>> )

"if = 0 coincident"

  

0

 (4)
> 

#This section is to derive a CrossProduct and DotProduct for row and column matrix inputs
> 

M:=<A[1,1]|A[1,2]|A[1,3]> ; #point

M := Vector[row](3, {(1) = A[1, 1], (2) = A[1, 2], (3) = A[1, 3]})

 (5)
> 

N:=<B[1,1]|B[1,2]|B[1,3]> ; #point

N := Vector[row](3, {(1) = B[1, 1], (2) = B[1, 2], (3) = B[1, 3]})

 (6)
> 

(M &x N)^%T ; #line thro' points

3

 (7)
> 

R:=<A[1,1],A[2,1],A[3,1]> ;# line

R := Vector(3, {(1) = A[1, 1], (2) = A[2, 1], (3) = A[3, 1]})

 (8)
> 

S:=<B[1,1],B[2,1],B[3,1]> ;# line

S := Vector(3, {(1) = B[1, 1], (2) = B[2, 1], (3) = B[3, 1]})

 (9)
> 

(R &x S)^%T  ;# intersect point

[`?`]

 (10)
> 

 

> 

T:=<A[1,1]|A[1,2]|A[1,3]>  ;#point

T := Vector[row](3, {(1) = A[1, 1], (2) = A[1, 2], (3) = A[1, 3]})

 (11)
> 

U:=<B[1,1],B[2,1],B[3,1]>  ;# line

U := Vector(3, {(1) = B[1, 1], (2) = B[2, 1], (3) = B[3, 1]})

 (12)
> 

DotProduct(T,U,conjugate=false) ; #check coincidence of point and line

A[1, 1]*B[1, 1]+A[1, 2]*B[2, 1]+A[1, 3]*B[3, 1]

 (13)
> 

V:=<A[1,1],A[2,1],A[3,1]> ;# line

V := Vector(3, {(1) = A[1, 1], (2) = A[2, 1], (3) = A[3, 1]})

 (14)
> 

 

> 

W:=<B[1,1]|B[1,2]|B[1,3]> ;#point

W := Vector[row](3, {(1) = B[1, 1], (2) = B[1, 2], (3) = B[1, 3]})

 (15)
> 

DotProduct(V,W,conjugate=false)  ; #check coincidence of line and point

A[1, 1]*B[1, 1]+A[2, 1]*B[1, 2]+A[3, 1]*B[1, 3]

 (16)
> 

# CrossProduct and DotProduct procedure for matrix inputs
> 

 

> 

ProjCp:=overload([

proc(a::Matrix(1,3),b::Matrix(1,3))
option overload;
description "join of points";
local A:=a,B:=b;
print("line thro' points");
<<A[1, 2]*B[1, 3] - A[1, 3]*B[1, 2],-A[1, 1]*B[1, 3] + A[1, 3]*B[1, 1],A[1, 1]*B[1, 2] - A[1, 2]*B[1, 1]>>;
end proc,
> 

 

> 

proc(a::Matrix(3,1),b::Matrix(3,1))
option overload;
description "meet of lines";
local A:=a,B:=b;
print("intersect point");
<<A[2, 1]*B[3, 1] - A[3, 1]*B[2, 1] | -A[1, 1]*B[3, 1] + A[3, 1]*B[1, 1] | A[1, 1]*B[2, 1] - A[2, 1]*B[1, 1] >>;
end proc,

proc(A::{Matrix(1,3),Matrix(3,1)},B::{Matrix(1,3),Matrix(3,1)})
option overload;
description "incidence of point and line";
local dp;
uses LinearAlgebra;
if type(A,'Matrix'(1,3))and type(B,'Matrix'(3,1))   then
dp:=A[1, 1]*B[1, 1] + A[1, 2]*B[2, 1] + A[1, 3]*B[3, 1];

elif type(A,'Matrix'(3,1))and type(B,'Matrix'(1,3)) then
dp:=A[1, 1]*B[1, 1] + A[2, 1]*B[1, 2] + A[3, 1]*B[1, 3];

end if;
print("if = 0 coincident");
return dp
end proc,

proc(A::{Matrix(1,3),list})
option overload;
description "convert to/from cartesian";
if type(A,'Matrix'(1,3)) and A[1,3]<>0 then
[A[1,1]/A[1,3],A[1,2]/A[1,3]];
elif A::list then
<<A[1]|A[2]|1>>;
end if;
end proc,

proc(A::{Matrix(3,1)},{vars:=[x,y]})
option overload;
description "convert to expression";

A[1,1]*vars[1]+A[2,1]*vars[2]+A[3,1];

end proc
 
> 

]):
> 

p1:=<<1|2|4>>:p2:=<<3|-5|7>>:p3:=a*p1-b*p2:
> 

L1:=<<5,7,-8>>:L2:=<<-1,4,-6>>:
> 

L3:=ProjCp(p1,p2)

"line thro' points"

  

L3 := Matrix(3, 1, {(1, 1) = 34, (2, 1) = 5, (3, 1) = -11})

 (17)
> 

ProjCp(L1,L2)

"intersect point"

  

Matrix([[-10, 38, 27]])

 (18)
> 

ProjCp(p1,L3)

"if = 0 coincident"

  

0

 (19)
> 

ProjCp(L3,p1)

"if = 0 coincident"

  

0

 (20)
> 

ProjCp(ProjCp(p1,p2),p3)

"line thro' points"

  

"if = 0 coincident"

  

0

 (21)
> 

Cartlist:=ProjCp~([p1,p2,p3])

[[1/4, 1/2], [3/7, -5/7], [(a-3*b)/(4*a-7*b), (2*a+5*b)/(4*a-7*b)]]

 (22)
> 

ProjCp~(Cartlist)

[Matrix(1, 3, {(1, 1) = 1/4, (1, 2) = 1/2, (1, 3) = 1}), Matrix(1, 3, {(1, 1) = 3/7, (1, 2) = -5/7, (1, 3) = 1}), Matrix(1, 3, {(1, 1) = (a-3*b)/(4*a-7*b), (1, 2) = (2*a+5*b)/(4*a-7*b), (1, 3) = 1})]

 (23)
> 

ProjCp~([L1,L2,L3])

[5*x+7*y-8, -x+4*y-6, 34*x+5*y-11]

 (24)

 

 

 

 

 

> 

 

> 

 


 

Download 2024-08-16_Projective_Vectors_as_Matrices.mw

print content of overloaded proc from a ...

Ronan 1401 Maple 2024
procedure module overload
August 12 2024
1 1

I use a procedure called Isee  to print to screen procedures from my packages.

However if the procedure is overloaded it Isee doesn't print it. Is there a way around this.
I have inserted screen shots to show the outputs for the package.

> 

restart
> 

Test:=proc(a::{vector})
print(a);
end proc;

proc (a::{vector}) print(a) end proc

 (1)
> 

Test1:=overload([ proc(a::{vector})
                  option overload;
                  print(a);
                  end proc,

                  proc(a::{Matrix})
                  option overload;
                  print(a);
                  end proc
                ])

proc () option overload; [proc (a::{vector}) option overload; print(a) end proc, proc (a::{Matrix}) option overload; print(a) end proc] end proc

 (2)
> 

Isee := proc(a)
interface(verboseproc = 3);
printf("%P", eval(a));
end proc;

proc (a) interface(verboseproc = 3); printf("%P", eval(a)) end proc

 (3)
> 

Isee(Test)

proc(a::{vector})
    print(a);
end proc

  
> 

Isee(Test1)

proc()
    option overload;
    [proc(a::{vector}) option overload; print(a); end proc,
        proc(a::{Matrix}) option overload; print(a); end proc];
end proc

  
> 

#with(Routines);
#

> 

#An non overloaded procedure in a package
> 

Isee(ConicMatrix);
#

> 

 

                                  ConicMatrix

  
> 

#An overloaded procecure in a package

> 

Isee(FactReduce);
#

                                  FactReduce

  
 

 

Download 2024-08-12_print_overloaded_procedure.mw

A simplification / reduction question...

Ronan 1401 Maple 2024
simplification
July 19 2024
1 1

I know this question has been asked time and time again. Starting of with the expr . That is the end goal I want to achieve.  How would I reduce the expansion to get it into 1-f(x,y,z)/g(x,y,z) format?. I have tried all sorts of approaches.
 

> 

restart
> 

#
> 

expr:=1 - (x__1*x__2 + y__1*y__2 - z__1*z__2)^2/((x__1^2 + y__1^2 - z__1^2)*(x__2^2 + y__2^2 - z__2^2))

1-(x__1*x__2+y__1*y__2-z__1*z__2)^2/((x__1^2+y__1^2-z__1^2)*(x__2^2+y__2^2-z__2^2))

 (1)
> 

normal( (1) );

(x__1^2*y__2^2-x__1^2*z__2^2-2*x__1*x__2*y__1*y__2+2*x__1*x__2*z__1*z__2+x__2^2*y__1^2-x__2^2*z__1^2-y__1^2*z__2^2+2*y__1*y__2*z__1*z__2-y__2^2*z__1^2)/((x__1^2+y__1^2-z__1^2)*(x__2^2+y__2^2-z__2^2))

 (2)
> 

simplify( (2) );

((y__2^2-z__2^2)*x__1^2-2*x__2*(y__1*y__2-z__1*z__2)*x__1+(y__1^2-z__1^2)*x__2^2-(y__1*z__2-y__2*z__1)^2)/((x__1^2+y__1^2-z__1^2)*(x__2^2+y__2^2-z__2^2))

 (3)
> 

radnormal(%)

(x__1^2*y__2^2-x__1^2*z__2^2-2*x__1*x__2*y__1*y__2+2*x__1*x__2*z__1*z__2+x__2^2*y__1^2-x__2^2*z__1^2-y__1^2*z__2^2+2*y__1*y__2*z__1*z__2-y__2^2*z__1^2)/((x__1^2+y__1^2-z__1^2)*(x__2^2+y__2^2-z__2^2))

 (4)
> 

Test:=combine(%)

(x__1^2*y__2^2-x__1^2*z__2^2-2*x__1*x__2*y__1*y__2+2*x__1*x__2*z__1*z__2+x__2^2*y__1^2-x__2^2*z__1^2-y__1^2*z__2^2+2*y__1*y__2*z__1*z__2-y__2^2*z__1^2)/((x__1^2+y__1^2-z__1^2)*(x__2^2+y__2^2-z__2^2))

 (5)
> 

 

> 

n:={op(numer(Test))}

{x__1^2*y__2^2, x__2^2*y__1^2, -x__1^2*z__2^2, -x__2^2*z__1^2, -y__1^2*z__2^2, -y__2^2*z__1^2, -2*x__1*x__2*y__1*y__2, 2*x__1*x__2*z__1*z__2, 2*y__1*y__2*z__1*z__2}

 (6)
> 

d:={op(expand(denom(Test)))}

{x__1^2*x__2^2, x__1^2*y__2^2, x__2^2*y__1^2, y__1^2*y__2^2, z__1^2*z__2^2, -x__1^2*z__2^2, -x__2^2*z__1^2, -y__1^2*z__2^2, -y__2^2*z__1^2}

 (7)
> 

d subset n

false

 (8)
> 

d intersect n

{x__1^2*y__2^2, x__2^2*y__1^2, -x__1^2*z__2^2, -x__2^2*z__1^2, -y__1^2*z__2^2, -y__2^2*z__1^2}

 (9)
> 

add(%[i],i=1..nops(%))

x__1^2*y__2^2-x__1^2*z__2^2+x__2^2*y__1^2-x__2^2*z__1^2-y__1^2*z__2^2-y__2^2*z__1^2

 (10)
> 

factor( (10) );

x__1^2*y__2^2-x__1^2*z__2^2+x__2^2*y__1^2-x__2^2*z__1^2-y__1^2*z__2^2-y__2^2*z__1^2

 (11)
> 

n minus d

{-2*x__1*x__2*y__1*y__2, 2*x__1*x__2*z__1*z__2, 2*y__1*y__2*z__1*z__2}

 (12)
> 

 


 

Download 2024-07-20_Q_Simplify_Reformat.mw

how to improve the typesetting of the ga...

Ronan 1401 Maple 2024
textplot plotting
July 09 2024
2 12

I am trying to improve the positioning of the gammas in the diagram (at bottom) as they are too close to the points and lines. gamma1 and gamma2 have been assigned values. This best I could come up with is shown below.

> 

restart;
> 

with(plots):with(plottools):
> 

with(Typesetting):
> 

`gamma2`:=<3|5|2>

Vector[row](3, {(1) = 3, (2) = 5, (3) = 2})

 (1)
> 

Pgamma2:=[3/2,5/2]

[3/2, 5/2]

 (2)
> 

display(point(Pbeta2,symbol=solidcircle,symbolsize=14),textplot([Pgamma2[],Typeset((`gamma2` )),align={above}]))

 
> 

 

 

Download 2024-07-10_Typeset_gamma.mw

Hyperlink Question...

Ronan 1401 Maple 2024
documentation hyperlink
July 07 2024
1 1

I have a command called Dual in a SubPackage. RationalTrigonometry:-UHG:-Dual(..). I cannot get the hyperlink from the overview page to work.i.e RationalTrigonometry,UHG,Dual If I use Dual on its own it finds another Maple command to do with boolean logic. What syntax should I use here? I have used RationalTrigonometry,Spread without a problem to avoid another Maple command.

 

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