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Question: Using row/column matrices as projective vectors

Question:Using row/column matrices as projective vectors

Ronan 1341 Maple 2024
vector physics dotproduct crossproduct
August 16 2024
0

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)

 

 

 

 

 

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