How to find the orientation of the line of the intersection between two planes?

Is there any method/indiacator that i can use to know the orientation of the the intersection line between two planes( using Dual Plucker Matrix )?

Dual Plücker matrix

I used the follwoing code get the line:

P1 =[177668442.453315 ,-102576923.076923,	0];
P2 =[ -102576923.076923	,177668442.453315 	,-102576923.076923];
P3= [0,	-102576923.076923,	88834221.2266576];

P11= [152763459.308716 ,	-102576923.076923,	0];
P22=[ -102576923.076923,	183536536.231793	,    -102576923.076923];
P33= [0,	-102576923.076923,	91768268.1158967];
 
A=null([[P1;P2;P3],ones(3,1)]);  %plane 1
B=null([[P11;P22;P33],ones(3,1)]);  %plane 2
L=A*B.' - B*A.' %line of intersection

L = 4×4

0 -0.0351 -0.0484 0.0000 0.0351 0 -0.0138 0.0000 0.0484 0.0138 0 0.0000 -0.0000 -0.0000 -0.0000 0

[Torsten]

What do you mean by "orientation of a line" ?

[M]

@Torsten any indicator of the direction/location...

Answer 1

[Matt J]

P1 =[177668442.453315 ,-102576923.076923,	0];
P2 =[ -102576923.076923	,177668442.453315 	,-102576923.076923];
P3= [0,	-102576923.076923,	88834221.2266576];
P11= [152763459.308716 ,	-102576923.076923,	0];
P22=[ -102576923.076923,	183536536.231793	,    -102576923.076923];
P33= [0,	-102576923.076923,	91768268.1158967];
 
A=null([[P1;P2;P3],ones(3,1)]);  %plane 1
B=null([[P11;P22;P33],ones(3,1)]);  %plane 2
L=A*B.' - B*A.' %line of intersection;

L = 4×4

0 -0.0351 -0.0484 0.0000 0.0351 0 -0.0138 0.0000 0.0484 0.0138 0 0.0000 -0.0000 -0.0000 -0.0000 0

N=null(L);

a=N(:,1)+N(:,2);
b=N(:,1)+2*N(:,2);

direction=normalize( b(1:3)/b(4)-a(1:3)/a(4) ,'n')

direction = 3×1

0.2242 -0.7894 0.5715

or,

direction=normalize( cross(A(1:3),B(1:3))  ,'n')

direction = 3×1

-0.2242 0.7894 -0.5715

[M]

@Matt J "direction=normalize( cross(A(1:3),B(1:3)) ,'n')"

This method is working good with my problem, but could you please tell me what is the relation between it and the intersected line between these two planes "L"?

I mean how this indicates about the direction of L ?

[Matt J]

The line is parallel to the vector direction.

[M]

@Matt J How to verify that the vector direction is parallel to the line "L" ?

[Matt J]

Plot the line using the parametric form mentioned by Torsten,

P(t) = Pstart + t*direction

and see if it gives the same line as you were getting before.

P1=[177668442.453315	-102576923.076923	0
-102576923.076923	177668442.453315 	-102576923.076923
 0	-102576923.076923	88834221.2266576]/100000000;

P2=[152763459.308716 	-102576923.076923	0
 -102576923.076923	183536536.231793	    -102576923.076923
 0	-102576923.076923	91768268.1158967]/100000000 *[0 1 0;1 0 0; 0 0 1];


A=null([P1,ones(3,1)]);  %plane 1
B=null([P2,ones(3,1)]);  %plane 2
L=A*B.' - B*A.'; %line of intersection

N=null(L);

Pstart=sum(N,2); Pstart=Pstart(1:3)/Pstart(4);
direction=normalize( cross(A(1:3),B(1:3))  ,'n');


plotLine(Pstart,direction); grid on; view(3)
plotPlane(A)
plotPlane(B)
axis([-1,1,-1,1,-1,1]*200)

function plotLine(Pstart,direction)

 t=linspace(-500,+500);

 xyz=num2cell(Pstart + direction(:)*t,2);

 line(xyz{:});
 
end

function plotPlane(H)

 hold on;

  fimplicit3(@(x,y,z)H(1)*x+H(2)*y+H(3)*z+H(4),'EdgeColor','none','FaceAlpha',0.3,'FaceColor','r')

 hold off

end

[M]

Thank you so much

[M]

@Matt J I have a question please if we have more than 3 dimensions, 4d 5d..

Would this method works?

because I tried 4d problem and 'N' size is 5*3

a=N(:,1)+N(:,2);

b=N(:,1)+2*N(:,2);

direction=normalize( b(1:3)/b(4)-a(1:3)/a(4) ,'n')

Also I want to ask why did you multiply by 2 here "b=N(:,1)+2*N(:,2);" ?

and what does a and b denote?

Answer 2

[Torsten]

If you look at the next lines in Matt's code, he creates 100 points on the line.

Thus in the modified code below, Pstart could be taken as a point on the line and d as a direction vector for the line emanating from Pstart.

Did you mean something like this ?

P1 =[177668442.453315 ,-102576923.076923,	0];
P2 =[ -102576923.076923	,177668442.453315 	,-102576923.076923];
P3= [0,	-102576923.076923,	88834221.2266576];
P11= [152763459.308716 ,	-102576923.076923,	0];
P22=[ -102576923.076923,	183536536.231793	,    -102576923.076923];
P33= [0,	-102576923.076923,	91768268.1158967];
 
A=null([[P1;P2;P3],ones(3,1)]);  %plane 1
B=null([[P11;P22;P33],ones(3,1)]);  %plane 2
L=A*B.' - B*A.'; %line of intersection

N=null(L);

t=linspace(-.1,.1);

xyz=N(:,1) + N(:,2)*t;
xyz = xyz(1:3,:)./xyz(4,:);
P1 = xyz(:,1);
P2 = xyz(:,2);

Pstart = P1

Pstart = 3×1

2.2415 -7.8929 5.7147

d = (P2-P1)/norm(P2-P1)

d = 3×1

0.2242 -0.7894 0.5715

[M]

@Torsten why when I change the end point P2 for example P2 = xyz(:,100);

the signs of 'd' changes ?

[Torsten]

t=linspace(-.1,.1);
xyz=N(:,1) + N(:,2)*t;

At t = 0, the orientation of xyz changes.

A line is characterized by a point on it and a direction vector.

The line in your problem e.g. is given by

P(t) = Pstart + t*d (-Inf < t < Inf)

So you see that it doesn't matter that the sign of d changes because t runs from -Inf to +Inf.

[M]

@Torsten Thank you so much, but suppose we have 4d and above, I think we cant use this method

[Torsten]

Thank you so much, but suppose we have 4d and above, I think we cant use this method

to do what ?

[M]

@Torsten to know the direction of the intersected line "L" .

I meant if the planes in 4d and above dimensions

[Torsten]

In 4d, you usually need three planes to get a line as intersection.

The planes are given by

a_i.'*x = b_i

Their intersection is given by the solution of the linear system

a_i.'*x = b_i (1<=i<=3)

Example:

A = rand(3,4);
b = rand(3,1);            % Define three planes in 4d
Pstart = A\b              % Determine a point on the intersection line

Pstart = 4×1

1.9104 0 0.8004 -1.8355

d = null(A)               % determine the direction vector of the intersection line

d = 4×1

-0.3941 0.8664 -0.2635 0.1567

line = @(t) Pstart + t*d   % Equation of the intersection line

line = function_handle with value:
    @(t)Pstart+t*d

line(25)                   % Just an arbitrary point on the intersection line

ans = 4×1

-7.9428 21.6605 -5.7870 2.0814

A*line(25)-b               %Check whether the arbitrary point on the line is in the intersection of the 3 planes

ans = 3×1

1.0e-14 * -0.3331 0.0666 0.2554

[M]

P1 =[252716585.970010	-136769230.769231	0	0];
P2 =[ -136769230.769231	252716585.970010	-136769230.769231	0];
P3= [0	-136769230.769231	252716585.970010 	-136769230.769231];
P4 = [0	0	-136769230.769231	126358292.985005];



P11= [191269260.712188	-136769230.769231	0	0];
P22=[ -136769230.769231	259653876.096803	-136769230.769231	0];
P33= [0	-136769230.769231	259653876.096803	-136769230.769231];
P44=[0	0	-136769230.769231	129826938.048402];

A=null([[P1;P2;P3;P4],ones(4,1)]);  %plane 1
B=null([[P11;P22;P33;P44],ones(4,1)]);  %plane 2

L=A*B.' - B*A.' %line of intersection

L = 5×5

0 -0.0368 -0.0622 -0.0712 -0.0000 0.0368 0 -0.0261 -0.0354 -0.0000 0.0622 0.0261 0 -0.0095 -0.0000 0.0712 0.0354 0.0095 0 -0.0000 0.0000 0.0000 0.0000 0.0000 0

N=null(L)

N = 5×3

-0.0113 -0.4053 0.0000 -0.1871 0.8354 -0.0000 0.7858 -0.0789 -0.0000 -0.5894 -0.3627 0.0000 0.0000 0.0000 1.0000

[M]

@Torsten the plucker matrix method works to get the intersection line between 2 planes in 4d

[Torsten]

2 planes in 4d usually don't have a line, but a plane (2d object in 4d) as intersection.

[M]

How can we relate between your previous parametric method and to get the direction of the intersected line in 4d? @Torsten

[M]

@Torsten, Why No?

@Matt J

[M]

@Torsten could you please suggest the methods that we can get the intersected plane (2d object in 4d) ?

[Torsten]

The usual representation of the plane of intersection is given by all solutions x of a linear system of the form

A*x = b

Here, A is a 2x4 matrix and b is a 2x1 vector.

The rows of A are the normal vectors of 2 hyperplanes in the 4d space.

The vector b somehow represents the distance of these hyperplances to the origin.