How to find the line of intersection between the following two planes and plot the intersected line on the same two planes?

How to find the line of intersection between the following two planes and plot the intersected line on the same two planes?

The following points contain the following points:

The first plane:

P1 = 177668442.453315 -102576923.076923 0

P2 = -102576923.076923 177668442.453315 -102576923.076923

P3= 0 -102576923.076923 88834221.2266576

The secod Plane:

P1= 152763459.308716 -102576923.076923 0

P2= -102576923.076923 183536536.231793 -102576923.076923

P3= 0 -102576923.076923 91768268.1158967

[Torsten]

Quite large, your point coordinates. You might want to scale in x,y and z direction by a factor of 1/100000000.

Answer 1

[Matt J]

You can get the line of intersection in Dual Plucker Matrix form as follows:

A=null([[P11;P21;P31],ones(3,1)]);  %plane 1

B=null([[P12;P22;P32],ones(3,1)]);  %plane 2

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

However, for this to work well, you must re-express your P data in larger units, as Torsten suggests.

[M]

@Matt J Thanks but how to plot this line (L) in 3d?

[M]

@Matt J I ploted the intersected Planes, but I want to plot the line that you got on the same figure to verify that this is the intersected line?

[Matt J]

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);

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

xyz=N(:,1) + N(:,2)*t;
xyz=num2cell(xyz(1:3,:)./xyz(4,:),2);

line(xyz{:}); grid on; view(3)
plotPlane(A)
plotPlane(B)
axis([-1,1,-1,1,-1,1]*200)

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]

@Matt J I cant see the line on the same graph, Do you know why?

[M]

@Matt J I dont to rescale the numbers, because they have meaning

[Torsten]

Hint: Look at how P1, P2 and P3 are scaled in Matt's code.

[Matt J]

I dont to rescale the numbers, because they have meaning

What meaning? That's like saying centimeters have more meaning than kilometers.

[Matt J]

In any case, it still works with the P data as originally scaled:

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];


hold on 

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(-0.1,0.1);

xyz=N(:,1) + N(:,2)*t;
xyz=num2cell(xyz(1:3,:)./xyz(4,:),2);

plot3(xyz{:}); grid on



line(xyz{:}); grid on; view(3)
plotPlane(A)
plotPlane(B)
axis([-1,1,-1,1,-1,1]*200)

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]

@Matt J many thanks, very helpful

[M]

@Matt J, I want to ask a question please, Do you know how to change the color of the line? because i want to plot multiple lines in different colors on the same plot. Thanks

[Matt J]

line() takes the same kind of arguments as patch()

Example: line(x,y,'Color','red','LineWidth',3) creates a red line that is 3 points wide.

[M]

@Matt J Thanks much

[M]

Hello @Matt J , I want to ask a question please, I increased the number of points of the planes ,

For example they became 4 instead of 3 , I edited the code as the follwoing, and I got the intersected line

P4 = [0 0 -136769230.769231 126358292.985005]

P44= [0 0 -136769230.769231 127179926.292009]

A=null([[P1;P2;P3;P4],ones(4,1)]); %plane 1

B=null([[P11;P22;P44],ones(4,1)]); %plane 2

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

But I faced errors in ploting part and I didnt know how to edit 'xyz' ?

could you please tell me what is the 'xyz' and based on what I should edit it ?

N=null(L);

t=linspace(-0.1,0.1);

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

xyz=num2cell(xyz(1:3,:)./xyz(4,:),2);

plot3(xyz{:}); grid on

line(xyz{:}); grid on; view(3)

plotPlane(A)

plotPlane(B)

axis([-1,1,-1,1,-1,1]*200)

[M]

@Matt JCould you please help, because you are the only one you know how to modify the code , because you have implemented. Thanks

[Matt J]

But I faced errors in ploting part and I didnt know how to edit 'xyz' ?

I don't think so. The code you posted doesn't even run to the creation of A and B. It also doesn't really make sense that you would be adding a 4th point. Four points are not generally coplanar.

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];
P4 = [0	0	-136769230.769231	126358292.985005]

P4 = 1×4

1.0e+08 * 0 0 -1.3677 1.2636

P44= [0	0	-136769230.769231	127179926.292009]

P44 = 1×4

1.0e+08 * 0 0 -1.3677 1.2718

A=null([[P1;P2;P3,P4],ones(4,1)])  %plane 1

Error using vertcat
Dimensions of arrays being concatenated are not consistent.

B=null([[P11;P22;P44],ones(4,1)])  %plane 2

[M]

P1 =[252716585.970010	-136769230.769231	0	0];
P2 =[ -136769230.769231	252716585.970010	-136769230.769231	0];
P3= [0	-136769230.769231	252716585.970010	-136769230.769231];
P11= [233844467.968633	-136769230.769231	0	0];
P22=[ -136769230.769231	254359852.584018	-136769230.769231	0];
P33= [0	-136769230.769231	254359852.584018	-136769230.769231];
P4 = [0	0	-136769230.769231	126358292.985005];
P44= [0	0	-136769230.769231	127179926.292009];

A=null([[P1;P2;P3;P4],ones(4,1)])  %plane 1

A = 5×1

0.2420 0.4472 0.5843 0.6325 0.0000

B=null([[P11;P22;P33;P44],ones(4,1)])  %plane 2

B = 5×1

0.2657 0.4542 0.5791 0.6227 0.0000

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

L = 5×5

0 -0.0089 -0.0151 -0.0173 -0.0000 0.0089 0 -0.0064 -0.0088 -0.0000 0.0151 0.0064 0 -0.0024 -0.0000 0.0173 0.0088 0.0024 0 -0.0000 0.0000 0.0000 0.0000 0.0000 0

[M]

@Matt J, But I couldnt modify the rest of code (ploting Part)

[M]

@Matt J, I just realized that this problem in 4D ....so we couldn't visualize it

[Matt J]

So problem solved, right?

[M]

Yes, thank you so much

[M]

Hi @Matt J, could you please elaborate to me why did you scale P1 and P2 as the following? why did you transform P2 only??

Also, Why do the planes have six vertices in the plot?

Thanks

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];

Hi @Torsten, if you are available please clarify. thanks

[M]

@Matt J

[Matt J]

could you please elaborate to me why did you scale P1 and P2 as the following?

Because in these lines, the ones(3,1) become vanishingly small compared to the Pij data if you don't scale them.

A=null([[P11;P21;P31],ones(3,1)]);  %plane 1
B=null([[P12;P22;P32],ones(3,1)]);  %plane 2

why did you transform P2 only??

I think I just didn't like the P2 that you gave. I wanted a different P2 data set for the example.

Also, Why do the planes have six vertices in the plot?

Becase the plot box has 12 edges, but the planes only intersect 6 of them.

[M]

@Matt J Thanks for your reply.

Becase the plot box has 12 edges, but the planes only intersect 6 of them.

Regarding this, Is there a way to control the vertices? for example to have only 4 vertices in the plot? or that will affect the meaning or so ? I need the plots only for visualization purposes in my research, having 6 vertices looks weird

[Matt J]

You can use patch to plot whatever portion of the plane you wish.