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

6 次查看（过去 30 天）

M 2023-2-5

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1906875-how-to-find-the-orientation-of-the-line-of-the-intersection-between-two-planes

编辑： Torsten 2023-2-6

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

2 个评论
显示无隐藏无

Torsten 2023-2-5

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

M 2023-2-5

编辑：M 2023-2-5

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

请先登录，再进行评论。

请先登录，再回答此问题。

采纳的回答

Matt J 2023-2-5

1
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1906875-how-to-find-the-orientation-of-the-line-of-the-intersection-between-two-planes#answer_1164220

编辑：Matt J 2023-2-5

在 MATLAB Online 中打开

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

6 个评论
显示 4更早的评论隐藏 4更早的评论

Matt J 2023-2-5

在 MATLAB Online 中打开

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 2023-2-5

Thank you so much

M 2023-2-5

编辑：M 2023-2-5

@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?

请先登录，再进行评论。

更多回答（1 个）

Torsten 2023-2-5

2
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1906875-how-to-find-the-orientation-of-the-line-of-the-intersection-between-two-planes#answer_1164190

编辑：Torsten 2023-2-5

在 MATLAB Online 中打开

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