Intersection of two lines

Hi, I have four coplanar points P1, P2, P3 and P4 in 3d. I would like to calculate the intersection point among the line passing through P1 and P2 and that passing through P3 and P4.

Moreover I would like to evaluate the angle between the vector connecting such intersection and the origin with z-axis and then plot the vector.

Answer 1

[Matt J]

You need a least squares solution, since lines in 3D do not generally intersect. Assuming your point data are in column-vector form, this is,

c=[P2-P1,P3-P4]\(P3-P1);

P_intersection=P1+c(1)*(P2-P1)

angle=acos(P_intersection(3)/norm(P_intersection))

[Gaetano Pavone]

I have edited my question @Matt J

[Matt J]

I don't see anything new in the phrasing of the post that would change my answer. Did you try it?

You seem to have added that your 4 points are ideally coplanar, but that simply isn't a reliable assumption, in general. If P1...P4 were generated by floating point calculations, they will contain floating point errors that effectively render the points non-coplanar. It doesn't matter, though. My solution should work whether the data is ideally coplanar or not.

[Gaetano Pavone]

I have added a request: plot the vector from the origin to the intersection point

[Matt J]

You can use the line() command to plot the vector. If you want the vector to appear as an arrow, you can use

https://www.mathworks.com/matlabcentral/fileexchange/14056-arrow3?s_tid=srchtitle

or other similar File Exchange offerings.

[Gaetano Pavone]

For

P1=[735;234;1397];
P2=[234;735;1397];
P3=[-265;234;1397];
P4=[234;-265;1397];
c=[P2-P1,P3-P4]\[-P1,P3];
P_intersection=P1+c(1)*(P2-P1);
angle=acos(P_intersection(3)/norm(P_intersection));

it doesn't give me the correct answer.

P_intersection should be [234;234;1397].

[Matt J]

You seem to have had P2 and P3 interchanged.

P1=[735;234;1397];
P3=[234;735;1397];
P2=[-265;234;1397];
P4=[234;-265;1397];
c=[P2-P1,P3-P4]\(P3-P1); %edited
P_intersection=P1+c(1)*(P2-P1)

P_intersection = 3×1

234 234 1397