Two plane intersection and angle between 2 planes

I am trying to implement my problems in different ways. So, may be though this question has some relation to some other questions, please answer me.

We know; Intersection of two planes will be given a 3D line. (In case of segments of planes, then we will have a 3D line segment for the sharing edge portion of both planes, and my question is referred with this).

If I need to assign weights for each line, then this can be achieved with respect to the degree of angle between two planes. So, I can say when two planes make 90 degrees, then the produced line is more precise than when the line subtends 150 or close to 180 degrees. So the 90 degree case give one weight and 150 case give another weight. (I think, this weight tells, how accurately model these lines. so, I can consider this also as quality of the lines. maybe, I am wrong.)

So, I want to know some degree of measure to assign weights in order to discriminate the lines in terms of angle between two planes. So, could you please elaborate how I can categorize different weight classes in terms of that angle as I am really poor in geometry? Or maybe there are some other measures like; length of line segment or size or plane segments. Thank you in advance.

• What, precisely, is the "quality" of a line? Sep 1 '11 at 11:35
• @ J. M. i think this term "quality" will not fit for my question (may be). so i modify my post. thanks
– niro
Sep 1 '11 at 11:46

Your considerations about the quality of the intersection of two planes are correct. The intersection will be computed most accurately if the planes are orthogonal. The interection is not defined for parallel planes and hence it will be inaccurate for almost parallel planes.

This can be quantified by looking at the normal vectors $n_1, n_2$ of the 2 planes. The angle $\alpha$ between the planes is the angle between their normal vectors, so: $$n_1 \cdot n_2 = |n_1| |n_2| \cos \alpha$$

In the following I assume that the normal vectors are normalized (their length is 1). One formula for the intersection computes the denominator $$q = 1 - (n_1 \cdot n_2)^2 = 1 - \cos^2 (\alpha)$$ so this $q$ could be a possible measure of the intersection quality: if $q < \epsilon$ then the quality is bad. The problem is then ill-conditioned. The best quality is $q=1$.

We get the same measure by considering the direction vector of the intersection line: $$u = n_1 \times n_2$$ It is $$|u| = |n_1| |n_2| \sin\alpha = \sin \alpha$$ $$|u|^2 = \sin^2 (\alpha) = 1 - \cos^2 (\alpha) = q$$

If $q$ is small then the length of the direction vector is small and the intersection line is poorly defined.

• thank you for the explanation. it is systematic and clear. thanks
– niro
Sep 1 '11 at 18:54
• just i have 2 things to clarify. (01) have you taken |n1| & |n2| as 1? i feel it is not common for all. is that? (02) further, i feel something like this (may be it is wrong). but to learn i would ask you, if i say; when 2 planes intersect at 90 degrees but if the length of the intersection line is shorter than the lenght of the line which makes 20 degree angle (between its 2 planes), then i feel 2nd line is accurate than first as the its length is higher. but, i feel strongest measure is the angle between plane. so then this will give us wrong interpretation. please make clear this. thanks
– niro
Sep 8 '11 at 10:50
• (1) I assumed imlicitely that the normal vectors are normalized, i.e. their length is 1. So, I could have simplified the calculation and concentrate on the angle between the planes. Also the mentioned denominator in the referenced Wikipedia makes this assumtion. I am adding this remark to my answer to make it clear. (2) Considering the intersection of 2 "plane sections" is much more difficult. You are right that if the interection segment is short then the intersection is poorly defined. You might look e.g. here or even at the 2-dim case first. Sep 8 '11 at 14:30
• thanks for the explanation. what about if i only use sin(angle)?, i think, if i can incorperate sine angle, lenght of intersection segment and area of (respective) 2 planar segments, then it would solve the case "plane sections". (actualy, my problem is related to sections.) but i do not know a proper way to combine all above factors together. any explanation plz.
– niro
Sep 8 '11 at 22:08
• I would use $\sin^2()$, but you can use $abs(\sin)$ as well. I think you are on the right track with the other characteristics. Sep 9 '11 at 13:33

Without further information, I would suggest you might take the sine of the angle between the two planes (or its absolute value) as your indicator of how they are perpendicular.

For $90^\circ$ this gives $1$; for $150^\circ$ or $30^\circ$ it gives $0.5$, and for close to $180^\circ$ or $0^\circ$ it gives close to $0$.

• it is simple and robust. i never think this. thank you very much.
– niro
Sep 1 '11 at 14:50