Math Is Fun Forum

Rouelibre

Member

Registered: 2009-07-14

Posts: 1

How to measure the maximum angle between 2 planes

Hello,
I would like to know if anyone can help me derive a function for estimating the maximum angle between two planes.

Here's the situation:
I have a tribrach, which is a piece of survey equipment that is used to plumb a level instrument atop a tripod.  It is essentially constructed of two equilateral triangles, one above the other, and connected at their vertices via turnbuckles that can be turned to change the vetical distance between them.  Please see uploaded image for a top and side view.

I would like to somehow measure the separation of the two planes (the two plates of the tribrach) with calipers, and compute the maximum angle between them.

I know that if I take any three measurements of distance, 120 degrees apart (lets call them z1, z2, and z3), their average is the distance between the two plates at their center (call it L).

I can come up with three functions describing the three angles relative to each of the three measuring points:
Say r is the distance between the center of the tribrach and the side of the tribrach that was measured.  I can come up with:

rsinθ1 = abs[L-z1]
rsinθ2 = abs[L-z2]
rsinθ3 = abs[L-z3]

Is there a function that can use this information to solve for the maximum angle between these two planes?

Thank you in advance for any insights!

juriguen

Member

Registered: 2009-07-05

Posts: 59

Re: How to measure the maximum angle between 2 planes

If I understood correctly, you need to calculate what I found it's called the "Dihedral angle" between two planes, defined as the one they form viewing the planes along their line of intersection:

http://en.wikipedia.org/wiki/Dihedral_angle

According to that article from Wikipedia, you can calculate the angle either from the pairwise cross product of vectors defining the planes, of from 3 points that characterize each plane.

The points that you can use would be those of the "sub-triangles" formed to measure z1, z2 and z3, using the system and origin of coordinates that you like (see attached picture).

With the 3 points for each plane you can directly use the "pseudo-code" that is included in the Wikipedia's article, which I haven't tried to understand yet, but hope will be straightforward!

Jose

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“Make everything as simple as possible, but not simpler.” -- Albert Einstein