Friday, April 27, 2012

Angle Between Intersecting Planes

In a practical sense, there are a few different ways that you can go about determining the angle between two planes.  But before we get to specific methods and tools, let's try to lay the ground work  let's try to understand what we really mean by an angle between two planes.

Definition:  Angle Between Intersecting Planes

Two non-parallel planes intersect at a line.  The angle between intersecting planes is measured perpendicularly to this line of intersection.  In more detail, we can say, for planes P1 and P2 with line of intersection L:
1. The vertex of the angle between the planes is any given point on the line of intersection (that is, just pick one).
2. One arm of the angle is coincident with P1 and the other arm is coincident with P2.
3. Both arms of the angle are perpendicular to L.
Discussion of Definition

We note in passing that there is no need for the two arms of the angle to have any certain length nor is there a need for the lengths to be constrained by any relationship concerning relative lengths  they may differ.  Items 1 and 2 (above) should come as no surprise, but not everyone will immediately understand item 3.  Indeed, if you consider only the two trivial cases  (i) two coincident planes (technically excluded from discussion here) or (ii) perpendicular planes  item 3 appears superfluous.  To see that it is necessary, consider  planes which meet at a 45° angle (as an example).  Imagine putting a 45-45-90 triangle in place so that two of its edges are in contact with a plane each.  Suppose you made one of those planes rest on the top of the triangle and allowed that plane to rotate freely around the line of intersection (L).  Now, keeping the vertex of the triangle (the vertex on the line of intersection) in place and rotating the triangle around that point, you will find that the plane which is resting on the triangle will lift up  the angle between the planes will increase.  By the time you have rotated the triangle so that the vertical edge is against the "resting" plane, the angle between the planes will be 90°.
Here is a video where I demonstrate this scenario:

Perpendicular Angle Between Planes vs. Non-perpendicular Angle Between Planes
(Criteria 1, 2, and 3) vs. (Criteria 1 and 2)

In the above scenario we used the same triangle at different angles and changed the angle between the planes.  Now let's leave the planes be and change the triangle.  Let's draw the orthographic views of planes (represented as rectangles) intersecting at angle i, according to our criteria and draw two triangles in place:  one at 90° to the line of intersection and one at θ to the line of intersection.

I've been lazy in my drawing as the angle for β that I really want is in the plane of the triangle that's on an angle and not the angle of the orthographic projection of that triangle (which is what I have just drawn).  But bear with me.  The angle of intersection is given as i and we take the dimensions L and θ to be given.  From this information we can find formulae for R, h, and β.  Here are some formulae we can take directly off of the drawing:

$\sin \theta = \frac{R}{L}$ $\tan \beta = \frac{h}{L}$ $\tan i = \frac{h}{R}$

Manipulating these equations gives

$R = L \sin \theta$ $h = L \sin \theta \tan i$ $\tan \beta = \sin \theta \tan i$

Observe that β  i.

Calculations Without a Scientific Calculator

You can obtain a rise and run for β without a scientific calculator by using a little measurement and geometry.  Suppose you are able to obtain a rise and run for each of angles i and θ.  You might do this by measuring either a shop drawing or a physical situation.  Choose the run arbitrarily and measure the rise which corresponds to it.  The hypotenuse (for θ) can either be measured or calculated using the Pythagorean theorem.  The formulae for the rise and run of β are simply

rise of β = (rise of θ) × (rise of i)

run of β = (hypotenuse of θ) × (run of i)