X Marks the Spot (VIII)

In the previous two posts in this series we have started developing the lines of intersection between the two sticks of wood that form the problem. Readers will have no doubt seen the similarity between the method used in post VII and post VI, as the planes and their traces were parallel to one another. Thus the development seen in post VII was quite easy to accomplish as all projection lines had to run parallel to those derived in the sixth post.

Today we will develop the planes and projection for the remaining two sides of piece ‘a’. You will see that the method is no different than what has been shown already, so I anticipate this post will no present an appreciable difficulty for those readers following along with the drawing work.

Here’s where things stood after the last session, where we developed the green face of stick ‘a’:

Which produced this series of lines on the elevation view of stick ‘b’:

Now, be forewarned that the drawing is about to become quite a bit more congested with lines, so you really have to stay on top of which line is which, and for those working with pencil and paper, it would be a good idea to use different colors for the projection lines from each given face.

Today, we will start by dealing with one of the two remaining faces of stick ‘a’ – a face which I have colored blue in the picture below, and extended into a larger plane, as before:

Just as before, I make the blue plane disappear, leaving its trace along the floor:

That trace along the floor intersects, just as the previous plane’s traces, the plan view of piece ‘b’ at elevation, giving points 9, 10, and 11:

Points 9, 10, and 11, in turn, project over at 90˚ to the axis of piece ‘b”s plan, to intersect the floor line of piece ‘b’ in elevation view, giving points 9′, 10′, and 11′.

Next, we need to obtain a point at the top of piece ‘b’ which we can connect to so as to obtain a line of the blue plane crossing piece ‘b’. The steps for this are just as we did in the previous two posting – we determine which line of the top cut of piece ‘a’ is to project, then run it over to the side of piece ‘b’ in plan:

In the above picture note point 12, which is the point where the projection from the white diamond (the top cut) meets the side arris of stick ‘b’. From point 12, the line reflects at 90˚ to the axis of stick ‘b”s plan, to produce point 12’:

Now all that remains is to connect point 12′ with the appropriate point out of the group 9′, 10′, and 11′ – the point which also corresponds to the same arris line as did point 12. Looking at it carefully, you should find that the correct choice was point 11′:

Now we can project from points 9′ and 10′ lines across the elevation of stick ‘b’, lines which are of course parallel to the line going from 11′ to 12′:

I have also labeled the pertinent points of intersection between these three parallel lines just drawn and the elevation view of stick ‘b’, namely points 9″, 10″ (2 points), and 11″. Note, as with the previous two rounds of establishing projection and points of intersection, that the line giving point 10 on the plan view is actually atop two points at once; thus, point 10, projected over to point 10′, then becomes two discrete points, both labeled 10″, on the elevation view.

if you look closely, you’ll see that one of the points, point 9″, seems to be in the same location as point 7″ – close up view reveals that while they are close, they are not in fact sharing the exact same territory:

Now I’ll repeat the entire process one more time for the one remaining face on stick ‘a’ not dealt with so far:

This face, and the plane I’ve formed off of it, is colored yellow in the above drawing.

Again, we remove the plane and follow its ground trace along to see where it meets the plan view of stick ‘b’ in elevation, like this:

This has formed points 13, 15, and 15, from which we turn 90˚ to the arris lines of the plan and project over to the ground line for stick ‘b’ in elevation view, giving points 13′, 14′, and 15′.

Now to find the lines on the elevation view- we could draw lines parallel to the one we found between points 11′ and 12, or we can project over from the white diamond that is the top cut of stick ‘a’, just as before, here forming point 16:

From point 16, the line reflects back 90˚ to the arris of the stick, in plan, and moves up to connect with the same line as our points 4′, 8′, 12′, and now 16′:

From there it is again the same matter of connecting the dots and establishing our parallel lines on the elevation view of stick ‘b’, giving points 13″, 14″, and 15″:

A close up – note points 13″, 14″ (2 places) and 15″:

Well that brings to a close that phase of the descriptive drawing process. We’ve established a whole mess ‘o points on that elevation view of stick ‘b’ – 12 points altogether. In the next post in this series we’ll start connecting those points to one another so you can see what, if any, point there has been to all this exercise. I realize that many might be feeling a little overwhelmed with the profusion of lines, and I ask for your patience and can assure you that these line do in fact ‘untangle’ and will make good sense soon enough.

Thanks for coming by the Carpentry Way today, checking in on the first post of 2011. Comments always welcome.