X Marks the Spot (XII)

It was so cold overnight that even with a block heater plugged in my diesel truck would simply not start this morning. So, I won’t be going into the shop today, or anywhere else for that matter, and have therefore some time to do another installment in this series. Previous posts are archived to the right of the page. This is the penultimate post, and is no more difficult than the preceding two posts in terms of what will be done today.

Where I last left off, I have completed the drawing of the lines for the front and rear planes of stick ‘a’ upon the unfolded view of stick ‘b’:

Now it is time to deal with the remaining two faces of stick ‘a’, and trace their projections onto that unfolded view of stick ‘b’.

Now in post 8, I developed one of the faces on stick ‘a’, a face which I colored blue. In the same post I developed the remaining face of stick ‘a’, and colored it yellow. Today I will likewise develop both of those faces, as the technique is identical for both and in fact the process is much that same as was detailed for the other two faces already developed. Here then are the two planes to be dealt with today:

Recall that the blue plane, traced along the ground, crossed the plan view of stick ‘b’ at slope, generating points 9, 10, and 11:

Likewise, the yellow plane, running parallel to the blue one, generates points 13, 14, and 15 from its ground trace intersecting the plan view of stick ‘b’ at slope.

Then, as the reader may recall, points 9, 10, and 11 projected 90˚ across to meet the ground line of stick ‘b”s elevation view, giving points 9′, 10′, and 11’. These points in turn were projected across the elevation view to give points 9″, 10″ and 11″. In the next drawing I connect points 9″, 10″, and 11″ to form a picture of the cutting plane, in red to the left, on the elevation view of sitck ‘b’:

A different vantage point to see the red cutting plane on stick ‘b”s elevation:

Now, in exactly the same manner as in the previous two posts, we project over from the red cutting plane to the unfolded view of stick ‘b’:

Where out projection lines from points 9″, 10″ and 11″ meet the arris lines on the unfolded view, I mark small circles to indicate the intersection points.

Next, we connect the dots as before, remembering that the arris on each side of the unfolded view is one and the same:

That zig-zagging line then is the red plane from stick ‘a’ as it crosses the various faces of stick ‘b’. If I clean up the view a little, you can see the three plane lines now formed on the unfolded view of stick ‘b'”

Next I do exactly the same thing with the green plane, which meets the elevation view of stick ‘b’ at points 13″, 14″, and 15″ – I connect these points to form the cutting plane:

Again, project over to the unfolded view and connect the dots:

There we have it, and as expected, these lines from the green plane run parallel to the lines from the blue one:

That completes the projection lines of all 4 planes from stick ‘a’ onto stick ‘b’. The final step, and subject of the next post in this series, is to mark out the material to be cut away from stick ‘b’ so that it can be fitted to stick ‘a’. That step is probably the hardest of all the steps in this problem, and requires careful consideration and visualization. See you next time.

2 Replies to “X Marks the Spot (XII)”

  1. The suspense is killing me! Hurry and give the cutting specs, the temporary supports are buckling! I hope you post more problems like this in the future. Am loving the Japanese Carpentry drawing I bought recently!

  2. Paul,

    I'm so pleased to read of your enthusiasm for this thread! I might do more like it in the future, though I, frankly, fear the audience for it is rather limited and may repel some readers altogether. Also, great to hear that you are enjoying the Carpentry Essay – more of those to come this year.


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