Old-school drafting meets complex curves
March 8, 2013 8:20 AM   Subscribe

How do I draw a 3D curve in 2D? Or rather, how do I algorithmically draft an arbitrary 3D curve from a 2D perspective?

I have a line that curves in three dimensions simultaneously and want to figure out how to determine how to display it in two dimensions (i.e. drafting on paper) without perspective (i.e. vanishing points).

I've run into this issue a couple of times and have a prior unfinished project waiting on developing this skill.

This time, I am designing a wooden window sash and have muntin sections with a moulding profile(the 1915 muntin profile shown here) that overlay and meet each other at various angles. I am trying to determine how to draft the coped profiles (this is the curve in 3D space) from an isometric view. The "bottom" section will be a straight cut with a coped overlay on the "top" section. To add insult to injury, I also have an area where three muntin sections meet in an even more complex profile.

I know craftsmen used to be able to do this by hand (so I should too, goddamnit!). I'm moving from 2D to 3D modeling of this design and want to determine the profile before modeling it. Again, I'm also interested in the big-picture method for my other stalled and future projects.

Yes, I could cheat and use a 3D composite of these shapes to figure out how it "should" look and back-fill the construction, but what I'm primarily interested in is how to do this by hand. My goal is to design forward instead of empirically determine how to do it and extract the design from that.

I've had some luck with "mechanical drawing" resources (mostly books from the early 1900s), but figured that, of anyone, MeFites will know who and where to look for the information.

For example, there is a way to use a compass at specific points to draw a (2D) circle isometrically, and I want to be able to do the same for an arbitrary 3D curve.

Anyone have resources, keywords, and/or tutorials on how to do this algorithmically?
posted by bookdragoness to Media & Arts (5 answers total) 4 users marked this as a favorite
posted by oceanjesse at 8:45 AM on March 8, 2013

My dim memory of drafting in high school tells me that it usually involved picking points along the curve from regular sections on the isometric views. Can I ask why you need to do this? Unless you're doing some CNC millwork, I'm wondering if you need it at all.

In joinery, most marking is transferred from one component to another, rather than being computed as separate, independent forms. In Roy Underhill's tutorial on simple sash making you can see creates his 45 degree angles with guide blocks and then copes one piece to fit around the other instead of trying to create two matching mitres.
posted by bonobothegreat at 8:50 AM on March 8, 2013 [1 favorite]

The math required to project 3 dimensions into two is really complex. Look at this page and see if you still want to do this.

You can use any 3D-modelling software package to model your curve and then project it into an isometric view if you just want the output.
posted by tylerkaraszewski at 8:54 AM on March 8, 2013

OpenGL implementations provide orthographic projections. Put your 3D object ("model") into an OpenGL world and apply, among other things, an orthographic projection transformation. So you might look at the source code to open source implementations (example) to see how the math is done.
posted by Blazecock Pileon at 9:09 AM on March 8, 2013

Fun to read, both the post and the comments. I'm old enough to have learnt to do this, and have done it a number of times. For me, it took years of practice. And then 3D-modelling came up. Basically bonobothegreat is right. 90% of the time, one would avoid the actual 3D joints, and leave that up to fitting on site. Cheaper and in many senses better. Obviously, that had a lot of design consequences, and sometimes, one would want to make a special detail. Then one would mold the detail in the workshop, measure it with tools for measuring points on a 3D surface, and draw the sections in a accordance with the measurements, maybe adjusting for machinery properties. One had a ton of curves for that - I don't know the English expression, here they are called ship-builders' curves, but after 10.000 hours of practice, people would be able to draw moldings and profiles on free hand. (As well as straight lines. Try that for starters. If you can't draw straight lines of 80 cm, you are not going to be able to draw curves). The furniture people were always the best at this
posted by mumimor at 9:46 AM on March 8, 2013 [2 favorites]

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