dimetric drafting, ellipses

[peterl95124]

As an admirer of Kozo Hiraoka's books I've been investigating how to do the drafting he does,
many of his 3-D draftings appear to be 7-deg / 41-deg "dimetric" draftings,
so I'm trying to do that for some of my engine designs

you can get a better idea what "dimetric" drafting means at this website,
one of the best I've found but like all the rest contains zero info on circles/ellipses
(and you have to skip over, don't get distracted by, the useless info on aspect ratios)
https://www.andrew.cmu.edu/user/ramesh/teaching/course/48-175/lectures/10.AxonometricProjections.pdf

the straight lines aren't a problem, 7-deg is 1:8 slope, and 41-deg is 9:10 slope,
so I don't even use a protractor, but circles have to be drawn as ellipses and there's the rub

strangely, even though 7-deg / 41-deg is pretty common, partly because the angles are easy,
and also party because the foreshortening factors are easy (Z = 1:1, X = 1:1, Y = 1:0.5),
I can't find any books or webpages that say what ellipse templates to use.

So I figured it out the hard way, I derived what rotation matrices result in the 7-deg and 41-deg
drafting angles, then what those rotation matrices did to the viewing angles to the circles
which dictates what ellipse template to use.

What I got out of the calculations was a "20 deg template" for the side and top,
and a "60 deg template" for the front face,
as you can see in the picture of my drafting below, and they seem to work pretty well.

I am still gobsmacked and befuddled that either no one has ever done this before,
or that my google searching skills are so bad, not sure which...

I drew this dimetric cube a long time ago (to prove to myself that the top and left side were identical parallelograms), and when the ellipse templates arrived in the mail they didn't go large enough to fill a face or even a quarter of a face, but I'm still pretty satisfied that the calculations appear to have worked.

PS, there's a secret to using the ellipse templates, they have to be angled correctly and its not obvious how, the "minor axis" of the ellipse (the short axis) has to be parallel to the "other" axis of the cube, for example the top of the cube is the X-Y plane so the minor axis of the ellipse has to be parallel to the Z axis, the left side of the cube is the Y-Z plane so the ellipse on that side has its minor axis parallel to the X axis, non-intuitive but it works, and makes sense after you realize what way the circle is tilted when becoming an ellipse.

 

[Jasonb]

I just went looking for LINEX templates as they used to do a whole range of different angles but it does not look like they are even made these days. ebay has listings for them as "Vintage" Used to have a Sataday job in a drawing office suppliers and sold loads but just like technical pens they have become obsolete.

The ones for this type of use were marked with major & minor axis as well as the tangent points much like the isometric ones that can still be found but not sure if they did one for this specific projection.

I suppose a couple of accurately constructed ellipses could then be enlarged & reduced then arranged on a sheet for water jet cutting from plastic or maybe even resin printed. Bot if your angled ones give a good enough result then stick with them

 

[gbritnell]

A looong time ago a drafting tool was manufactured. It was called an ellipsograph. It had 2 rails at right angles to each other. On the rails were followers. Attached to the followers were arms that could be adjusted along their length. The 2 rails had 3 legs, one at each corner. They had tiny points on them that would keep them in place while using the tool. On the one arm was a head that held an inking pen. You could draw any size or angle of ellipse within the limits of the tool. I built one many years ago just to see how it would work. Being in the design/drafting CAD field for most of my life i was always in sheer admiration of Kozo's work. Not just his drawings but the processes and calculations he composed.

 

[Jasonb]

just found these which has the 7.5 & 42 deg angles though limited range of sizes

https://www.amazon.co.uk/Perspective-Isometric-Dimetric-Cavalier-Projection/dp/B003MFM1MU

https://www.amazon.co.uk/Aristo-AR5085-Axonograph-Dimetric-Template/dp/B004FQVQBW