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michel123456

Why 41,25 stencil ?

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Posted (edited)

IMG_20180525_105954x.thumb.jpg.877dba24590d217a49859a76cc3030c1.jpg

Today I retrieved from a drawer this drawing instrument, called "stencil" or Schablon  in German, that we used when drawing by hand (we stopped about twenty years ago).

Looking at it, I noticed this strange angle 41,25 degrees. Combined with the other one 7,10 degrees.

And I really could not recall why? What is so special with those 2 angles?

Internet came to the rescue and I found it, but just by curiosity, can you figure out what is so special with those 2 angles?

IMG_20180525_110006.thumb.jpg.10941f12e28b5ebdb123778888f830c0.jpgIMG_20180525_110020.thumb.jpg.3ee1fb6ee89bc6cdea94bd6266f853bd.jpg

Edited by michel123456
adding images

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Posted (edited)

42/7 are the standard angles for dimetric projections, not sure about what is on there though. Also, it is a "Schablone" (with "e", though technically I should not complain about typos, if it was one). 

Edit, just checked some drawings and it seems that there are variations in dimetric projections, including 41.25/7.11.

Edited by CharonY

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Posted (edited)

Right. Sorry for the missing e in Schablone.

I couldn't figure the reason for those angles, but the German Wiki helped a bit.

Quote

 

Mathematischer Hintergrund:

  • Eine Ingenieur-Axonometrie entspricht einer senkrechten Parallelprojektion auf eine Ebene mit dem Normalenvektor (= negativer Projektionsrichtung) (7,1,1)⊤{\displaystyle ({\sqrt {7}},1{,}1)^{\top }}{\displaystyle ({\sqrt {7}},1{,}1)^{\top }} mit anschließender Skalierung um den Faktor 322≈1,06{\displaystyle {\tfrac {3}{2{\sqrt {2}}}}\approx 1{,}06}{\displaystyle {\tfrac {3}{2{\sqrt {2}}}}\approx 1{,}06}. Der Grundriss des Normalenvektors schließt mit der x-Achse einen Winkel von arccos⁡(722)≈20,7∘{\displaystyle \arccos({\tfrac {\sqrt {7}}{2{\sqrt {2}}}})\approx 20{,}7^{\circ }}{\displaystyle \arccos({\tfrac {\sqrt {7}}{2{\sqrt {2}}}})\approx 20{,}7^{\circ }} ein. Der Winkel gegenüber der x-y-Ebene beträgt arccos⁡(223)≈19,47∘{\displaystyle \arccos({\tfrac {2{\sqrt {2}}}{3}})\approx 19{,}47^{\circ }}{\displaystyle \arccos({\tfrac {2{\sqrt {2}}}{3}})\approx 19{,}47^{\circ }}. Die exakten Winkel zwischen den Achsen sind:
α=arccos⁡(−74)≈131,4∘{\displaystyle \alpha =\arccos(-{\tfrac {\sqrt {7}}{4}})\approx 131{,}4^{\circ }}{\displaystyle \alpha =\arccos(-{\tfrac {\sqrt {7}}{4}})\approx 131{,}4^{\circ }}
β=arccos⁡(−18)≈97,18∘{\displaystyle \beta =\arccos(-{\tfrac {1}{8}})\approx 97{,}18^{\circ }}{\displaystyle \beta =\arccos(-{\tfrac {1}{8}})\approx 97{,}18^{\circ }}.

 

 
(edit) oops, mathematics destroyed. See below & wiki article
5b0a7084ae49d_ScreenShot05-27-18at11_46AM.thumb.JPG.77523bde08548ca4aa9f68d2946f300a.JPG
Where 2α + β =360 but I must admit the German explanation is far from clear to me.
Edited by michel123456
math destroyed

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