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What'southward the best way to depict two planes intersecting at an bending that isn't $\pi /2$?

If I make them both vertical and vary the angle between them, the diagram always looks as though our viewpoint has changed but the planes are all the same intersecting at $\pi /2$.

I can't quite work out how to draw i or both of them non-vertical in such a way as to make the bending between them appear to be obviously not a right angle.

Thanks for any help with this!

asked Apr 17, 2012 at nine:35

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  • $\begingroup$ One good method is to have the dot product of their unit normal vectors, and have the arc cosine of that to get the angle between the planes, every bit in this related question. In particular, the planes are perpendicular iff the dot product of their normal vectors is zero. Also, the plane $ax+by+cz=d$ has normal vector $(a,b,c)$. $\endgroup$

    Apr 17, 2012 at nine:38

  • $\begingroup$ If y'all were to look at the intersection from the line of intersection, the planes would clearly appear to intersect at an angle other than ninety degrees(provided they don't intersect at 90 degrees). $\endgroup$

    Apr 17, 2012 at ix:42

  • $\begingroup$ @bgins - apologies for causing confusion - I meant to ask almost drawing them, not 'showing' non-orthogonality in the mathematical sense. I've now amended the title and question to make this clearer $\endgroup$

    Apr 17, 2012 at nine:55

  • $\begingroup$ @BenEysenbach - unfortunately I can't practise that, because I need to prove two distinct points on the line of intersection $\endgroup$

    Apr 17, 2012 at 9:56

  • $\begingroup$ One way would exist to accept an acute triangle and extend the larger sides into planes, sometthing like hither. Another would exist to draw several intersecting radial lines and extend them all to planes, possibly using color, something like here or here. Lastly, you might try drawing a parallelopiped (similar hither) and refer to the planes of the faces. $\endgroup$

    April 17, 2012 at 10:07

ane Answer i

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Here'due south my try, along with a few ideas I've applied in my drawings for multivariable calculus.

  • It helps to start with one of the planes completely horizontal, or at least close to horizontal-- then everything else you draw volition be judged in relation to that.
  • Probably the most of import matter is to use perspective. Parallel lines, similar reverse 'edges' of a plane, should not be fatigued as parallel. In an image correctly drawn in perspective, lines that meet at a mutual, far-off point will announced to be parallel. Notice the three lines in my horizontal aeroplane that will meet far abroad to the upper-left of the drawing. This forces you lot to interpret the lower-correct edge every bit the virtually edge of the airplane. I sometimes employ thicker or darker lines to point the most border, but perspective is a much more than dominant force. It helps you interpret the drawing even if it's not perfectly done, equally often happens when I'grand cartoon on the board.
  • You lot can 'cheat' by copying existent objects. I started this drawing by studying my laptop from an odd angle, and reproducing the planes defined by the keyboard and screen.
  • Whatever extra lines showing the 'filigree lines' of each plane will help. Whenever I talk about normal vectors, I always depict a petty plus sign on the plane to anchor them.
  • The intersection line of the two planes can be totally arbitrary- find that mine appears parallel with edges of the horizontal plane, simply not quite parallel with any edges of my skew airplane. Y'all can experiment with different angles and lines of intersection; many of them will yield nice drawings.

not normal planes

answered April 17, 2012 at 14:46

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