What's the best way to draw two planes intersecting at an angle 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 still intersecting at $\pi /2$.

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

Thanks for any help with this!

  • $\begingroup$ One good method is to take the dot product of their unit normal vectors, and take the arc cosine of that to get the angle between the planes, as 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 $(a,b,c)$. $\endgroup$
    – bgins
    Apr 17 '12 at 9:38
  • $\begingroup$ If you were to look at the intersection from the line of intersection, the planes would clearly appear to intersect at an angle other than 90 degrees(provided they don't intersect at 90 degrees). $\endgroup$
    – Ben
    Apr 17 '12 at 9:42
  • $\begingroup$ @bgins - apologies for causing confusion - I meant to ask about 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 '12 at 9:55
  • $\begingroup$ @BenEysenbach - unfortunately I can't do that, because I need to show two distinct points on the line of intersection $\endgroup$ Apr 17 '12 at 9:56
  • 1
    $\begingroup$ One way would be to take an acute triangle and extend the larger sides into planes, sometthing like here. Another would be to draw several intersecting radial lines and extend them all to planes, perhaps using color, something like here or here. Lastly, you might try drawing a parallelopiped (like here) and refer to the planes of the faces. $\endgroup$
    – bgins
    Apr 17 '12 at 10:07

Here's my attempt, 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 will be judged in relation to that.
  • Probably the most important thing is to use perspective. Parallel lines, like opposite 'edges' of a plane, should not be drawn as parallel. In an image correctly drawn in perspective, lines that meet at a common, far-off point will appear to be parallel. Notice the three lines in my horizontal plane that will meet far away to the upper-left of the drawing. This forces you to interpret the lower-right edge as the near edge of the plane. I sometimes use thicker or darker lines to indicate the near edge, but perspective is a much more dominant force. It helps you interpret the drawing even if it's not perfectly done, as often happens when I'm drawing on the board.
  • You can 'cheat' by copying real objects. I started this drawing by studying my laptop from an odd angle, and reproducing the planes defined by the keyboard and screen.
  • Any extra lines showing the 'grid lines' of each plane will help. Whenever I talk about normal vectors, I always draw a little plus sign on the plane to anchor them.
  • The intersection line of the two planes can be totally arbitrary- notice that mine appears parallel with edges of the horizontal plane, but not quite parallel with any edges of my skew plane. You can experiment with different angles and lines of intersection; many of them will yield nice drawings.

not normal planes


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