How to Draw Multiple Planes
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What's the best mode to draw two planes intersecting at an angle that isn't $\pi /2$?
If I make them both vertical and vary the angle betwixt them, the diagram ever looks as though our viewpoint has inverse but the planes are still intersecting at $\pi /two$.
I can't quite work out how to draw one or both of them not-vertical in such a way every bit to make the angle between them appear to exist obviously non a right angle.
Thanks for any assistance with this!
asked Apr 17, 2012 at 9:35
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1 Reply one
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Here's my endeavour, along with a few ideas I've practical in my drawings for multivariable calculus.
- Information technology helps to offset with one of the planes completely horizontal, or at least close to horizontal-- then everything else you depict volition be judged in relation to that.
- Probably the most important matter is to use perspective. Parallel lines, like opposite 'edges' of a plane, should not be fatigued as parallel. In an image correctly drawn in perspective, lines that meet at a mutual, furthermost point will appear to exist parallel. Notice the 3 lines in my horizontal plane that volition meet far away to the upper-left of the drawing. This forces you lot to interpret the lower-right edge as the nigh edge of the plane. I sometimes apply thicker or darker lines to signal the near edge, simply perspective is a much more dominant forcefulness. It helps you interpret the drawing even if it's not perfectly done, equally often happens when I'thou drawing on the board.
- You can 'cheat' by copying real objects. I started this cartoon by studying my laptop from an odd angle, and reproducing the planes defined by the keyboard and screen.
- Whatsoever extra lines showing the 'grid lines' of each plane volition help. Whenever I talk about normal vectors, I ever describe a niggling plus sign on the plane to anchor them.
- The intersection line of the 2 planes can be totally capricious- discover that mine appears parallel with edges of the horizontal plane, but not quite parallel with whatsoever edges of my skew plane. You tin experiment with different angles and lines of intersection; many of them will yield nice drawings.
answered April 17, 2012 at xiv:46
Jonas KibelbekJonas Kibelbek
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Source: https://math.stackexchange.com/questions/132881/how-best-to-draw-two-planes-intersecting-at-an-angle-which-isnt-pi-2
$\begingroup$ I proficient 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 detail, the planes are perpendicular iff the dot production of their normal vectors is zero. Also, the plane $ax+by+cz=d$ has normal vector $(a,b,c)$. $\endgroup$
Apr 17, 2012 at ix:38
$\begingroup$ If yous were to look at the intersection from the line of intersection, the planes would clearly appear to intersect at an angle other than xc degrees(provided they don't intersect at xc degrees). $\endgroup$
Apr 17, 2012 at 9:42
$\begingroup$ @bgins - apologies for causing defoliation - I meant to enquire about drawing them, not 'showing' non-orthogonality in the mathematical sense. I've now amended the title and question to brand this clearer $\endgroup$
April 17, 2012 at 9:55
$\begingroup$ @BenEysenbach - unfortunately I can't do that, because I need to evidence two distinct points on the line of intersection $\endgroup$
Apr 17, 2012 at 9:56
$\begingroup$ 1 way would be to take an astute triangle and extend the larger sides into planes, sometthing like here. Another would exist to describe several intersecting radial lines and extend them all to planes, mayhap using color, something like here or hither. Lastly, yous might try drawing a parallelopiped (like here) and refer to the planes of the faces. $\endgroup$
Apr 17, 2012 at 10:07