How to Draw 2 Intersecting Planes
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What's the all-time way to draw 2 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 every bit though our viewpoint has changed simply the planes are nonetheless intersecting at $\pi /2$.
I can't quite piece of work out how to draw 1 or both of them non-vertical in such a fashion equally to brand the angle between them announced to be evidently not a right angle.
Thanks for any help with this!
asked Apr 17, 2012 at 9:35
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ane Respond one
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Here's my attempt, along with a few ideas I've practical in my drawings for multivariable calculus.
- It helps to start with 1 of the planes completely horizontal, or at to the lowest degree close to horizontal-- then everything else y'all draw volition be judged in relation to that.
- Probably the most important thing is to use perspective. Parallel lines, similar opposite 'edges' of a plane, should not be drawn every bit parallel. In an image correctly drawn in perspective, lines that run across at a common, furthermost 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 equally the about edge of the plane. I sometimes utilize thicker or darker lines to indicate the near edge, but perspective is a much more dominant forcefulness. It helps you lot translate the drawing even if it's not perfectly done, equally often happens when I'm cartoon on the board.
- Yous 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 almost normal vectors, I always draw a little plus sign on the plane to anchor them.
- The intersection line of the ii planes can be totally capricious- notice that mine appears parallel with edges of the horizontal plane, but non quite parallel with any edges of my skew airplane. Yous can experiment with different angles and lines of intersection; many of them will yield nice drawings.
answered April 17, 2012 at 14:46
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
$\begingroup$ One skilful method is to take the dot product of their unit normal vectors, and take the arc cosine of that to get the angle betwixt the planes, as in this related question. In particular, the planes are perpendicular iff the dot product of their normal vectors is zippo. Too, the plane $ax+by+cz=d$ has normal vector $(a,b,c)$. $\endgroup$
Apr 17, 2012 at nine:38
$\begingroup$ If you were to look at the intersection from the line of intersection, the planes would clearly announced to intersect at an angle other than 90 degrees(provided they don't intersect at 90 degrees). $\endgroup$
April 17, 2012 at 9:42
$\begingroup$ @bgins - apologies for causing confusion - I meant to inquire almost drawing them, non 'showing' non-orthogonality in the mathematical sense. I've now amended the championship and question to make this clearer $\endgroup$
April 17, 2012 at 9:55
$\begingroup$ @BenEysenbach - unfortunately I can't do that, because I demand to show two distinct points on the line of intersection $\endgroup$
April 17, 2012 at 9:56
$\begingroup$ One way would exist to accept an astute triangle and extend the larger sides into planes, sometthing similar hither. Some other would be to draw several intersecting radial lines and extend them all to planes, perhaps using color, something like here or hither. Lastly, yous might endeavour drawing a parallelopiped (like here) and refer to the planes of the faces. $\endgroup$
Apr 17, 2012 at 10:07