How to Draw Intersecting Vertical Planes
$\begingroup$
What's the all-time way to draw ii planes intersecting at an bending that isn't $\pi /2$?
If I make them both vertical and vary the bending betwixt them, the diagram e'er looks every bit 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 one or both of them non-vertical in such a way as to make the angle between them appear to be patently not a correct angle.
Cheers for any aid with this!
asked Apr 17, 2012 at 9:35
$\endgroup$
5
1 Answer 1
$\begingroup$
Here'south my endeavor, along with a few ideas I've applied in my drawings for multivariable calculus.
- Information technology helps to start with 1 of the planes completely horizontal, or at least shut to horizontal-- then everything else you draw will exist judged in relation to that.
- Probably the near important matter is to use perspective. Parallel lines, like opposite 'edges' of a aeroplane, should not be drawn as parallel. In an epitome correctly drawn in perspective, lines that meet at a common, far-off bespeak will appear to be parallel. Discover the three lines in my horizontal plane that will see far away to the upper-left of the drawing. This forces you to translate the lower-correct border as the near edge of the plane. I sometimes use thicker or darker lines to point the near edge, but perspective is a much more than dominant force. It helps you lot interpret the cartoon fifty-fifty if it's not perfectly washed, equally often happens when I'm 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 actress lines showing the 'filigree lines' of each plane will help. Whenever I talk about normal vectors, I always draw a niggling plus sign on the aeroplane to anchor them.
- The intersection line of the ii planes can be totally arbitrary- notice that mine appears parallel with edges of the horizontal airplane, just not quite parallel with any edges of my skew aeroplane. You can experiment with different angles and lines of intersection; many of them will yield prissy drawings.
answered April 17, 2012 at 14:46
Jonas KibelbekJonas Kibelbek
vi,760 iii gilt badges 27 silver badges 31 statuary badges
$\endgroup$
Not the reply you're looking for? Browse other questions tagged geometry euclidean-geometry or ask your own question.
Source: https://math.stackexchange.com/questions/132881/how-best-to-draw-two-planes-intersecting-at-an-angle-which-isnt-pi-2
$\begingroup$ I adept method is to have the dot product of their unit normal vectors, and take the arc cosine of that to get the angle between the planes, equally in this related question. In particular, the planes are perpendicular iff the dot product of their normal vectors is naught. Besides, 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 await 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 ninety degrees). $\endgroup$
April 17, 2012 at 9:42
$\begingroup$ @bgins - apologies for causing defoliation - I meant to ask virtually drawing them, non 'showing' not-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 tin't do that, because I need to show two distinct points on the line of intersection $\endgroup$
Apr 17, 2012 at 9:56
$\begingroup$ I way would exist to take an acute triangle and extend the larger sides into planes, sometthing like hither. 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 cartoon a parallelopiped (similar here) and refer to the planes of the faces. $\endgroup$
Apr 17, 2012 at x:07