I am looking for textbooks pertaining to drawing intersecting planes, intersecting point (among others) as follows.

enter image description here

I am not sure this topic is taught in high school around the globe. That is why I am a bit difficult to find the book in bookstores.


As a reference:

In some secondary mathematics curricula, there is a topic called Stereometry that deals with investigating the position and finding the intersection, angle, and distance of lines and planes defined within a prism or pyramid. Coordinate system is not used. The metric tasks are solved using Pythagoras’ theorem, trigonometric functions, and sine and cosine rules. The basic problem is to find the section of the figure by a plane that is defined by three points related to the figure. In this article, a formula is derived that gives the positions of the intersection points of such a plane and the figure edges, that is, the vertices of the section polygon. Spreadsheet implementations of the formula for cuboid and right rectangular pyramids are presented. The user can check his/her graphical solution, or proceed if he/she is not able to complete the section.

It is quoted from this link.

  • $\begingroup$ Back in the Olden Days (before desktop computers) there would be one or more courses on "Mechanical Drawing" which taught how to do this. (Now completely replaced by courses on CAD/CAM.) So, maybe look for textbooks called "Mechanical Drawing" from before 1980, if you can still find them. $\endgroup$ – Gerald Edgar May 16 at 13:08
  • $\begingroup$ Or perhaps "Engineering Drawing." $\endgroup$ – Joel Reyes Noche May 16 at 13:22
  • 1
    $\begingroup$ @JoelReyesNoche Engineering Drawing is more about correct labelling, about how the arrows look and where the numbers are placed and how to indicate say a section vs an inner cutout. I think that the OP asks more about solving where the section plane will cut a 3D figure, which is more about stereometry or solid geometry, and less about drafting to industry standards. $\endgroup$ – Rusty Core May 16 at 16:25
  • $\begingroup$ You mean drawing by hands or with a computer like tikz or geogebra? $\endgroup$ – Paracosmiste May 16 at 19:36
  • $\begingroup$ @Paracosmiste: Tools do not matter. Algorithm is the main. I did the above diagram with PSTricks. $\endgroup$ – Money Oriented Programmer May 17 at 4:11

A quick Google search or two (various keyword combinations) shows some old pedagological articles on teaching solid geometry and the importance of drawing. You will need to create a free JSTOR profile to read them. Also mechanical drafting textbooks (more often the ones called intermediate) treat solid geometry drawing as a typical test problem (and show how to do it using construction lines). There's a YouTube video of a drafting teacher showing how to make a correctly dimensioned drawing of a square pyramid truncated by an angled plane.

I didn't find a quick canned answer to your question. But there are several decent resources that if you parse/combine will help you if you take the effort.

One example: https://www.youtube.com/watch?v=Cghf62w6f_k

Another: https://www.youtube.com/watch?v=kohKsPo59Ts (at second 14, he mentions a textbook that perhaps you can track down)

I don't have enough of an account to assemble all the links...but really I think you benefit from doing the simple Google research/reading yourself.

I think the ones from a drafting coverage are better and more detailed than those from a geometry slant. There is a little assumption that you will have learned, be interested in overall drafting though.

  • 1
    $\begingroup$ What textbook at 14s? $\endgroup$ – Paracosmiste May 17 at 17:22

This is not an answer and will not help. But it may be useful to know there are webpages that need nothing more than a browser to interactively explore cross-sections: Here is one based on Geogebra.

          Image from here.
The underlying calculations, however, are not evident.


Can't help myself. Second answer.

One more cool video that even almost has calculus insights. Or anyhow a bunch of math-y insights. Instructor works more on the drawing, doesn't discuss the math insights...but y'all can see them.


He's drawing a helix. To start with there is an issue of projecting a cylinder (which is a circle in reality (extended up) onto a page (2-D surface). So he draws the construction lines. The key insight is that when you stare at a cylinder (beer can, e.g), you see half of it and you see the "middle" straight on. But as you move closer to the edges, there is more and more of a ratio between apparent length and length on the cylinder. For example, the lettering is easy to read in the middle, but more and more visually distorted because of foreshortening as you move to can edges. At the edge (or tangent of your sightline if you are close enough for noticeable parallax), this foreshortening becomes infinite.

So anyhow, these construction lines he draws, help to give the varying distances in flat view, for the curved surface. Thus when he draws the helix, it has to be curved, not a straight line to connect the construction lines. He could draw segments, but knows that he is approximating a curve of steepening slope as he nears the "edge". Of course in reality, the helix has a constant vertical slope. But as viewed or projected, it becomes infinite (vertical), right at the edge of the cylinder.

P.s. You might just track the YT video guy down and send an email and ask for book advice. He's not just a shop teacher. Has other videos showing pentagon construction with compass. So I think he will be sympathetic to your geometry aims, not just machine design drawing skills.


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