There is a tradition in the use of orthogonal projections to represent 3D objects in printed school math textbooks. On the other hand, perspective projections represent better the way as we "see" real objects with our eyes. Take a look to compare:

cubes in perspective and orthogonal projections

Some people argue that orthogonal projections are better because it is easier to draw with them by hand. But, today, we have technology that allows the construction of 3D objects in our smartphones.

Well, our team is writing a (free) mathematics textbook for High School and this question appeared: is there any research or study that support the choice of a given projection in textbooks?

Any suggestions are welcome!

Thanks, Humberto.

  • 2
    $\begingroup$ Orthogonal is easy to understand, does not differ from your "focusing distance", does not change its shape when rotated in a 3D visualization, skills gathered when reading and drawing orthogonal views can be applied to drafting, although the latter is less important now as drafting is often done on a computer. Orthogonal view is like looking at an object from great distance using telephoto lens. I suggest sticking to orthogonal and leaving fancy perspectives and Dutch angles to artists and architects. $\endgroup$
    – Rusty Core
    Mar 4, 2020 at 21:51
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    $\begingroup$ @RustyCore: Your comment above looks like an answer to the question. Perhaps you could write it as an answer so that answer acceptance and voting can be used. Unless, that is, you feel that it is not sufficiently based on research or a study that the OP requests. $\endgroup$
    – J W
    Mar 5, 2020 at 5:58
  • $\begingroup$ I think, the "drawing by hand argument" is a valid one, even though we do have smartphones. It is important that the students are able to quickly draw a quick&dirty sketch on paper as a thinking aid when solving problems. If they only can do this with a smartphone, they are dependent on the smartphone when solving a problem. $\endgroup$
    – Photon
    Mar 5, 2020 at 7:36
  • $\begingroup$ Orthogonal projections better* preserve the things we care about in mathematics, like angles and parallel lines. *: Better in the sense that, for example, angles aren't preserved, but at least the way they aren't preserved is consistent. $\endgroup$
    – Adam
    Mar 5, 2020 at 14:54

2 Answers 2


I for some reason seem to aesthetically like perspective projection more, but, now that I think about it, I always sketch on the board using orthogonal perspective if I have straight edges. (Though, I might use perspective projection if I'm trying to show an osculating plane?)

I don't know what kind of book you're making but if it is very easily real-world application and could include photographs of real-world objects, then you might consider perspective projection since then you could superimpose the real-world-accurate line drawings/sketch over the photograph and then have the sketch next to it?

enter image description here


I haven't researched the topic yet but if your "front" and "back" are important, then it might be beneficial to consider dotted lines or even no lines for more clear perspectives of what vertex is closest to the viewer.

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Stick with the orthogonal projection. Your students are going to see that much more in courses before/after your course. And in literature in general. It is mildly disconcerting to see the alternate projection when so used to the standard one.

Note, that this argument applies even if the alternate projection is "better". And the same thing applies with the many other questions we have with people obsessed with changing notation or the like. Or on Wikipedia with changing English grammar, spelling, etc. to something they think makes more sense (but is non standard).

There are many areas you can still innovate and improve. One is cost, of course. But I would challenge you to push for quality composition (ZERO typos, grammar errors, etc.). It is frustrating to use even purchased texts that contain errors. And unfortunately, amateur texts tend to be even worse on average. Make it an objective to be four dot oh!

Another area is quality pedagogy. Many amateur texts suffer from poor explanation (for NEW LEARNERS, not math colleagues!), even worse than the norm in professional texts, which have issues also. They tend to read like the grad student or new faculty explaining things to himself, not to his audience (with awareness of new learner limitations). Same thing for exercises/answers/examples/hints. (Squared! Since "doing" is how we learn, not monograph explication.)

Furthermore, you should have a sophisticated knowledge from assessing several other books of ways to skin the cat. This doesn't mean you can never have a new idea. But it is important if you're going to do "one more attempt at an X text" that you have looked at more than one. In other words, not just trying the project from scratch, with no perspective, learning as you go.

And resist the temptation (probably prone by personality to proofy mathematicians, but not people-savvy educators) of wanting to convert the world back to Beta from VHS.


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