I keep forgetting I am writing for an international audience here! Having referred to Descriptive Geometry (DG) often, I doubt that everybody associates similar mental images with it. Anybody remembering the trace of a plane, for example?
Learning about the history and culture of Descriptive Geometry, I realize it is rooted in a European tradition – maybe even central and Eastern European specifically. Descriptive – a term introduced by the French founder of DG – is not meant to just describe or depict 3D objects – rather to represent them, to allow for systematically investigating the properties of 3D structures.
In Austrian high schools, DG is a self-contained subject separate from mathematics and fine art classes. I had two years of DG classes in last two years of high school (age 16-18). Every time somebody calls my geometrical images math art, my category-obsessed brain wants to inject: No, that’s not math – it is DG! :)
I think of DG as the only practical skill I obtained as part of my well-rounded high-school education – practical at the time software was not yet eating the world.
Little did I know back then that I will use DG in a very special way decades later! Today I am happy to announce the release of my latest drawing Rolling Not Slipping as a canvas print:
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In the often quoted article Descriptive Geometry in Today’s Engineering Curriculum (2005) Austrian professor Hellmuth Stachel, gives this definition of Descriptive Geometry:
Descriptive Geometry is a method to study 3D geometry through 2D images. It provides insight into structure and metrical properties of spatial objects, processes and principles. Typical for Descriptive Geometry is the interplay
a) between the 3D situation and its 2D representation, and
b) between intuitive grasping and rigorous logical reasoning
He argues that the efficient use of 3D software requires the analytical skills trained in a DG education.
There have been attempts to modernize the name – to call it technical or applied, or add visualization or modeling.
Quoting Stachel again:
Descriptive Geometry’ is more than ‘descriptive’ geometry as well as ‘Geometry’ is more than its literal sense, i.e., ‘measuring the earth’.
As an example, he shows how to answer this question geometrically Where does the sun rise earlier on June 21, in Oslo or in Vienna. I think this is a good choice as the geometric solution is obtained way easier than manipulating mathematical symbols and functions.
Stachel also says he himself is not able to keep the different views of 3D objects in this mind, and he shows the – potentially surprising – views of a rhombic dodecahedron as an example. I second that – seeing a “simple” 3D Lissajous Figure from three different direction never ceases to amaze me:
I have been so used to thinking of mathematical objects only through the lens of equations and code, and that it took me quite a while to get back into DG-style thinking. There is a also lot of jargon that needs to be learned – things you don’t hear in a math class. I often find it easier to “reverse engineer” an example in a book and ponder about first principles, rather than plowing through the description using all the jargon – Hauptgerade, Deckgerade, Spurendreieck, projizierend machen…
The Wikipedia article about DG is also on point, though a bit technical. The examples show typical DG problems, such finding the shortest line segment connecting two skew lines. You do not need to follow the examples in detail (and they do not show all “helper lines”), but you get a feeling for the connection of different views of a 3D object: You turn or fold down the object several times until a certain condition is met – typically a planar figures displayed true to size without distortions.
I like that they emphasize the set of procedures:
Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions by using a specific set of procedures.
The geometry of DG is not complicated on principle – it is just about projections using parallel rays (orthographic) or rays emerging from a point (perspective). All the jargon is there is formulate these procedures in a succinct way, allowing to keep a mental cheat sheet where you can pick the right little trick from!
When constructing 3D Lissajous Figures, the projections onto the co-ordinate planes also requires analog calculations. But these constructions are something completely different, not related to 3D space. I am usually divide the unit circle into equally sized parts to geometrically obtain the values for sine functions at “interesting points in time” (where the 2D Lissajous figures would intersect themselves, for example).
In some of my Lissajous drawings I’ve highlighted these circles:
I tend to search for specific properties of a scene that allow for a shortcut: When thinking about the intersection of a sphere and an ellipsoid I came up with this argument that I did not consider a typical DG trick:
Two curved shapes are intersecting each other – how can you find the intersection? None of the usual tricks in descriptive geometry works: You cannot simply rotate a plane so that you see the plane as a line.
I think everybody has an idiosyncratic way of looking at things like that: this is how I view it! I pick a prolate ellipsoid for this visualization, a cigar. Gird the ellipsoid with a ring, around its “belly”, inside the sphere.
Now move the ring outward without distorting or bending it, in the direction perpendicular to the circle spanned by the ring. The ring becomes smaller and smaller as you move along the surface of the ellipsoid.
Now watch where this ring hits the surface of the sphere first. It has to be a pair of points in one of the planes that are perpendicular to the ring and contain the axis you move the ring along. It is a pair of points for reasons of symmetry.
If you see this “first” intersection point right in the axis in one view, you will see it on the outer edges when rotating the ellipsoid by 90°.
It is very likely that I have re-invented the wheel here (it happened before). But the more software – and AI – are progressing, the more I find it intriguing to “re-invent the wheel” and to limit myself to a small set of “allowed” resources. I‘ve started to do the same with software, too. Any of these geometric puzzles have sure been solved before 100 times over. The pleasure is in re-discovering it for yourself “offline”, not by googling or asking AI bots!
When creating Rolling Not Slipping, I’ve used a combination of re-inventing such procedures and orthodox :) orthographic projection:
The apparent circumference of the ellipsoids is kind of the union of a scaffolding of the three principle ellipses, plus additional information about the largest extension of the ellipsoid. This is not 100% rigorous: I do not track down the exact elliptical shape of the projection of the ellipsoid, but rather infer its outline from three ellipses and a few tangents.
To build that scaffolding, I had to construct the projections of the main ellipses onto the coordinate planes … and I had underestimated this task in particular for the red ellipsoid. Only for the view shown in the upper right corner was there enough symmetry to make the task straight-forward. For the other views, I needed some helper constructions to obtain the sizes of projected line segments. E.g. the semi-circles in the bottom right are actually part of a fourth view – looking straight onto the invariable plane. This is the essence of all the DG procedures: Turn or tilt the planes and lines step-by-step, and “read off” the length of a segment using the compass.
I am aiming at creating art, not illustrations for textbooks. So, I give myself permissions to using these poetic shortcut arguments., like moving the “ring around the belly of the ellipsoid”. But I make it harder by not using annotations (There are actually standards for that!), and by overlapping all the shapes. If I really wanted to display the scenario clearly, I would have to use much larger paper and avoid the overlaps.
Creating these drawings is a journey of self-discovery. Why am I doing this?
The article on the role of DG linked above mentions that these things obsolete in the era of software:
~ complicated manual constructions,
~ hard theoretical proofs,
~ the theory of how to obtain images of particular 3D objects
… while these skills are timeless and also apply to usage of software:
~ the capability to comprehend virtual 3D situations from given images,
~ mental orientation in 3-space (e.g., user coordinate system),
~ basic knowledge of 3D geometry,
~ promoting creativity and problem-solving skills,
~ applications of geometry,
~ producing attractive illustrations
For better of for worse, I am unfortunately also drawn :) to the tasks considered obsolete (if you replace “hard theoretical” with “typical physicist’s hand waving math intuition”). I am giving myself permission to indulge in an outdated craft.
I once read a brief account by a teacher on a social network: They had shown their student artworks created by humans in the past – from oil paintings to photography. Students’ default response was: That must be AI – humans cannot do this!
I am very well aware of the limitations I have as a human geometer. But I want to keep pushing the limits of what we humans are able to achieve with bare hands: Certainly without AI, but also without software, and in general without constantly “looking up” something – be in in a book or online.
Once – in the heyday of internet optimism – it was said that it is not important to know anything anymore, you just have to know how to find it. The easier is becomes to look things up, the stronger I disagree.
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See work-in-progress photos for Rolling not Slipping and read more about the math/physics background here:
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