My drawing abilities are quite limited but I always figured it was simply due to a lack of practice. I spent my formative years on reading, writing, and arithmetic, not on drawing. So I never appreciated this really simple fact; drawing three-dimensional objects and scenes is extremely difficult. I studied a fair amount of drawing manuals and the best advice I encountered was “In order to draw an object in general attitude, you have to think three dimensionally”. Not very helpful.
I also discovered something about geometry. First of all, I totally love it. I really enjoy the combination of geometric constructions and algebraic equations, that can express the same thing in different, but equally precise, languages. It is very fulfilling to draw a construction of lines and points, to guess at a deeper structure, then prove it in algebra, and go back to the sketch and turn it into an exact drawing. Everything in this little universe works like a charm.
The second thing I discovered is that three-dimensional geometry is extremely difficult. And I have no lack of experience here. I am trained in physics which involves dealing with topics like mechanics and electrodynamics that have 3D vector calculus at their cores. I also spend many years studying the human capabilities of perceiving the 3D world. And still. I get easily confused while trying to solve basic problems in 3D Euclidean geometry.
Why is drawing in three dimensions so hard? Why is geometry in three dimensions so hard? I can safely say I am not the only person with deficiencies in these areas. The main problem is in dealing with 3D rotations. We can perceive rotations of objects without a problem. But imagining these rotations correctly can easily blow our mental fuses. Luckily we have 3D computer graphics to help us.