. The initial preprint is from 2015, and it went through several versions, but the main results are unchanged, though some figures and section/theorem numbering are different, and there are several minor modifications in the text.
You can cite the published version as:
António Bandeira Araújo (2018) Ruler, compass, and nail: constructing a total spherical perspective, Journal of Mathematics and the Arts, 12:2-3, 144-169, DOI: 10.1080/17513472.2018.1469378
Please let me know if you find any typos or errors in the article. At the moment the known corrections are as follows:
1 - On page 153, line 19 (the line above equation 1), where it reads
" the perspective map f " it should read " the perspective map p "
What is the article about?
In the 1960s, Barre and Flocon wrote a seminal book ( La perspective curviligne, 1967, Flammarion, Paris) wherein they defined a spherical perspective in a way that allowed for systematic drawing by elementary means, that is, by ruler and compass. However, "spherical perspective" was a bit of a misnomer since those rules were only specified for half the sphere. In this paper I extend those systematic constructions to the whole sphere. This extension had been attempted before, but either not successefuly or not systematically/completely. With this paper I solve the total spherical perspective in the same sense that Barre and Flocon solved the hemisphere: that is, I give a systematic method to calculate and draw all lines and vanishing points with ruler and compass (or even freehand). I also summarize Barre and Flocon's method in a very simple way (reduced to about two pages). Furthermore, I take the opportunity to consider what is the nature of a perspective as a mathematical object. I think it should be seen as a game of compactification that has anamorphosis as its central player (see the paper for details).
What's with the nail ?
The initial title of the paper was "A construction of a spherical perspective in ruler, compass, and nail". It alludes to the fact that a set of (abstracted) mechanical apparatus defines a scope - it permits a certain class of points to be obtained from another through its action. Although you can construct the total spherical perspective with ruler and compass operations, I found that a more natural way to think about it is to imagine that you are using (or actually use) a third mechanical implement - namely, a nail fixed at the center of the page. This nail, interacting with a ruler, allows the construction of the antipodal image of a given meridian, and hence to construct the total spherical perspective by piggybacking on the hemsipherical constructions of Barre and Flocon (you'll have to read the paper for details).
What's the purpose of this web page?
I intend to place here several helpful files:
All of these need some work before I place them here, so I'll ask you to come around somewhat later (hopefully not infinitely long); I'll be working on it as my other duties allow.
- Additional illustrations and comments that I didn't fit onto the paper due to space limitations.
- High resolution charts and grids.
- Animations and videos of constructions in the paper.
- Geogebra tools and other software that I developed along with the paper.
It goes without saying (but I'll say it nonetheless) that
this page is under construction.
António Araújo, DceT, Universidade Aberta