# Perspective and Projective Geometry

This is a workbook to learn the techniques of perspective drawing and the theory of perspective and projective geometry. The exercises range from very practical to proving theorems, but it is essentially based on experiments and discovering the theory by practicing on the examples.

The book starts with an example of a very practical experiment. One person (the director) has to stand immobile with one eye closed in front of a big window at a few meters distance. He or she is looking at the landscape or the buildings outside. The view should have preferably many straight lines. The window is used as a canvas, on which the outside world has to be projected and the director has to instruct other students (the artists) to fix waste tape on the window where the director sees the projected straight lines of the outside world. This is a way to detect how a 3D world is represented on a 2D window-canvas. The first module has several questions that can be answered in the blank space that is left open for that purpose or one has to check true/false answers or choose from multiple choice possibilities. All the pages of the book are perforated so that they can be torn out and handed in for correction or feedback. There is also a section for homework with more questions to answer (marked with circled E: Ⓔ) and art assignments like make drawings or take pictures (marked with a triangulated A) or more theoretical exercises related to theorems and proofs (marked with a squared P). In an appendix to this first module some explanation is given that should lead to a definition of sketches in n-point perspective, which is what the subsequent modules will work to on a more theoretical basis.

This first module described above is an example of how all the 13 modules are organized, although some are more theoretical, and none of the others have an appendix. Some of the questions are incomplete and the student has to guess what the question is. That has of course been prepared in previous questions, but still it may be a problem to correctly complete the sentence, which makes it impossible to continue with the next questions. Therefore, I believe this is not a workbook for self-study, a teacher should be guiding the process but it remains a challenge for the student whose responsibility is to discover the proper way to go or to detect the concepts and the theorems that support the constructions.

To illustrate how one moves from the window taping experiment to the theory, we note that in section 2, one has to analyse and complete the graphic representation of a 3-dimensional construction of two thick tiles on top of each other in the form of the letter T, drawn in (a 1-point) perspective. And then module three is stuffing up the theory with definitions and properties of points, lines, segments, planes, and module four is introducing geometry in $\mathbb{R}^2$ and $\mathbb{R}^3$ with the announcement of Ceva's and Menelaus's theorems. Module 5 extends the Euclidean space in (2 and 3 dimensions) by introducing ideal points, lines and planes, that are essentially the points at infinity. This gives the extended spaces $\mathbb{E}^2$ and $\mathbb{E}^3$. This seems to complicate things, but it actually simplifies life since no exception has to be made for these ideal objects which is the whole idea of projective geometry. To formalize the perspective drawings, meshes and maps are defined in $\mathbb{E}^3$

. The latter allow to project a 3D scene onto a 2D plane and it can be used for example to correctly project equispaced segments (say vertical poles of a fence or a square tiling of a floor) onto non-equispaced distances on the canvas or how to correctly draw a poster on a wall that is represented on a canvas in perspective.

The theory continues as above, but always in connection with practical problems related to the plotting of 3D scenes on a 2D canvas. The next issue is Desargues theorem (two triangles are in line perspective if and only if they are in point perspective). It is formulated in terms of meshes in $\mathbb{E}^3$ and a proof is to be derived. On the other hand, this module also introduces exercises in GeoGebra (a free interactive software package for geometry, algebra, and other computations). An elementary introduction to using GeoGebra is added in an appendix. Now the student should know enough about projective geometry to move objects around with concepts like (perspective) collineations, homologies and how these connect with harmonic sets. One may now experiment by moving points in a GeoGebra plot.

Herewith the modules move somewhat towards numerics. For example the position of the designer in the taped window example could be found by someone who moves herself into a position where the scene outside aligns with the tapes on the window. But now, at this stage of the book, it is possible to compute the distance of the viewer's eye to the canvas from the perspective drawing. Or it should be possible to derive from the 1-point perspective projection of a box whether it is a cube or not. Module 10 allows to draw boxes in 2-point perspective, respecting the actual distances and module 11 catches the cross ratio of four points as a numerical invariant allowing to draw lines in perspective that are equidistant in reality, a problem that was also previously considered in module 6. Another invariant is a more complex $h$-expression relating distances between 8 points in a rectangular configuration. This is named after and proved by Howard Eves (1911-2004 — the authors give wrong dates 1913-2000). Also the Casey angle of 4 collinear points is another invariant as proved by M. Frantz and named after John Casey (1820-1891) (the theorem here is not to be confused with what is generally known as Casey's theorem).

The last two modules introduce the Cartesian coordinate system, projective coordinates and linear algebra and even some topological concepts like the Möbius band and the possible shape of $\mathbb{E}^2$ (the four colour theorem does not hold for this space since six colours are necessary in general).

Every module starts with a one-page graphic, which is to be worked on as one progresses in the module, and thus it is somehow the target and motivation for the module that is coming up. For reference, the main definitions and theorems are summarized in an appendix. A last appendix deals with writing mathematical prose, and that includes style, punctuation, use of formulas and words, etc. This is of course important for any student who has to learn to write mathematics, but even more so it will be important for students who follow the course with an artistic background, and who have not been exposed very often to mathematical texts. I assume however that they may find the abstract mathematics of this text a bit hard to do.

The book is a nice mixture of mathematics (with rather abstract concepts and proofs), but with practical introductions of these concepts and interesting applications in art as well as in practical situations. Moreover, it gives an introduction to the computer system GeoGebra, and a hint towards linear algebra, analytic geometry and topology. There are excursions into historical aspects, music, photography, and many other tracks. But foremost it is a workbook with an extensive list of many different assignments ("cool" problems to use the language of the authors). More material related to these modules is available at the Futamura website.

**Submitted by Adhemar Bultheel |

**20 / Dec / 2019