Projective Geometry applied to Perspective art: A proofs-based course

Projective Geometry applied to Perspective Art is an inquiry-based course designed for sophomore- and junior-level mathematics majors.   The materials on this page -- developed as a collaborative project by Marc Frantz in Indiana, Fumiko Futamura in Texas, and Annalisa Crannell in Pennsylvania -- come from a project is supported by NSF TUES Grand DUE-1140135, so thank you for your tax dollars!

We presented an MAA Minicourse in January 2014; here are the slides and handouts from that minicourse.

The materials below in particular are ones that Crannell has used in her own course.  (The exception is "Geometric Division", which was a worksheet she used in MAA Minicourses).  Because they are still in draft form, comments are welcome!  Please feel free to contact us!

Each worksheet, once completed, will have

  • a one-page picture that is also a math/art puzzle
  • a module with IBL-style in-class worksheet
  • a homework set (with short exercises, an art project, and a proof/counter-example)
  • an instructor's section, with a guide to using the module, and 
  •  answers to the module / homework questions.



1. Window Taping: the After Math


Instructor's manual for Chapter 1

• basic understanding of parallel lines and planes; 

• images of lines in a picture plane; 

• plan views; 

• definition of vanishing point;

• importance of the notion of “parallel” in determining the existence and location of the vanishing point.

2. Drawing ART


Instructor's manual for Chapter 2

• problem solving

• artistic application of the rules “Lines parallel to the picture plane have parallel images; lines not parallel to the picture plane but parallel to one another converge to the same vanishing point”.

3. Image of a line



Instructor's manual for Chapter 3

Students explore possible definitions of projection of points and lines onto a plane.


• visualizing projections of points and lines in R2 and R3; 

•considering “special cases” of projection; 

•understanding the difference between artistic applications and mathematical definitions.

4.  Intro to Geogebra, plan views

Dynamic Cubes (has both GeoGebra & non-GeoGebra version)

Fieldtrip (Museum and/or poster viewing)


Instructor's manual for Chapter 4

• introduction to viewing distance for one-point perspective cubes and boxes

• introduction to GeoGebra 

• review of plan views, review definition of vanishing points 

5. Extended Real Space



Instructor's manual for Chapter 5




(5.  The Euclidean Plane)

(5.  Euclidean Space)

• The definition of ideal points,  the extended plane, and extended real space; 

• logical consequences of those definitions; 

• proving statements by using  definitions; 

• visualizing points and lines at infinity.

Optional: review of Euclidean Geometry, with Ceva's and Menelaus's Theorems, and a foreshadowing of the harmonic set).

6. Meshes and Mesh Maps


Follow up:
Fieldtrip to draw a "poster"


Instructor's manual for Chapter 6

Students explore the basic structure of a mesh and a formal definition of projection onto a plane. As an application, they draw and devise solutions to geometric division problems.

The notion of a mesh is not a standard one in projective geometry. A mesh is useful, however, in artistic applications such describing a 3-d object like a house: for this object, we want to draw some lines (such as the top edge of the roof) but not others (we don’t draw the line that connects the front top of the roof to the back lower left corner of the house). Similarly, for this object we care about some points of intersection (the four corners of the floor) but not others (the front left edge of the roof does not intersect the back right edge of the roof in a point in R3, even though their images intersect in Figure 2).

7. Squares in 2-point perspective


Instructor's manual for Chapter 7

Students discover the viewing circle; they locate the viewing target and viewing distance for a square in two-point perspectives. From there, they learn to draw a box and then a cube in 2-point perspective. 

We foreshadow the need for harmonic sets of points.

8. Desargues Discovery


• How to draw shadows; 

• defining perspective from a point and perspective from a line; 

• exploring Desargues’s Theorem.

8. Desargues Proof

Instructor's manual for Chapter 8

In which we prove the theorem.  (Uses a "modified Moore-method" approach).

9. Colineations

•An intro to colineations, in particular perspective colineations in the plane

9. Harmonic Homologies

We show that period-2 homologies (like reflections) relate to the above worksheet on squares in 2-point perspective. 

9.  Elations


Instructor's manual for Chapter 9

We practice constructing elations; these are like the fencepost-repeating problems above

10: Numerical Invariants: Cross ratio

An introduction to cross ratio, with empirical evidence that it is a projective invariant. Harmonic sets of points and relationship to vanishing points of images of a rectangle. Applications to fence division and to photogrammetry. 

10.  Numerical Invariants:  Circular products and Eves's Theorem

Circuluar products, h-expressions, Eves’s theorem, and its applications to perspective problems. Students are guided through a sketch of the proof of Eves’ Theorem. 

10.  Numerical Invariants:  Casey Angles

Instructor's manual for chapter 10

An introduction to Casey’s angles; proof of Casey’s Theorem; perspective application to drawing and deducing angles. 

11.  Coordinate Geometry




11.  Homogeneous Coordinates


Instructor's manual for Chapter 11

In the Cartesian coordinate worksheet, we  connect the making of perspective pictures with a review of three-dimensional Cartesian coordinate space. In other words, we do perspective ``by the numbers''. 

The homogeneous coordinate worksheet introduces a more useful kind of coordinate system for perspective and projective images. 

12.  The shape of Extended Real Space

Instructor's manual

We create Möbius bands, Möbius shorts, and punctured versions of the projected plane.

Appendix:  Writing mathematical prose

1. Getting started
2. Pronouns and active voice
3. Introducing variables, constants, and other mathematical symbols
4. Punctuation with algebraic expressions in the sentence
5. Paragraphs and lines
6. Figures 

#. Geometric Division 

#. Geometric Division (follow up)

Fence division problems and their relation to the cross-ratio   (Not part of the regular sequence of modules; out of sequence  with the rest of the modules)

Followed by a presentation developed for our minicourse participants to expand on the worksheet above