Projective Geometry applied to Perspective art: A proofsbased course
Projective Geometry applied to Perspective Art is an inquirybased course designed for sophomore and juniorlevel 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 DUE1140135, so thank you for your tax dollars!
We presented an MAA Minicourse in January 2014; here are the slides and handouts from that minicourse.
http://people.southwestern.edu/~futamurf/minicourse2014.html
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 onepage picture that is also a math/art puzzle
 a module with IBLstyle inclass worksheet
 a homework set (with short exercises, an art project, and a proof/counterexample)
 an instructor's section, with a guide to using the module, and
 answers to the module / homework questions.
Worksheets
1. Window Taping: the After Math

• 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

• 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”. 

Students explore possible definitions of projection of points and lines onto a plane. Topics: • 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 & nonGeoGebra version) Fieldtrip (Museum and/or poster viewing)

• introduction to viewing distance for onepoint perspective cubes and boxes • introduction to GeoGebra • review of plan views, review definition of vanishing points 
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). 
Follow up:

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. 
7. Squares in 2point perspective

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

• How to draw shadows; • defining perspective from a point and perspective from a line; • exploring Desargues’s Theorem. 
In which we prove the theorem. (Uses a "modified Mooremethod" approach). 

9. Colineations 
•An intro to colineations, in particular perspective colineations in the plane 
We show that period2 homologies (like reflections) relate to the above worksheet on squares in 2point perspective. 

9. Elations

We practice constructing elations; these are like the fencepostrepeating problems above 
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, hexpressions, Eves’s theorem, and its applications to perspective problems. Students are guided through a sketch of the proof of Eves’ Theorem. 
An introduction to Casey’s angles; proof of Casey’s Theorem; perspective application to drawing and deducing angles. 


In the Cartesian coordinate worksheet, we connect the making of perspective pictures with a review of threedimensional 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. 
We create Möbius bands, Möbius shorts, and punctured versions of the projected plane. 

Appendix: Writing mathematical prose 
1. Getting started 
Fence division problems and their relation to the crossratio (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 