# 28-30.10 More fun with projectivities

It’s Halloween!  And so we spent the lecture exploring some strange and eerie applications of projective transformations.  Here are the rough notes Dr. Gunn used to prepare the lecture.

• Semester Projects.
• Possible themes.  Balancing mathematical content with practical implementation (“theory and practice”).  Rough suggested time-table.
• Lecture write-up from 28.10.2013.  Are there volunteers to write up the lecture from Tuesday?  Recall: we went through all the gory details required to construct a projectivity (as 3×3 matrix) which maps the unit square onto a trapezoid.  Either directly enter into blog (which supports LaTeX formatting directly) or prepare a pdf file for inclusion.
• State of gitorious.  This week we have the modest goal of being able to “push” student projects back onto your gitorious accounts, and from there notify me (by e-mail or “merge request” from gitorious) that there is something to look at.  Next week we’ll conclude the gitorious work by showing how to update to receive new assignments from me, and how to update jReality.  See this blog post for details.
• The Internship trailer.
• Applications of projective transformations:  the Ames room.
• Noneuclidean geometry via projective geometry.
• TriangleGroup demo
• Klein’s Erlangen program: preserved properties determine subgroup of PGL(n+1).
• The subgroups SO(3) and SO(2,1) lead to spherical/elliptic and hyperbolic geometry, resp.
• Distance via inner product, cos and cosh. Examples.

## 2 thoughts on “28-30.10 More fun with projectivities”

1. i thought it would be nice to add that similar techniques with perspectivity have been utilizd in baroque architecture, a famous example would be, of course, the palazzo spada in rome. and a little earlier, another, yet far more sophisticated architectural history example, that actually accounts for the fact that straight lines exist nowhere but in our imagination.

• Thanks for this interesting historical footnote. One could also here include the role that “perspective relief” sculpture played (such as the Florence Baptistry doors by Ghiberti) in the development of perspective painting.

On another note, I would find it good when students would also identify themselves by signing their comments, either with a real name or the 2-digit number from the course. It’s unfortunate that there’s only a single student account — I find it more like a real discussion when each contribution is associated to a person.