Curves and Surfaces I: Affine and Projective Spaces, and the Problem of Lines in 3D

Abstract:

We aim to give a gentle introduction to 19th - century geometric developments in affine and projective settings that set the stage for the classical study of curves and surfaces, using the 20th - century framework of Linear Algebra. To be concrete we will focus on an elementary topic of considerable importance in graphics and robotics: the efficient representation of lines in 3D space. This is not at all so easy as our first and second year courses might suggest: especially when we are interested in extending the theory to k-dimensional flats in n-dimensional space.

We will meet Plucker co-ordinates, the Grassmann determinant principle, the Klein quadric, and the role of skew-symmetric bilinear forms, the last being joint work with Chris Halse-Rogers.