### Hongbo Li: On Advanced Projective Geometric Invariants

Key Laboratory of
Mathematics Mechanization

Academy of Mathematics and Systems Science

Chinese Academy of Sciences

Beijing 100190, P. R. China

### Abstract

Geometric invariants refer to expressions whose variables represent
geometric entities and whose values are "invariant" under geometric
transformations of the geometric entities represented by the variables. They
play important roles in pattern recognition, machine learning, as well as
geometric reasoning and geometric constraint solving. Particularly in geometric
reasoning, geometric invariants have the benefit of simplifying symbolic
manipulations by avoiding the introduction of coordinates in symbolic form.

Projective geometry is mathematically the simplest classical geometry in
that its transformation group is the general linear group. The basic projective
geometric invariants are determinants of the homogeneous coordinates of points
in projective space, called "brackets". While all invariants are generated
by brackets by multiplication and addition, these basic invariants are often too
low level to describe and manipulate projective geometric problems effectively.
Grassmann-Cayley algebra provides a means of constructing advanced
projective geometric invariants by its meet product and outer product. The
so-called "Cayley bracket algebra" is a non-associative algebra of advanced
invariants. Symbolic manipulations based on the non-associative product in this
algebra are usually very difficult.

On the contrary, in Euclidean
geometry, with the introduction of Euclidean distance structure based on the
conformal model, "conformal Geometric Algebra" induces an algebra of advanced
Euclidean invariants called "null bracket algebra", where null vectors
representing points are multiplied by the associative C lifford product.
Symbolic manipulations in null bracket algebra ar e much easier than those in
the algebra of inner products and determinants, or geometrically, the algebra of
squared distances and signed volumes.

When it comes back to projective
geometry, our recent research shows that by taking lines in space as basic
geometric entities, and introducing an R(3,3) inner product into the 6D linear
space of bivectors representing lines, the Clifford structure based on the
6D inner product space conforms perfectly to the projective structure based on
the 4D linear space representing points of the projective space. This agreement
suggests the possibility of constructing advanced projective geometric
invariants in the null Geometric Algebra over R(3,3) by taking lines as basic
geometric objects. Generating projective transformations and advanced projective
invariants will be explored in this new algebraic framework of projective line
geometry, and be presented in this talk.