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


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.