Applications of a Particular Four-Dimensional Projective Geometry to Galactic Dynamics

Projective geometry - from foundations to applications

Four-Dimensional Projective Orbifold Hypersurfaces

We classify four-dimensional quasismooth weighted hypersurfaces with small canonical class, and verify a conjecture of Johnson and Kollár on infinite series of quasismooth hypersurfaces with anticanonical hyperplane section in the case of fourfolds. By considering the quotient singularities that arise, we classify those weighted hypersurfaces that are canonical, Calabi–Yau, and Fano fourfolds. ...

Three-Dimensional Projective Geometry with Geometric Algebra

The line geometric model of 3-D projective geometry has the nice property that the Lie algebra sl(4) of 3-D projective transformations is isomorphic to the bivector algebra of Cl(3, 3), and line geometry is closely related to the classical screw theory for 3-D rigid-body motions. The canonical homomorphism from SL(4) to Spin(3, 3) is not satisfying because it is not surjective, and the projecti...

Catadioptric Projective Geometry: Theory and Applications

CATADIOPTRIC PROJECTIVE GEOMETRY: THEORY AND APPLICATIONS Christopher Michael Geyer Supervisor: Kostas Daniilidis Catadioptric cameras are devices which use both mirrors (catadioptric elements) and lenses (dioptric elements) to form images. In computer vision and robotics, catadioptric cameras have been used for their wide field of view. Such cameras often have a field of view greater than 180Æ...

Applications of Hofer’s Geometry to Hamiltonian Dynamics

We prove the following three results in Hamiltonian dynamics. • The Weinstein conjecture holds true for every displaceable hypersurface of contact type. • Every magnetic flow on a closed Riemannian manifold has contractible closed orbits for a dense set of small energies. • Every closed Lagrangian submanifold whose fundamental group injects and which admits a Riemannian metric without closed ge...