Deformations of flags and convex sets in \(\mathbb{RP}^2\)

Deformation spaces have always been of importance in geometry. By understanding how objects change one can also find properties that help understand if two objects are the same in a certain sense, e.g. two triangles in the euclidian plane are the same if there is an isometry sending one onto the other.

In this thesis we want to investigate deformations of n-tuples of flags in the projective plane and provide visualizations of the changes. This will allow us to parameterize the space of positive n-tuples of flags by considering internal parameters induced by a triangulation of suitably nested polygons. By introducing the eruption, shearing and bulging flows discussed by Wienhard and Zhang [WZ18] we are then able to fully understand and visualize how these parameters change.

We will also draw a connection to hyperbolic geometry through showing that ideal polygons in properly convex sets are of finite volume with regard to the Hilbert metric. Furthermore we will extend the deformations to marked strictly convex domains with C1 boundary.
But the main goal of this thesis remains to visualize the deformations in some of the discussed cases.