This paper offers an approach of developing an order- theoretic structure theory of two-dimensional convex continuum structures. The chosen approach is based on convex planar continua and their subcontinua as primitive notions. In a first step convex planar continua are mathematized and represented by ordered sets. In a second step "points" are deduced as limits of continua by methods of Formal Concept Analysis. The convex continuum structures extended by those points give rise to complete atomistic lattices the atoms of which are just the smallest points. Further research is planned to extend the approach of this paper to higher dimensional continuum structures.