  This paper presents a spline-based parameterisation framework for plane
graphs. The plane graph is characterised by a collection of curves forming
closed loops that fence-off planar faces which have to be parameterised
individually. Hereby, we focus on parameterisations that are conforming across
the interfaces between the faces. Parameterising each face individually allows
for the imposition of locally differing material parameters which has many
applications in engineering applications, such as elasticity and heat transfer.
For the parameterisation of the individual faces, we employ the concept of
harmonic maps. The plane graph's spline-based parameterisation is suitable for
numerical simulation based on isogeometric analysis or can be utilised to
extract arbitrarily dense classical meshes. Application-specific features can
be built into the geometry's mathematical description either on the spline
level or in the mesh extraction step.
