  In this note, we propose a straightforward method to produce an straight-line
embedding of a planar graph where one face of a graph is fixed in the plane as
a star-shaped polygon. It is based on minimizing discrete Dirichlet energies,
following the idea of Tutte's embedding theorem. We will call it a geodesic
triangulation of the star-shaped polygon. Moreover, we study the homotopy
property of spaces of all straight-line embeddings. We give a simple argument
to show that this space is contractible if the boundary is a non-convex
quadrilateral. We conjecture that the same statement holds for general
star-shaped polygons.

Computational Geometry;General Topology