Geodesic triangulations of a star-shaped polygon

Let X(Ω, T) denote the space of geodesic triangulations of Ω with combinatorial type T, where T is a triangulation of a convex polygon Ω embedded in ℝ^2, and let W(Ω, T) be the weight space of (Ω, T). Using a framework established by Luo which gives a relation between Tutte’s Embedding Theorem and X(Ω, T), we prove several new results regarding the motion and limiting behavior of a geodesic triangulation τ ∈ X(Ω, T), which is the image of a weight w ∈ W(Ω, T) under the Tutte map ψ: W(Ω, T) → X(Ω, T), as we map τ to another geodesic triangulation of Ω by varying an interior-to-boundary edge weight within w. Using these aforementioned results, we then give a possible approach toward studying the homotopy type of X(Ω, T), where Ω is strictly star-shaped with one reflex vertex and T is a triangulation of Ω which contains no dividing edges.