Terrains are often modeled by triangulations and one of the criteria is that: triangulation should have nice shape.Delaunay triangulation is a good way to formalize nice shape. Another criterion is slope fidelity in terrains. In natural terrains there are no abrupt changes in slope. A triangulation for a terrain should use triangles of nice shape and have slope fidelity. To achieve these characteristics, higher-order Delaunay triangulations are used. Two methods are presented and the result of implementations and visualizations show they perform very well on real-world data.