Explain the concept of planar graph embeddings and their applications in Computational Geometry.

Planar graph embeddings refer to the representation of a graph in a two-dimensional plane such that no edges intersect each other, except at their endpoints. In other words, it is a way of drawing a graph on a plane without any edge crossings. This concept is extensively used in computational geometry to solve various problems related to graph theory and geometric algorithms.

One of the key applications of planar graph embeddings is in the study of planar graphs. A planar graph is a graph that can be embedded in a plane without any edge crossings. By representing a planar graph as a planar graph embedding, we can analyze its properties and characteristics more effectively. For example, planar graph embeddings can be used to determine the connectivity of a graph, identify cycles and paths, and calculate the face and vertex degrees.

Planar graph embeddings are also crucial in solving geometric problems. Many geometric algorithms rely on the concept of planar graph embeddings to efficiently solve problems such as finding the convex hull, computing the Voronoi diagram, and constructing Delaunay triangulations. By representing the geometric problem as a planar graph and utilizing its embedding, we can apply graph algorithms to solve the problem more efficiently.

Furthermore, planar graph embeddings are used in graph drawing and visualization. They provide a way to visually represent complex graphs in a two-dimensional space, making it easier to understand and analyze the structure of the graph. Various graph drawing algorithms utilize planar graph embeddings to create aesthetically pleasing and informative visualizations.

In addition to these applications, planar graph embeddings have connections to other areas of computer science and mathematics. They are used in network routing algorithms, circuit design, graph theory algorithms, and even in the study of DNA sequences.

Overall, planar graph embeddings play a fundamental role in computational geometry by providing a powerful tool to analyze and solve problems related to graph theory and geometric algorithms. They enable efficient computation, visualization, and understanding of complex graphs and geometric structures.