In particular, the thickness of a given graph is two, if the graph's edges can be partitioned into two subgraphs that have a simultaneous embedding, and the geometric thickness is two, if the edges can be partitioned into two subgraphs with simultaneous geometric embedding. In simultaneous geometric embedding each graph must be drawn as a planar graph with line segments representing its edges rather than more complex curves restricting the two given graphs to subclasses of the planar graphs. Many results on simultaneous geometric embedding are based on the idea that the Cartesian coordinates of the two given graphs' vertices can be derived from properties of the two graphs. One of the most basic results of this type is the fact that any two path graphs on the same vertex set always have a simultaneous embedding. To find such an embedding, one can use the position of a vertex in the first path as its "x"-coordinate, and the position of the same vertex in the second path as its "y"-coordinate. In this way, the first path will be drawn as an "x"-monotone polyline, a type of curve that is automatically non-self-crossing, and the second path will similarly be drawn as a "y"-monotone polyline. This type of drawing places the vertices in an integer lattice of dimensions linear in the graph sizes. Similarly defined layouts also work, with larger but still linear grid sizes, when both graphs are caterpillars or when both are cycle graphs. A simultaneous embedding in a grid of linear dimensions is also possible for any number of graphs that are all stars. Other pairs of graph types that always admit a simultaneous embedding, but that might need larger grid sizes, include a wheel graph and a cycle graph, a tree and a matching, or a pair of graphs both of which have maximum degree two. However, pairs of planar graphs and a matching, or of a tree and a path exist, that have no simultaneous geometric embedding. Testing whether two graphs admit a simultaneous geometric embedding is NP-hard. More precisely, it is complete for the existential theory of the reals. The proof of this result also implies that for some pairs of graphs that have simultaneous geometric embeddings, the smallest grid on which they can be drawn has doubly exponential size. When a simultaneous geometric embedding exists, it automatically is also a simultaneous embedding with fixed edges. In simultaneous embedding with fixed edges, curves or bends are allowed in the edges, but any edge present in both graphs must be represented by the same curve in both drawings.