Examples of "planarly"
In contrast to Fáry's theorem for planar graphs, not every 1-planar graph may be drawn 1-planarly with straight line segments for its edges. However, testing whether a 1-planar drawing may be straightened in this way can be done in polynomial time. Additionally, every 3-vertex-connected 1-planar graph has a 1-planar drawing in which at most one edge, on the outer face of the drawing, has a bend in it. This drawing can be constructed in linear time from a 1-planar embedding of the graph. The 1-planar graphs have bounded book thickness, but some 1-planar graphs including "K" have book thickness at least four.
Restricted models of simultaneous embedding have also been studied. In simultaneous geometric embedding, each graph must be drawn planarly with line segments representing its edges rather than more complex curves. In order to guarantee that such an embedding exists, one must restrict the two given graphs to subclasses of the planar graphs. In another restricted model of simultaneous embedding, simultaneous embedding with fixed edges, curves or bends are allowed in the edges but any edge that is present in both of the given graphs must be represented by the same curve in both drawings. When a simultaneous geometric embedding exists, it automatically is also a simultaneous embedding with fixed edges.
Two-page book embeddings with a fixed partition of the edges into pages can be interpreted as a form of clustered planarity, in which the given graph must be drawn in such a way that parts of the graph (the two subsets of edges) are placed in the drawing in a way that reflects their clustering. Two-page book embedding has also been used to find simultaneous embeddings of graphs, in which two graphs are given on the same vertex set and one must find a placement for the vertices in which both graphs are drawn planarly with straight edges.
However influential Malfatti's debut into the Modernist art scene was in Brazil, her later pieces of work seem to revert to an older and more serene style. No longer are her pieces shocking or with as intense of a mixture of Cubism and Impressionism. Her painting "O Canal e a Ponta" done in the 1940s is not nearly as stunning in style or character as "The Idiot". The brush-strokes are much smaller and identical in relation to one another. They all go in the same direction, merely seen as a transitory means for color to be expressed. Malfatti's color palette is much less stunning as well. Although the colors in the painting are still lovely, they lack the vivid contrast which makes "The Idiot" so memorable. Instead of playing off of opposite colors, "O Canal e a Ponta" is one swell of deep shades that vary slightly from light reflections. The painting is still beautiful, it is simply much more traditional. The style is much more reminiscent of a toned down expressionism and does not utilise any qualities of Cubism which made her first pieces so planarly abstract. Even the subject matter is much more traditional and "European". It is a very tranquil setting of a bridge over a river with two trees on its sides a set of houses down the way and a serene sky. It is very harmonious but lacking in zest which would make her painting unique.