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The output of the construction process is a planar graph. To be sure, the graph has an intrinsic beauty that holds up when it is rendered as simple line art. Historically, however, these designs were never merely drawn as lines. Islamic star patterns are typically used as a decoration for walls and floors. The faces of the planar graph are realized as a mosaic of small terracotta tiles in a style known as ``Zellij''. Often, the edges are thickened and incorporated into the mosaic with narrow tiles, sometimes broken up to suggest an interlacing pattern. Islamic designs can also be found carved into wood or stone and built into trellises and latticework.

Figure 7: Rendering styles.
\includegraphics [width=0.8in]{figures/render_plain.eps} \includegraphics [width=0.8in]{figures/render_outline.eps} \includegraphics [width=0.8in]{figures/render_embossed.eps} \includegraphics [width=0.8in]{figures/render_interlaced.eps} \includegraphics [width=0.8in]{figures/render_checker.eps} \includegraphics [width=0.8in]{figures/render_overlay.eps}
Plain Outline Emboss Interlace Checkerboard Outline and

To increase the aesthetic appeal of our implementation, we provide the ability to render the planar graph in a manner reminiscent of some of these techniques (see Figure 7). The outline style thickens the edges of the planar graph, adding weight and character to the lines of the plain style. The emboss style adds a 3D effect to the outline style, simulating the appearance of a wooden trellis; the centre of each thickened edge is raised towards the viewer and the graph is rendered by specifying the direction of a fictitious light source. The interlace style adds line segments at each crossing to suggest an over-under relationship between the crossing edges. When every vertex in the graph has degree two or four, the crossings can always be chosen so that the graph is broken into strands that adhere to a strict alternation of over and under in their intersections with other strands. The final style, checkerboard, renders the faces of the graph and not the edges. When all vertices have even degree (as they must in an interlace design), it is always possible to colour the faces with only two colours in such a way that faces with the same colour never share an edge. The checkerboard style walks the graph, creating a consistent 2-colouring.

A further enhancement can be achieved by layering one of the edge-based rendering styles on top of the checkerboard style. This combination comes closest to the appearance of Zellij.

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Craig Kaplan 2000-08-16