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Regular and Wirtinger Projections

It would be nice if all the significant features of the 3D objects are visible in the projections. In the projection of a polygon, we do not expect some vertices or edges disappeared. Let S be a set of n distinct points and disjoint line segments in the space, and H be the plane where we will operate the projection. Let SH be the parallel projection of S onto H.

Definition: A parallel projection of S is said to be regular if no three points of S project to the same point on H and no vertex of S projects to the same point on H as any other point on S.

The following picture shows three different projections of a cube, one of them is not a regular projection.

3 projections of a cube

For a regular projection of disjoint line segments, we may conclude from the definition:

  1. No point of SH corresponds to more than one vertex of S.
  2. No point of SH corresponds to a vertex of S and an interior point of an edge of S.
  3. No point of SH corresponds to more than two interior points of edges of S.
Hence, the only crossing points we will find in a regular projection, will come from the intersection of the interiors of two edges of S. Thinking about it, you will see that almost every polygon has a regular projection. Nevertheless, a regular projection may lose some important information of the original object. Let's have a look at the next picture.

It is indeed a regular projection, but two edges have collinear projections. Thus we lose precious information about the shape of the polygon. That kind of loss is not good for visualisation, so we introduce a restricted class of regular projections called Wirtinger projections, for which no two consecutive edges of the 3D polygon have collinear projections.

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