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.
For a regular projection of disjoint line segments, we may conclude
from the definition:
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
Nevertheless, a regular projection may lose some important information
of the original object.
Let's have a look at the next picture.
- No point of SH corresponds to more than one vertex of
- No point of SH corresponds to a vertex of S and an
interior point of an edge of S.
- No point of SH corresponds to more than two interior
points of edges of S.
It is indeed a regular projection, but two edges have collinear
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
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