Visual Pascal Configuration and Quartic Surface
For any two lines and a point in R 3, there is a line having intersections with these two lines and the point. This fact implies that two lines in R 3 make a visual intersection from any viewpoint even if these lines are in twisted position. In this context, the well-known Pappus’ theorem in R 2 is simply extended as that in R 3, i.e., if the vertices of a spatial hexagon lie alternately on two lines, then from any viewpoint, three visual intersections of opposite sides are visual collinear. In a similar way, Pascal’s theorem is also extended in R 3, i.e., if the vertices of a spatial hexagon lie on a cone, three visual intersections of opposite sides are visual collinear from the viewpoint at the vertex of the cone. In this case, for six vertices in R 3 we obtain a quartic surface as the set of viewpoints. We will investigate this surface depending on the vertices of a spatial hexagon. A relation between non-singular cubic curve and complete quadrilateral is naturally and geometrically derived.
KeywordsConic Section Cross Ratio Visual Intersection Regular Octahedron Quartic Surface
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