JCDCG 2004: Discrete and Computational Geometry pp 143-150

# Visual Pascal Configuration and Quartic Surface

• Yoichi Maeda
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3742)

## Abstract

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.

## Keywords

Conic Section Cross Ratio Visual Intersection Regular Octahedron Quartic Surface
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

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Silverman, J.H., Tate, J.: Rational Points on Elliptic Curves. Springer, New York (1992)