# Visual Pascal Configuration and Quartic Surface

## 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## Preview

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## References

- 1.Berger, M.: Geometry II. Springer, Heidelberg (1987)zbMATHCrossRefGoogle Scholar
- 2.Jennings, G.: Modern Geometry with Applications. Springer, New York (1994)zbMATHGoogle Scholar
- 3.Silverman, J.H., Tate, J.: Rational Points on Elliptic Curves. Springer, New York (1992)zbMATHGoogle Scholar