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.