Abstract
AbstractThe impulse to study this topic came from a variant of the Wallace-Simson theorem, which deals with the locus of the point P such that the points that are symmetric to P with respect to three lines in the plane are collinear. A 3D generalization can be as follows: Given four straight lines which are parallel to a plane. Determine the locus of the point P such that points that are symmetric to P with respect to these four lines are coplanar. Surprisingly, the locus of P is a cylinder of revolution with the axis which is perpendicular to the fixed plane. Moreover, all planes given by points that are symmetric with an arbitrary point P of the locus with respect to the given four lines pass through a fixed line f. While in the planar version the fixed element is the orthocenter of the triangle given by the three lines, the role of the fixed line f with respect to the four given lines is not obvious.
Funder
University of South Bohemia in České Budějovice
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Mathematics (miscellaneous)
Reference14 articles.
1. Altshiller-Court, N.: College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle. Barnes & Noble, Inc., New York (1952)
2. Blažek, J., Pech, P.: A spatial generalization of Wallace–Simson theorem on four lines. Adv. Intell. Syst. Comput. 1296, 103–114 (2021)
3. Capani, A., Niesi, G., Robbiano, L.: CoCoA, a System for Doing Computations in Commutative Algebra. http://cocoa.dima.unige.it (1995)
4. Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, 2nd edn. Springer, New York (1997)
5. Coxeter, H.S.M., Greitzer, S.L.: Geometry Revisited. Mathematical Association of America, Toronto (1967)