标题： 十个点何时位于一个二次曲面上？ 显示英文标题

标题： When do Ten Points Lie on a Quadric Surface?

摘要： 提供了布鲁塞尔问题的解决方案，这是一个最初于1825年提出的几何决策问题，询问何时三维空间中的十个点位于二次曲面上，该曲面由一个二次多项式的消失给出。 该解决方案构造了四个新点，这些点共面当且仅当原来的十个点位于二次曲面上。 该解决方案仅使用通过两个已知点构造的直线、通过三个已知点构造的平面以及这些对象的交点。 涉及的工具包括将面积原理扩展到三维空间、括号多项式和格拉斯曼-凯莱代数，以及冯·斯托尔特关于几何算术的结果。 许多特殊情况被直接处理，从而得到一般情况，其中三对点生成异面直线，其余四个点处于一般位置。 在一般情况下，关键步骤是为通过十个点中的六个点的二次曲线找到一个优美的基，这利用了从Macaulay2得出的见解，这是一个十九世纪未出现的计算代数包。 显示英文摘要

摘要： A solution is provided to the Bruxelles Problem, a geometric decision problem originally posed in 1825, that asks for a synthetic construction to determine when ten points in 3-space lie on a quadric surface, a surface given by the vanishing of a degree-2 polynomial. The solution constructs four new points that are coplanar precisely when the ten original points lie on a quadric surface. The solution uses only lines constructed through two known points, planes constructed through three known points, and intersections of these objects. The tools involved include an extension of the Area Principle to three-dimensional space, bracket polynomials and the Grassmann-Cayley algebra, and von Staudt's results on geometric arithmetic. Many special cases are treated directly, leading to the generic case, where three pairs of the points generate skew lines and the remaining four points are in general position. A key step in the generic case involves finding a nice basis for the quadrics that pass through six of the ten points, which uses insights derived from Macaulay2, a computational algebra package not available in the nineteenth century.