Many properties of finite point sets only depend on the relative position of the points, e.g., on the order type of the set. However, many fundamental algorithms in computational geometry rely on coordinate representations. This includes the straightforward algorithms for finding a halving line for a given planar point set, as well as finding a point on the convex hull, both in linear time. In his monograph Axioms and Hulls, Knuth asks whether these problems can be solved in linear time in a more abstract setting, given only the orientation of each point triple, i.e., the setʼs chirotope, as a source of information. We answer this question in the affirmative. More precisely, we can find a halving line through any given point, as well as the vertices of the convex hull edges that are intersected by the supporting line of any two given points of the set in linear time. We first give a proof for sets realizable in the Euclidean plane and then extend the result to non-realizable abstract order types.

有限点集的许多性质仅取决于点的相对位置，例如，取决于集合的顺序类型。然而，计算几何中的许多基本算法依赖于坐标表示。这包括用于为给定的平面点集找到一半线的直接算法，以及在线性时间内找到凸包上的一个点。在他的专着Axioms and Hulls 中, Knuth 询问是否可以在更抽象的设置中在线性时间内解决这些问题，仅给定每个点三元组的方向，即集合的 chirotope，作为信息源。我们肯定地回答这个问题。更准确地说，我们可以找到一条通过任何给定点的等分线，以及在线性时间内与集合中任何两个给定点的支撑线相交的凸包边缘的顶点。我们首先给出在欧几里得平面中可实现的集合的证明，然后将结果扩展到不可实现的抽象顺序类型。