三维空间中八个点的翻转图是连通的
Flip Graphs for Eight Points in Three Dimensions Are Connected
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中文总结 AI 辅助
本文证明三维空间中四至八个点（无四点共面）的完全几何翻转图在2↔3翻转下连通，通过径向投影和归纳法，最多35次翻转可达放置形式。
中文摘要 AI 辅助
我们证明了三维空间中任意四个到八个点的配置（无四点共面）在 $2 \leftrightarrow 3$ 翻转下具有连通的完全几何翻转图。在整个序列中，每个点保持固定且始终存在。证明将每个四面体剖分转化为凸包顶点处的放置形式：不与该顶点关联的四面体填充剩余点的凸包。这一化简使我们能够利用正则四面体剖分的连通性，通过归纳法建立连通性。主要的几何工具是径向投影，它将与凸包顶点关联的四面体转化为平面三角剖分。当该平面三角剖分是正则的时，改变其提升高度会产生合法的空间翻转，逐步缩小关联四面体所占区域，并达到放置形式。平面提升准则和不同凸包顶点处投影之间的相容性约束解决了剩余的小规模情形。对于八个点，最多需要35次翻转即可达到放置形式。
英文摘要
We prove that every configuration of four through eight points in three-dimensional space, with no four points coplanar, has a connected full geometric flip graph under $2 \leftrightarrow 3$ flips. Every point remains fixed and present throughout the sequence. The proof brings each tetrahedralization into placing form at a convex hull vertex: the tetrahedra not incident to that vertex fill the convex hull of the remaining points. This reduction allows us to establish connectivity by induction, using the connectivity of regular tetrahedralizations. The main geometric tool is radial projection, which turns the tetrahedra incident to a hull vertex into a planar triangulation. When this planar triangulation is regular, varying its lifting heights produces legal spatial flips that progressively shrink the region occupied by the incident tetrahedra and reach placing form. Planar lifting criteria and compatibility constraints between the projections at different hull vertices resolve the remaining small cases. For eight points, at most 35 flips are needed to reach placing form.
