Motivated by the successful application of geometry to proving the Harary-Hill Conjecture for "pseudolinear" drawings of $K_n$, we introduce "pseudospherical" drawings of graphs. A spherical drawing of a graph $G$ is a drawing in the unit sphere $\mathbb{S}^2$ in which the vertices of $G$ are represented as points -- no three on a great circle -- and the edges of $G$ are shortest-arcs in $\mathbb{S}^2$ connecting pairs of vertices. Such a drawing has three properties: (1) every edge $e$ is contained in a simple closed curve $\gamma_e$ such that the only vertices in $\gamma_e$ are the ends of $e$; (2) if $e\ne f$, then $\gamma_e\cap\gamma_f$ has precisely two crossings; and (3) if $e\ne f$, then $e$ intersects $\gamma_f$ at most once, either in a crossing or an end of $e$. We use Properties (1)--(3) to define a pseudospherical drawing of $G$. Our main result is that, for the complete graph, Properties (1)--(3) are equivalent to the same three properties but with "precisely two crossings" in (2) replaced by "at most two crossings". The proof requires a result in the geometric transversal theory of arrangements of pseudocircles. This is proved using the surprising result that the absence of special arcs ( coherent spirals) in an arrangement of simple closed curves characterizes the fact that any two curves in the arrangement have at most two crossings. Our studies provide the necessary ideas for exhibiting a drawing of $K_{10}$ that has no extension to an arrangement of pseudocircles and a drawing of $K_9$ that does extend to an arrangement of pseudocircles, but no such extension has all pairs of pseudocircles crossing twice.

翻译：由成功应用几何来证明 Harary- Hill 螺旋曲线曲线曲线的边缘值为 $\ mathb{S\\2$ 以证明 $K_n美元 的“ 假数字线性” 绘图值。 我们引入了“ 假数字” 图形图的“ 假数字” 。 图形$G$ 的球形图是单位域 $\ mathbb{S\\2$ 美元 的绘图 $G$ ; 如果 美元以美元表示为点, 则在大圆上没有三千美元, 美元是最短的 美元 。 如果 美元以美元表示的直径值表示直角值, 则以美元表示的直角值表示 。 如果 美元表示的直径是直径, 则以美元表示的直角值表示的直径值 。