Computing the Voronoi diagram of mixed geometric objects in $R^3$ is challenging due to the high cost of exact geometric predicates via Cylindrical Algebraic Decomposition (CAD). We propose an efficient exact verification framework that characterizes the parameter space connectivity by computing certified topological transition sets. We analyze the fundamental non-quadric case: the trisector of two skew lines and one circle in $R^3$. Since the bisectors of circles and lines are not quadric surfaces, the pencil-of-quadrics analysis previously used for the trisectors of three lines is no longer applicable. Our pipeline uses exact symbolic evaluations to identify transition walls. Jacobian computations certify the absence of affine singularities, while projective closure shows singular behavior is isolated at a single point at infinity, $p_{\infty}$. Tangent-cone analysis at $p_{\infty}$ yields a discriminant $Δ_Q = 4ks^2(k-1)$, identifying $k=0,1$ as bifurcation values. Using directional blow-up coordinates, we rigorously verify that the trisector's real topology remains locally constant between these walls. Finally, we certify that $k=0,1$ are actual topological walls exhibiting reducible splitting. This work provides the exact predicates required for constructing mixed-object Voronoi diagrams beyond the quadric-only regime.

翻译：在 $R^3$ 中计算混合几何对象的 Voronoi 图面临挑战，因为通过柱形代数分解（CAD）的精确几何谓词成本高昂。我们提出一种高效的精确验证框架，通过计算经过认证的拓扑转移集来刻画参数空间的连通性。我们分析基本的非二次曲面情形：$R^3$ 中两条异面直线与一个圆构成的三平割线。由于圆与直线的平割线并非二次曲面，先前用于三条直线三平割线的二次曲面束分析不再适用。我们的流程采用精确符号评估来识别转移壁。Jacobian 计算确保不存在仿射奇点，而射影闭包表明奇异行为仅孤立于无穷远点 $p_{\infty}$。在 $p_{\infty}$ 处的切锥分析导出了判别式 $\Delta_Q = 4ks^2(k-1)$，识别出 $k=0,1$ 为分岔值。利用定向爆炸坐标，我们严格验证了三平割线的实拓扑在这些壁之间保持局部不变。最后，我们证明 $k=0,1$ 是呈现可约分裂的实际拓扑壁。这项工作提供了构建超越二次曲面情形的混合对象 Voronoi 图所需的精确谓词。