Motivated by applications in CNC machining, we provide a characterization of surfaces which are enveloped by a one-parametric family of congruent rotational cones. As limit cases, we also address ruled surfaces and their offsets. The characterizations are higher order nonlinear PDEs generalizing the ones by Gauss and Monge for developable surfaces and ruled surfaces, respectively. The derivation includes results on local approximations of a surface by cones of revolution, which are expressed by contact order in the space of planes. To this purpose, the isotropic model of Laguerre geometry is used as there rotational cones correspond to curves (isotropic circles) and higher order contact is computed with respect to the image of the input surface in the isotropic model. Therefore, one studies curve-surface contact that is conceptually simpler than the surface-surface case. We show that, in a generic case, there exist at most six positions of a fixed rotational cone that have third order contact with the input surface. These results are themselves of interest in geometric computing, for example in cutter selection and positioning for flank CNC machining.

受CNC加工中应用程序的激励，我们提供了由一参数全等旋转圆锥包围的表面的特征。作为极限情况，我们还讨论了直纹曲面及其偏移。这些特征是高阶非线性PDE，它们分别由Gauss和Monge将其推广到可展曲面和直纹曲面。该推导包括通过旋转锥对表面进行局部近似的结果，这些结果由平面空间中的接触顺序表示。为此，使用Laguerre几何形状的各向同性模型，因为那里的旋转锥对应于曲线（各向同性圆），并且在各向同性模型中相对于输入表面的图像计算出更高阶的接触。因此，一项研究的曲线表面接触在概念上比表面情况更简单。我们证明，在一般情况下，固定旋转锥的至多六个位置与输入表面具有三次接触。这些结果本身在几何计算中很有趣，例如在侧翼CNC加工的刀具选择和定位中。