Abstract

Computer-aided design (CAD) typically deals with geometries (points, lines, and planes) subject to constraints on distances and angles. Simple counting of freedoms and constraints, as used in the analysis of engineering structures, also provides a useful condition on the residual freedoms of a CAD drawing. Here, we derive general symmetry-extended counting equations to account more fully for the balance of freedoms and constraints in 2D point–line and 3D point–line–plane frameworks. General forms are given for symmetries of the freedoms of points, lines, and planes, and constraints based on distances and angles. The resulting toolkit can be used to give stronger conditions on dimensioning of CAD drawings. This importation to CAD of a physical point of view, in which residual freedoms correspond to the mechanisms and redundant constraints to the states of self-stress of a structure composed of bodies and joints, can often reveal hidden freedoms and redundancies in CAD systems. Point-group symmetry is not a panacea: Mechanisms that depend on specific geometries may escape detection by symmetry alone. One systematic limitation of this type is proved for polyhedra with planar faces and prescribed edge lengths.

中文翻译：

---

对称性在CAD的点线平面框架中的应用

摘要

计算机辅助设计（CAD）通常处理受距离和角度约束的几何形状（点，线和平面）。用于工程结构分析的自由度和约束的简单计数也为CAD工程图的剩余自由度提供了有用的条件。在这里，我们得到一般对称扩展对方程进行计数，以更充分地说明2D点线和3D点线平面框架中的自由度和约束度之间的平衡。给出了点，线和面自由度的对称性以及基于距离和角度的约束的一般形式。生成的工具包可用于为CAD工程图的尺寸标注提供更强的条件。物理观点的这种CAD输入，其中剩余的自由度对应于由机构和关节组成的结构的自应力状态的机制和冗余约束，通常可以揭示出CAD系统中隐藏的自由度和冗余。点组对称性不是万能药：依赖于特定几何形状的机制可能无法通过对称性单独进行检测。