Applications of symmetry in point–line–plane frameworks for CAD

Abstract

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.