Geometric constraint modelling: Boundary planar B-spline curves and control polyhedra for 5-axes response surface graph

摘要

This paper introduces a geometric constraint modelling system for 5-axes response surface graph based on free-form boundary planar curves and cross-boundary curves. Both are developed on the basis of nonperiodic B-spline curves. It is evident that generating the boundary planar curves is better done symmetrically on the main planes and the 45°-angular planes. While, generating the cross-boundary curves require to additionally identify and involve control points on the 30°- and 60°-angular planes. As a result, a mesh 3×4 size on parametric R~2 Euclidean space is obtained that symmetry and equally spaced interval from the design center. Advancement of the mesh onto R~3 Euclidean space forms the control polyhedra. Then, the B-spline surface will shape uniquely to follow the polyhedra. The emerging B-spline surfaces are estimated in partial on four different R~3 Euclidean spaces that, ultimately, converge as a composite 5-axes response surface graph.