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

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 R2 Euclidean space is obtained that symmetry and equally spaced interval from the design center. Advancement of the mesh onto R3 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 R3 Euclidean spaces that, ultimately, converge as a composite 5-axes response surface graph.