Kernel-based construction operators for Boolean sum and ruled geometry

Boolean sum and ruling are two well-known construction operators for both parametric surfaces and trivariates. In many cases, the input freeform curves in [Formula presented] or surfaces in [Formula presented] are complex, and as a result, these construction operators might fail to build the parametric geometry so that it has a positive Jacobian throughout the domain. In this work, we focus on cases in which those constructors fail to build parametric geometries with a positive Jacobian throughout while the freeform input has a kernel point. We show that in the limit, for high enough degree raising or enough refinement, our construction scheme must succeed if a kernel exists. In practice, our experiments, on quadratic, cubic and quartic Bézier and B-spline curves and surfaces show that for a reasonable degree raising and/or refinement, the vast majority of construction examples are successful.