Constructive Sculpting using 4D Spline Volumes

This paper presents an approach to constructive modelling of FRep solids defined by real-valued functions using Bspline volumes as primitives. A 4D uniform rational cubic Bspline volume is employed to define a 3D solid. While the first three coordinates are used to represent the spatial component of the volume to be sculpted, the fourth coordinate is used as a scalar, which corresponds to a function value or a volume density. Thus, the shape can be manipulated by changing the scalar control coefficients of the spline volume. This modelling process is interactive as the isosurface can be polygonized and visualized in real time. The distance property we obtain, combined with the properties of the spline volumes, allow us to use the resulting 3D solid as a leaf of a constructive modelling tree and to apply to it set-theoretic, blending and other operations defined using R-functions. Additional deformations can be achieved by moving arbitrary points in the coordinate space and applying space mapping at any level of the constructive tree. The final constructive solid is defined by a single real-valued function evaluated by the tree traversing procedure.