Geometric- and parametric-tolerance constraints in variational design of multiresolution curves and surfaces

The paper introduces constraints termed tolerance constraints, which specify several types of variations, in the design of smooth curves and surfaces at multiresolution levels. The mathematical model for such tolerance constraints is implemented by extending Welch and Witkin's (1992) work on linear constraints in variational shape sculpting. The tolerance constraints presented in this paper are classified into two types: geometric-tolerance and parametric-tolerance constraints. The geometric-tolerance constraints serve as constraints that allow variations in geometric size, while the parametric ones introduce variations in the parametric domain where the shape is defined. These two types are employed as not only finite-dimensional constraints, such as points and tangents, but also transfinite constraints, such as curves and areas. In order to find a smooth shape of a curve or a surface, the multiplier method is used that seeks to minimize the function subject the shape deformation, along with the penalty terms derived from the tolerance constraints. An optimal solution is then found easily because the derivatives of the penalty terms can be evaluated using vector and matrix calculations. Several design, examples are presented to show that the tolerance constraints are powerful tools for finding optimal shapes of multiresolution curves and surfaces.