Variable-radius blending by constrained spine generation

Abstract

Radius blends, very important in geometric and solid modeling, can be seen as the trimmed envelope of a rolling sphere or a sweeping circle with a constant or variable radius that centers on a spine curve and touches the surfaces to be blended along the linkage curves. Usually, in variable-radius blending, the radius is difficult to specify, and the spine curve is hard to trace. We propose several geometric constraints to specify the variable radius, which we then translate to a nonlinear system to represent the spine curve exactly. This is finally traced numerically in a high-dimensional space. We also propose a paradigm that implements the constraints while tracing along the spine curve in 3D space. We represent the result in parametric form.