Blends of canal surfaces from polyhedral medial transform representations

Area of Science:

• Computer-Aided Design
• Geometric Modeling
• Computational Geometry

Background:

• Constructing smooth blending surfaces between complex geometric shapes is a fundamental challenge in geometric modeling.
• Existing methods often struggle with blending multiple surfaces or handling complex topological relationships.

Purpose of the Study:

• To present a novel and robust method for generating blend surfaces between an arbitrary number of canal surfaces.
• To leverage a generalization of the medial surface transform for advanced blend surface construction.

Main Methods:

• The topological relationships of canal surfaces are defined using a convex polyhedron.
• The core technique involves a generalization of the medial surface transform.
• The blend surfaces are constructed as trimmed envelopes of one- and two-parameter families of spheres.

Main Results:

• A new method for constructing blend surfaces between multiple canal surfaces is successfully developed.
• The approach utilizes the medial surface transform of the input surfaces, rather than the surfaces themselves.
• The resulting blend surfaces are composed of trimmed envelopes of spherical families.

Conclusions:

• Blending the medial surface transform provides a powerful and elegant solution for generating complex blend surfaces.
• The proposed method offers a significant advancement in the field of geometric modeling and surface construction.
• The effectiveness of the approach is validated through various practical examples.