Medial axis transform to boundary representation conversion
The medial axis transform (MAT) has potential as a powerful representation for a conceptual design tool for objects with inherent symmetry or near-symmetry. The medial axis of two-dimensional objects or medial surface of three-dimensional objects provides a conceptual design base, with transition to a detailed design occurring when the radius function is added to the medial axis or surface, since this additional information completely specifies a particular object. To make such a design tool practicable, however, it is essential to be able to convert from an MAT format to a boundary representation of an object. In this thesis, we provide the details for the conversion of the MAT of a set of two- and three-dimensional objects to a boundary representation. We demonstrate certain smoothness properties of the MAT and show the relationship between the tangent to the MAT at a point and the boundary points related to that MAT point. We classify the MAT points based on the tangency conditions at the point, and for each type of point, we detail the method for obtaining the boundary points related to it. We discuss requirements for an MAT to be locally valid in the sense that the given curves could actually be the MAT of an allowable object. We also provide a theoretical error bound on the computation for the two-dimensional case. Finally, we discuss an implementation of our algorithm both for piecewise linear two-dimensional MATs and for piecewise planar and linear three-dimensional MATs, and demonstrate some results we have obtained.
Hoffmann, Purdue University.
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