A Theory and Methods for Three Dimensional Free Form Shape Construction

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This paper describes a new mathematical theory for expressing three dimensional free forms of curves and surfaces, and also methods for application in computer aided design with the results obtained from the theory. First, generative expressions for shape segments of curves and surfaces are deduced from an assumption that a shape segment exists within the smallest convex hull spanned by the control points of the segment. These expressions have simple construction and are easy to be manipulated. Then, shape synthesis and control by partitioning, order raising of a segment and/or by smooth connection of segments, are explained with the use of control points. Tangents, osculating planes, torsions and curvatures at any points on curves or surfaces are simply expressed analytically as well as graphically by the local control points of the shape. As for the shape synthesis by connection, detailed discussion and design methodology are given, especially for non-regular patch connection. Equations for vector interpolation with smooth and natural transition are deduced. A curve constructed from the parametric B-spline curve segments is proved to be a special case of this interpolation. This theory and its application methods aid man's comprehension and visualization of three dimensional shapes and control of their geometric characteristics, and accordingly they are useful for interactive shape design and geometric model construction.
一般社団法人情報処理学会の論文
1980-09-30

A Theory and Methods for Three Dimensional Free Form Shape Construction

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