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This paper describes a computer method for transforming an arbitrary developable surface into a flattened pattern. Developable surfaces are a special class of ruled surfaces. The Gaussian curvature of any point of the developable surfaces is zero. Some curves on developable surfaces are geodesic curves and the geodesic curvatures of all the points on these curves are zero. The flattening technique introduced in this paper is based on the geodesic curve length preservation and linear mapping principles. Both trimmed and untrimmed developable surfaces can be flattened by the algorithm.

There is an increasing demand in the conceptual design for more intuitive methods for creating and modifying free-form curves and surfaces in CAD modeling systems. The methods should be based not only on the change of the mathematical parameters but also on the user’s specified constraints and shapes. This paper presents a new surface representation model for free-form surface deformation representation. The model is a combination of two functions: a displacement function and a function for representing an existing NURBS surface called parent surface. Based on the surface model, the authors develop two deformation methods which are named SingleDef (Single-point constraint based deformation method), and MultiDef (Multiple-points constraints based deformation method). The techniques for free-form surface deformation allow conceptual designer to modify a parent surface by directly applying point constraints to the parent surface. The deformation methods are implemented and taken in an experimental CAD system. The results show that the designer can easily and intuitively control the surface shape.

In this paper, methods for generating and flattening developable surfaces by means of two given directrix curves have been investigated. By finding the two end points of every generatrix, a developable surface may be formed. Such a developable surface may be flattened by rotating part of the surface about a generatrix. A new approach based on the principle of geodesic curve length preservation for flattening developable surfaces is also introduced. In this technique, an algorithm for evaluating the geodesic curvature of a point of a curve that lies on a surface is utilized.