Design of solids with free-form surfaces

Abstract

I. IntroductionWe propose a unified method of generating a wide range of three dimensional objects from polyhedra to solids with free-form surfaces.Modeling systems for polyhedra and systems for free-form surfaces have been developed independently in the past because of the difference in their underlying theory and practices. However, this is not desirable for a designer.So in this paper, we have shown a method in which a wide range of shapes are generated in one system by using local modifications. Local modifications are procedures used to change the shape of solids locally. The construction and the modification of three dimensional shapes by these procedures are natural and easy for a designer in many cases. The implementation of these procedures in a computer is easy and their execution does not require much time~ Our method to construct a solid with free-form surfaces consists of following three phases, i) A solid which serves as a basis of free-form shape design is generated by local modifications. Edges of this solid are straight lines but its faces are not necessarily flat planes. 2) From this model, a curve model which adequately represents the characteristics of a free-form shape is generated.3) Surface equations interpolating over the curve model are generated.We have made a geometric modeling system MODIF. Using this system, a complicated solid with free-form surfaces can be designed easily. MODIF can generate color shaded pictures and cutter path data for making a real object model by NC machining tool.

Revisiting Traditional Curve Lofting to Improve the Hull Surface Design Process

“…Methods such as the Coons Patch with Gregory Square or Browns Square or the Gregory Bezier patch, Chiyokura (1983), can handle twist. Methods such as NURBS, with regular control polygon grids, are particularly poor at representing twist resulting in subtle unfairness due to loss of continuity between patches and warping of the surface.…”

Splines and geometric modeling

“…Chiyokura and Kimura [24] construct a solid with free-form surfaces. They start with a polyhedral solid which can be edited usi[lg local modifications and generate a curve model for the solid consisting of cubic Bezier curves and straight lines.…”

Smooth Parametric Surfaces and n-Sided Patches

“…A rationally corrected form of the bicubic Bernstein-Bézier patch can also be derived, by introducing two different control points for each of the four interior control points of the patch, as shown in Figure 3.4, see Chiyokura and Kimura [19] and Chiyokura [18].…”