Shape Modification Tools

Abstract

The purpose of this chapter is to develop tools which allow a designer to interactively make local modifications to an existing NURBS curve or surface, in a way that is natural and intuitive. A NURBS curve or surface is defined by its control points, weights, and knots; modifying any of these changes the shape of the curve or surface. Generally designers do not want to work with such concepts; they prefer to specify constraints and shapes which apply directly to the curve or surface. In this chapter we develop tools which take such constraints and shape specifications and convert them into modifications in control point locations or weight values. Specifically, Sections 11.2 and 11.3 are based on specifying point constraints (i.e., the curve or surface is to change locally so that it passes through a given point); multiple point and derivative constraints are handled in Section 11.5, allowing control of tangent and curvature; and the methods of Section 11.4 allow a designer to specify the “shape” that a curve or surface is to assume in a selected region. These geometrically intuitive inputs are converted into control point relocations in Sections 11.2, 11.4, and 11.5, and into weight modifications in Section 11.3. The influence of control points and weights on shape is geometrically clear and intuitive. This was already shown, qualitatively, in previous figures, e.g., Figures 3.7, 3.23, 4.4, 4.6, 4.11, and 4.12; this chapter quantifies this influence. Although knot locations also affect shape, we know of no geometrically intuitive or mathematically simple interpretation of this effect, hence we present no techniques for moving knot locations.