Developable surfaces have many desired properties in the manufacturing process. Since most existing CAD systems utilize tensor-product parametric surfaces including B-splines as design primitives, there is a great demand in industry to convert a general free-form parametric surface within a prescribed global error bound into developable patches. In this paper, we propose a practical and efficient solution to approximate a rectangular parametric surface with a small set of 􀀀-joint developable strips. The key contribution of the proposed algorithm is that, several optimization problems are elegantly solved in a sequence that offers a controllable global error bound on the developable surface approximation. Experimental results are presented to demonstrate the effectiveness and stability of the proposed algorithm.

This paper addresses the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. The basic idea of the approach is to find the conditions for precise merging of two B-spline curves, and perturb the control points of the curves by constrained optimization subject to satisfying these conditions. To obtain a merged curve without superfluous knots, we present a new knot adjustment algorithm for adjusting the end k knots of a kth order B-spline curve without changing its shape. The more general problem of merging curves to pass through some target points is also discussed.

The ability to directly manipulate an embedded object in the free-form deformation (FFD) method improves controllability. However, the existing solution to this problem involves a pseudo-inverse matrix that requires complicated calculations. This paper solves the problem using a constrained optimization method. We derive the explicit solutions for deforming an object which is to pass through a given target point. For constraints with multiple target points, the proposed solution also involves simple calculations, only requiring solving a system of linear equations.We show that the direct manipulations exhibit the commutative group property, namely commutative, associative, and invertible properties, which further enhance the controllability of FFD. In addition, we show that multiple point constraints can be decomposed into separate manipulations of single point constraints, thus providing the user the freedom of specifying the constraints in any appropriate order.

Identifying multiple deformable parts on meshes and establishing dense correspondences between them are tasks of fundamental importance to computer graphics, with applications to e.g. geometric edit propagation and texture transfer. Much research has considered establishing correspondences between non-rigid surfaces, but little work can both identify similar multiple deformable parts and handle partial shape correspondences. This paper addresses two related problems, treating them as a whole: (i) identifying similar deformable parts on a mesh, related by a non-rigid transformation to a given query part, and (ii) establishing dense point correspondences automatically between such parts. We show that simple and efficient techniques can be developed if we make the assumption that these parts locally undergo isometric deformation. Our insight is that similar deformable parts are suggested by large clusters of point correspondences that are isometrically consistent. Once such parts are identified, dense point correspondences can be obtained by an iterative propagation process. Our techniques are applicable to models with arbitrary topology. Various examples demonstrate the effectiveness of our techniques.

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