Relative blending
Introduction
Blending replaces portions of the boundary near the sharp features or constrictions of a solid model by smooth surfaces (blends) that connect tangentially to the remaining part. Some blending operations are associated with one or several edges of a solid. Others produce smooth connections between non-intersecting portions of a solid or of two different solids. The formulation proposed here handles both types. In the absence of aesthetic or dynamic flow considerations, which may impose functional constraints on the nature of the blends, drafting and manufacturing practices often call for circular blends, which have circular cross-sections [1]. Such blends are part of the boundary of a canal surface [2], which is swept [3] by a ball as it rolls while maintaining two or more tangential contacts with the boundary of . When the ball rolls in the interior of , the removed material is called a rounding. When the ball rolls in the exterior of , the added material is called a fillet. If the radius of the ball is constant the corresponding r-rounding and r-filleting operations have a set-theoretic formulation [4] as morphological opening and closing [5] with an r-ball. They may be applied to the entire solid or combined with Boolean operations to restrict their effect to a portion of the solid. In this paper, we propose a set-theoretic formulation for variable-radius blending. The formulation of the proposed relative blending uses a “bounding” solid to control the radius of the rolling ball locally. Intuitively, the rolling ball is required to stay in the symmetric difference, , and to be touching the boundary of and at all times, as shown in 2D by the disks in Fig. 1.
As such, our relative blending is a global operation that blends the entire solid. Hence the designer may not need to invoke specialized blending operations for corners where several blends meet. Because it offers a set theoretic formulation for the resulting set, relative blending may be incorporated within a CSG design paradigm [6], [7] and, if needed, combined with Boolean operations to localize the effect of blending.
Note that when is the Boolean combination of two balls and is a third ball (Fig. 2), the blending surface is a portion of a Dupin cyclide [8]. Dupin cyclides have been successfully used to compute smooth approximations of individual blends. Hence, one may view the proposed approach as an extension of the Dupin cyclide construction to more complex shapes and .
Observe that the spine (curve traced by the center of the rolling ball) must lie on the medial axis (also called Voronoi) surface [9] of (for rounding) or of its complement, !A (for filleting). It has been suggested that variable-radius blends could be specified by defining such a spine, either by drawing an approximating space curve and snapping it to the medial axis surface [10] or by defining it as the intersection of the medial axis surface with a control surface [11]. Another option is to draw one contact curve (also called a linkage curve or spring curve [12]) on and compute the spine from it. The relative blending approach proposed here offers a different specification mechanism, which may prove more convenient for the designer. For example, the designer may start by setting to be identical to and then modifying it locally through precise CSG operations or through less precise warping by pulling its boundary away from the boundary of . Intuitively, pulling further increases the radius of the blend.
Finally, note that our relative blending may be used to perform both filleting and rounding in a single blending operation.
In conclusion, we suggest that the proposed relative blending paradigm offers a useful alternative to previously proposed approaches to the specification of variable radius blending, because it offers a simple set-theoretic formulation, operates globally on an entire solid, and simplifies the specification of complex blends.
To validate and demonstrate the proposed formulation, we have implemented it in two dimensions for cases where both and are bounded by PCCs (curves made of smoothly-connected line segments and circular arcs) [13] and in three dimensions by voxelizing [14], [15] and , computing the volumetric model of the relative blending of with respect to and generating the result as an iso-surface. Both implementations are straightforward. Note that the first one is exact, while the second one is an approximation.
The computation of the exact shape of these blends in 3D is tractable for simple situations where the blend is part of a Dupin cyclide, such as the example in Fig. 2. In general, however, the resulting canal surfaces are highly complex [16]. Hence, we advocate approximation or discretization, as was previously done for constant and variable radius blends [13], [16]. Because the combination of the voxelization process and its reverse (the iso-surfaces extraction) perform a resampling, they will typically alter away from the blends. Hence, a more precise approach is to trace the variable-radius canal surfaces by “rolling” a variable radius ball that maintains two tangential contacts with the boundary of and one with the boundary of . The center of the rolling ball (i.e. the spine of the blend) follows an edge (seam [17]) of the Voronoi (medial axis) surface complex of . For simplicity, throughout the paper, we assume that and are closed-regularized (i.e. equal to the closure to their interior).
The remainder of the paper is organized as follows. We first discuss prior art. Then we present the set theoretic formulation of relative blending. Then, we discuss two different implementations. Finally, we propose user interface options and mention applications.
Section snippets
Related work
To put our contribution in perspective with a vast body of prior art on blending, we organize prior approaches into two categories: (1) The proposed relative blending approach is a ball-rolling technique, hence we contrast it with previously proposed constant-radius and variable-radius ball-rolling techniques. (2) A broad variety of other techniques for smoothing the sharp features of 2D shapes or 3D solids have been proposed. We briefly mention a few examples to provide a global context of our
Set-theoretic formulation of relative blending
Consider two shapes, and . We use respectively , , , , and , to denote the complement, interior, exterior, boundary, and closure of set . As mentioned previously, we assume that and are each closed-regularized. We use the term moat and symbol to refer to . Note that is the union of the boundaries and with the symmetric difference, , of the two shapes. Let be the set of all “maximal” balls of possibly zero radius in that tangentially intersect both
Computation
Even though implementation details are not the focus of the present paper, we include below an outline of two different implementations of relative blending. Many others are possible.
Suggested interface and applications
We briefly discuss three applications of relative blending.
Conclusions
The contributions reported here include the invention of the concept of relative blending and a set theoretic formulation of it. We outline practical approaches for implementing relative blending and propose a novel interface for specifying the bounding shape that controls the radius. We discuss several applications of relative blending, including interactive design and shape comparison. We hope that these initial contributions will fuel further research on efficient algorithms for computing
Acknowledgements
The authors wish to thank Dr. Lieutier and Dr. Chazal for their suggestions and contributions to the ideas reported here.
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