HelSweeper: Screw-sweeps of canal surfaces

doi:10.1016/j.cad.2011.09.014

Computer-Aided Design

Volume 44, Issue 2, February 2012, Pages 113-122

https://doi.org/10.1016/j.cad.2011.09.014 Get rights and content

Abstract

A tube is a solid bounded by the union of a one-parameter family of circles that may be decomposed into canal-surfaces and planar disks or annuli. A screw-sweep is the region swept by a shape during a screw motion. HelSweeper computes the boundary of a screw-sweep of an arbitrary union of tubes and polyhedra. To do so, it generates a superset of faces, splits them at their intersections, and selects the face portions that form the desired boundary. The novelty of the proposed approach lies in the fact that the faces contributed to this superset by a tube are each a screw-sweeps of a rigid curve (generator), which is the locus of grazing points, and that each grazing point is formulated as the intersection of a circle of the tube with a corresponding screw-plane. Hence, each such face is a one-parameter family of helices, each being the screw-sweep of a grazing point.

Highlights

► We compute the boundary of a screw-sweep supporting a broad set of canal surfaces. ► We approximate the motion as polyScrew and compute the swept solid as a union of screw-sweeps. ► We formulate generators as intersections between circles and screw-planes.

Introduction

A tube is a solid bounded by a surface that is the union of a one parameter family of disjoint circles and that can be decomposed into the closure of the union of relatively open canal surfaces, disks, and annuli. Our use of the term “tube” should not be confused with the term “tubular surface”. A screw-sweep is the region swept by a shape during a screw motion. We propose a novel formulation for a superset of faces, the union of which contains the boundary of the screw-sweep of a tube. We present the use of this formulation in the HelSweeper extension of a commercial CAD system to support screw-sweeps of unions of possibly interfering tubes and polyhedra (Fig. 1).

Supporting screw-motion sweeps of tubes is important for a variety of applications. Indeed, the tube is a common solid modeling primitive in CAD/CAM/CAE (cones, cylinders, tori, Dupin cyclides [1], constant and variable radius blends [2], [3], pipes, ducts, wires [4], [5], [6]), in computer vision (generalized cylinders [7]), and in medicine (vasculature [8], baffles [9]). Sweeps are important for modeling manufacturing processes [10], [11], [12], [13], [14], for planning and testing assembly/disassembly motions [15], [16], [17] and in robotics [18], [19]. Finally, the screw is the fundamental instantaneous rigid motion and has been widely used in mechanical engineering [20], [16], animation [21], [22], shape editing [23], [24], robotics [25], and CAD/CAM [17], [26].

Section snippets

Background and prior art

In this section, we provide definitions of our terms and notation and highlight the novelty and importance of our contribution in the context of prior art.

Proposed solution

Here, we explain the details of the derivation of our construction of $\partial T$ for a single screw or for a single screw-segment of a polyScrew motion.

Implementation and results

The HelSweeper system described in the paper was implemented in $C$ using CAA (CATIA Application Architecture), the API of CATIA [43], [44]. Through the CAA API, HelSweeper has control over the major functions of CATIA and access to the geometric entities created, such as generators and extrusion faces. To compute grazing points of the primitives in Fig. 5, we used a plane/circle intersection function provided by CATIA. We sampled the generators and screws using spline functions in CATIA.

Before

Conclusions

We propose an efficient technique for computing the boundary of the region swept by a tube undergoing a screw or piecewise-screw motion. ‘Our approach computes sweeps of solid objects that are unions of polyhedra, spheres, cylinders, cones, tori, Dupin cyclides, and more general tubes. It computes the parametric curves of the generators of the grazing points on each tube using circle/plane intersections and exploits the fact that they are invariant during screw motions. Our HelSweeper

Acknowledgments

Rossignac’s involvement in this work was supported in part by the CPA-G & V-T grant number 0811485 from the National Science Foundation and Kim’s by the Survivability Technology Defense Research Center of Agency for Defense Development and Defense Acquisition Program Administration of Korea under the contract No. UD090090GD.

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View full text

HelSweeper: Screw-sweeps of canal surfaces

Abstract

Highlights

Introduction

Section snippets

Background and prior art

Proposed solution

Implementation and results

Conclusions

Acknowledgments

Computer-Aided Design

Computer Aided Geometric Design

Computer-Aided Design

Computer-Aided Design

International Journal of Machine Tools & Manufacture

Computer Aided Design

Computer-Aided Design

Computer-Aided Design

Journal of Symbolic Computation

Computer-Aided Design

Computer-Aided Design

Computer Aided Geometric Design

On the geometry of dupin cyclides

The Visual Computer

Constant radius blending in solid modeling

Computers in Mechanical Engineering

Geometric modeling

Patient-specific surgical planning and hemodynamic computational fluid dynamics optimization through free-form haptic anatomy editing tool (SUGEM)

Medical and Biological Engineering and Computing

Geometric modeling for swept volume of moving solids

IEEE Computer Graphics and Applications

Five-axis swept volumes for graphic NC simulation and verification

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Swept volume determination and interference of moving 3-D solids

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JSME International Journal Series C