Efficient offset trimming for deformable planar curves using a dynamic hierarchy of bounding circular arcs

Lee, Jae‐Wook; Kim, Yong-Joon; Kim, Myung-Soo; Elber, Gershon

doi:10.1016/j.cad.2014.08.031

Cited by 12 publications

(4 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In 1995, Meek and Walton [21] introduced the concept of BCA as a handy tool for proving the cubic convergence of G 1 biarc approximation of planar B-spline curves. To the best of our knowledge, this bounding volume has never been used for the purpose of accelerating geometric algorithms until Lee et al [19] first used BCA as a tool for speeding up the offset trimming algorithm for planar freeform curves. We believe that BCA has a great potential in accelerating many other geometric algorithms for planar curves, which is the main motivation of the current work.…”

Section: Related Workmentioning

confidence: 99%

“…Nevertheless, it is time consuming to measure the maximum deviation of biarcs from the given curve and thus to determine the thickness of fat arcs. In a recent work on trimming planar offset curves, Lee et al [19] showed that the bounding circular arcs (BCA) of Meek and Walton [21] is considerably more efficient to generate and measure the error bound than fat arcs. In this paper, we compare the relative performance of the BCA-bounding region against spiral fat arcs (SFA) [4] (each constructed with two circular arcs and two line segments) and bilens [17] (with two biarcs, i.e., four circular arcs), all of which are planar curve approximation methods of cubic convergence.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Comparison of three bounding regions with cubic convergence to planar freeform curves

Lee

Kim

et al. 2015

Vis Comput

Self Cite

View full text Add to dashboard Cite

We compare the relative performance of bounding regions generated by three different curve-bounding methods with cubic convergence to planar freeform curves: spiral fat arcs (SFA) (Barton and Elber in Graph Models 73(2):50-57, 2011), bilens (Kumosenko in Comput Aided Geom Des 30(3):310-330, 2013), and bounding circular arcs (BCA) (Meek and Walton in J Comput Appl Math 59 (2): [221][222][223][224][225][226][227][228][229][230][231] 1995). For quantitative comparison, we consider three different criteria: geometric complexity (the number of circular arcs and line segments), construction time, and numerical stability. The BCA construction after one-step refinement (producing four circular arcs) is almost comparable to the other two methods in geometric complexity: the SFA with two circular arcs and two line segments, and the bilens with four circular arcs. In other comparison criteria, the BCA approach is more efficient and stable than the other two methods in producing a hierarchy of bounding regions that approximate a family of freeform planar curves within a given error bound.

show abstract

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Comparison of three bounding regions with cubic convergence to planar freeform curves

Lee

Kim

et al. 2015

Vis Comput

Self Cite

View full text Add to dashboard Cite

show abstract

“…Based on the fat-arc BVH, [Kim et al (2012)] demonstrated highly improved performance in the efficiency as well as the robustness in trimming the planar curve offset self-intersections. [Lee et al (2015a)] further improved the efficiency of planar offset trimming by using a BVH of bounding circular arcs (BCA). [Meek and Walton (1995)] introduced the concept of BCA for the purpose of proving the cubic convergence of their biarc approximation to planar curves.…”

Section: Related Workmentioning

confidence: 99%

“…In the planar case, non-convex bounding regions such as fat arcs (FA) and bounding circular arcs (BCA) are highly effective in accelerating geometric algorithms (such as intersection [Sederberg et al (1989)], offset [Lee et al (2015a)], medial axis, and Voronoi diagram computations [Lee et al (2016)]) for planar freeform curves. (Fat arc is an expansion of arc.)…”

Section: Introductionmentioning

confidence: 99%

Efficient Minimum Distance Computation for Solids of Revolution

Son

Yoon

Kim

et al. 2020

Computer Graphics Forum

View full text Add to dashboard Cite

We present a highly efficient algorithm for computing the minimum distance between two solids of revolution, each of which is defined by a planar cross‐section region and a rotation axis. The boundary profile curve for the cross‐section is first approximated by a bounding volume hierarchy (BVH) of fat arcs. By rotating the fat arcs around the axis, we generate the BVH of fat tori that bounds the surface of revolution. The minimum distance between two solids of revolution is then computed very efficiently using the distance between fat tori, which can be boiled down to the minimum distance computation for circles in the three‐dimensional space. Our circle‐based approach to the solids of revolution has distinctive features of geometric simplification. The main advantage is in the effectiveness of our approach in handling the complex cases where the minimum distance is obtained in non‐convex regions of the solids under consideration. Though we are dealing with a geometric problem for solids, the algorithm actually works in a computational style similar to that of handling planar curves. Compared with conventional BVH‐based methods, our algorithm demonstrates outperformance in computing speed, often 10–100 times faster. Moreover, the minimum distance can be computed very efficiently for the solids of revolution under deformation, where the dynamic reconstruction of fat arcs dominates the overall computation time and takes a few milliseconds.

show abstract

Automatic design for trimming die insert of automotive panel

Kong

Zhou

2020

Int J Adv Manuf Technol

View full text Add to dashboard Cite

Efficient offset trimming for deformable planar curves using a dynamic hierarchy of bounding circular arcs

Cited by 12 publications

References 28 publications

Comparison of three bounding regions with cubic convergence to planar freeform curves

Comparison of three bounding regions with cubic convergence to planar freeform curves

Efficient Minimum Distance Computation for Solids of Revolution

Automatic design for trimming die insert of automotive panel

Contact Info

Product

Resources

About