Spiral fat arcs - Bounding regions with cubic convergence

Michael Bartoň; Gershon Elber

doi:10.1016/j.gmod.2010.10.005

Spiral fat arcs - Bounding regions with cubic convergence

Michael Bartoň^*, Gershon Elber

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

A bounding region for spiral curve segments shaped by two circular arcs, parts of the osculating circles at the spiral's endpoints, and two lines is introduced. This bounding region, denoted spiral fat arc (SFA) is simple to construct and process, and shows a cubic approximation order to a given spiral curve. Given a general planar parametric curve, it can be split at curvature extrema (and inflection points), solving for the parametric locations for which κ′ = 0 (and κ = 0), κ being the signed curvature field, to yield a set of spiral curves. Each of the spirals is then fitted with a bounding SFA. Finding the intersection locations of two free-form planar curves is a fundamental task in geometric computing and computer aided design, and can immediately benefit from this new SFA bounding region. A recursive curve-curve intersection (CCI) algorithm that efficiently computes the intersection location of two parametric curves using SFAs is also introduced.

Original language	English (US)
Pages (from-to)	50-57
Number of pages	8
Journal	Graphical Models
Volume	73
Issue number	2
DOIs	https://doi.org/10.1016/j.gmod.2010.10.005
State	Published - Mar 2011

Keywords

Bounding regions
Curve-curve intersection
Fat arcs
Monotone curvature
Spiral

ASJC Scopus subject areas

Software
Modeling and Simulation
Geometry and Topology
Computer Graphics and Computer-Aided Design

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10.1016/j.gmod.2010.10.005

Cite this

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title = "Spiral fat arcs - Bounding regions with cubic convergence",

abstract = "A bounding region for spiral curve segments shaped by two circular arcs, parts of the osculating circles at the spiral's endpoints, and two lines is introduced. This bounding region, denoted spiral fat arc (SFA) is simple to construct and process, and shows a cubic approximation order to a given spiral curve. Given a general planar parametric curve, it can be split at curvature extrema (and inflection points), solving for the parametric locations for which κ′ = 0 (and κ = 0), κ being the signed curvature field, to yield a set of spiral curves. Each of the spirals is then fitted with a bounding SFA. Finding the intersection locations of two free-form planar curves is a fundamental task in geometric computing and computer aided design, and can immediately benefit from this new SFA bounding region. A recursive curve-curve intersection (CCI) algorithm that efficiently computes the intersection location of two parametric curves using SFAs is also introduced.",

keywords = "Bounding regions, Curve-curve intersection, Fat arcs, Monotone curvature, Spiral",

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AU - Bartoň, Michael

AU - Elber, Gershon

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AB - A bounding region for spiral curve segments shaped by two circular arcs, parts of the osculating circles at the spiral's endpoints, and two lines is introduced. This bounding region, denoted spiral fat arc (SFA) is simple to construct and process, and shows a cubic approximation order to a given spiral curve. Given a general planar parametric curve, it can be split at curvature extrema (and inflection points), solving for the parametric locations for which κ′ = 0 (and κ = 0), κ being the signed curvature field, to yield a set of spiral curves. Each of the spirals is then fitted with a bounding SFA. Finding the intersection locations of two free-form planar curves is a fundamental task in geometric computing and computer aided design, and can immediately benefit from this new SFA bounding region. A recursive curve-curve intersection (CCI) algorithm that efficiently computes the intersection location of two parametric curves using SFAs is also introduced.

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Spiral fat arcs - Bounding regions with cubic convergence

Abstract

Keywords

ASJC Scopus subject areas

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