Analytic properties of plane offset curves

doi:10.1016/0167-8396(90)90023-K

Computer Aided Geometric Design

Volume 7, Issues 1–4, June 1990, Pages 83-99

https://doi.org/10.1016/0167-8396(90)90023-K Get rights and content

Abstract

We survey the principal geometric and topological features of plane offset curves. With appropriate sign conventions, the irregular points of the offset at distance d from a regular generator curve arise where the generator has curvature $κ=− 1 d$ . Usually, this induces a cusp on the offset, but if κ is also a local extremum, we observe instead a tangent-continuous extraordinary point of infinite curvature. Such irregular points are intimately related to the evolute, or locus of centers of curvature, of the generator. Certain special regular points are then identified: those of horizontal or vertical tangent, and those where the curvature or its derivative vanish. A one-to-one correspondence (with due allowance for irregularities) is established between such characteristic points on the generator and its offsets at each distance d. In the absence of irregular points, simple relations between certain global properties of the generator and offset curves, such as their arc length, the area they bound, and their mean square curvature or “smoothness” may be derived. The self-intersections of offset curves, and the trimming of certain extraneous loops they delineate, are also addressed.

References (20)

R.T. Farouki
Hierarchical segmentations of algebraic curves and some applications
R.T. Farouki et al.
Algebraic properties of plane offset curves
Computer Aided Geometric Design
(1990)
J. Hoschek
Offset curves in the plane
Computer Aided Design
(1985)
J. Hoschek
Spline approximation of offset curves
Computer Aided Geometric Design
(1988)
J. Hoschek et al.
Optimal approximate conversion of spline curves and spline approximation of offset curves
Computer Aided Design
(1988)
R. Klass
An offset spline approximation for plane cubic splines
Computer Aided Design
(1983)
B. Pham
Offset approximation of uniform B-splines
Computer Aided Design
(1988)
J.H. Ahlberg et al.
The Theory of Splines and their Applications
(1967)
T. Banchoff et al.
Cusps of Gauss Mappings
(1982)
J.W. Bruce et al.
Curves and Singularities
(1984)

There are more references available in the full text version of this article.

Cited by (168)

Partition of the space of planar quintic Pythagorean-hodograph curves
2023, Computer Aided Geometric Design
The quintics are the lowest–order planar Pythagorean–hodograph (PH) curves suitable for free–form design, since they can exhibit inflections. A quintic PH curve $r (t)$ may be constructed from a complex quadratic pre–image polynomial $w (t)$ by integration of $r^{'} (t) = w^{2} (t)$ , and it thus incorporates (modulo translations) six real parameters — the real and imaginary parts of the coefficients of $w (t)$ . Within this 6–dimensional space of planar PH quintics, a 5–dimensional hypersurface separates the inflectional and non–inflectional curves. Points of the hypersurface identify exceptional curves that possess a tangent–continuous point of infinite curvature, corresponding to the fact that the parabolic locus specified by the quadratic pre–image polynomial $w (t)$ passes through the origin of the complex plane. Correspondingly, extreme curvatures and tight loops on $r (t)$ are incurred by a close proximity of $w (t)$ to the origin of the complex plane. These observations provide useful insight into the disparate shapes of the four distinct PH quintic solutions to the first–order Hermite interpolation problem.
Generalized plane offsets and rational parameterizations
2023, Computer Aided Geometric Design
In the first part of the paper a planar generalization of offset curves is introduced and some properties are derived. In particular, it is seen that these curves exhibit good regularity properties and a study on self-intersection avoidance is performed. The representation of a rational curve as the envelope of its tangent lines, following the approach of Pottmann, is revisited to give the explicit expression of all rational generalized offsets. Other famous shapes, such as constant width curves, bicycle tire-tracks curves and Zindler curves are related to these generalized offsets. This gives rise to the second part of the paper, where the particular case of rational parameterizations by a support function is considered and explicit families of rational constant width curves, rational bicycle tire-track curves and rational Zindler curves are generated and some examples are shown.
Comparing Accuracies of Length-Type Geographic Atrophy Growth Rate Metrics Using Atrophy-Front Growth Modeling
2022, Ophthalmology Science
To compare the accuracies of the previously proposed square-root-transformed and perimeter-adjusted metrics for estimating length-type geographic atrophy (GA) growth rates.
Cross-sectional and simulation-based study.
Thirty-eight eyes with GA from 27 patients.
We used a previously developed atrophy-front growth model to provide analytical and numerical evaluations of the square-root-transformed and perimeter-adjusted growth rate metrics on simulated and semisimulated GA growth data.
Comparison of the accuracies of the square-root-transformed and perimeter-adjusted metrics on simulated and semisimulated GA growth data.
Analytical and numerical evaluations showed that the accuracy of the perimeter-adjusted metric is affected minimally by baseline lesion area, focality, and circularity over a wide range of GA growth rates. Average absolute errors of the perimeter-adjusted metric were approximately 20 times lower than those of the square-root-transformed metrics, per evaluation on a semisimulated dataset with growth rate characteristics matching clinically observed data.
Length-type growth rates have an intuitive, biophysical interpretation that is independent of lesion geometry, which supports their use in clinical trials of GA therapeutics. Taken in the context of prior studies, our analyses suggest that length-type GA growth rates should be measured using the perimeter-adjusted metric, rather than square-root-transformed metrics.
Offsets and front tire tracks to projective hedgehogs
2022, Computer Aided Geometric Design
Citation Excerpt :
The movement of a bicycle, represented as an oriented segment in the plane, can be described by the trajectories of these rear and front wheel track curves. Offset curves have a wide range of applications, from Computer Aided Geometric Design (CAGD) and numerical-control machining to robot path-planning (see Farouki (2008) or Farouki and Neff (1990) and the references therein). The study of the front and rear track curves, besides its obvious engineering applicability, has a strong relationship with other famous problems in Physics such as the floating body problem or the motion of an electron in a parabolic magnetic field (see e.g. Bor et al. (2020)).
There are some known properties on curves of constant width and Zindler curves and their relationship with offsets and front tire-track curves in convex geometry. In this work, a generalization of all these concepts and results to hedgehogs is presented. Discontinuity issues coming from the common parametric definition of offsets and front track curves are solved with parameterizations by a support function. Finally, it is seen that there is a one-to-one correspondence between hedgehogs of constant width and generalized Zindler curves.
Classification of polynomial minimal surfaces
2022, Computer Aided Geometric Design
Citation Excerpt :
Owing to their graceful geometric properties and broad applications, introducing minimal surfaces into Computer-Aided Design (CAD) is meaningful. Pythagorean hodograph (PH) curves are special polynomial curves with polynomial hodographs (Farouki and Neff, 1990). In other words, the Euclidean norms of the first derivative of PH curves are real polynomials.
Minimal surfaces are widely applied in Computer-Aided Design and architecture due to their elegant shapes and remarkable geometric properties. On the basis of the theoretical results of Pythagorean hodograph (PH) curves, we provide a complete classification of the polynomial surface of degrees three to five. We point out the existence of a unique cubic minimal surface, one family of quartic polynomial minimal surfaces, and three families of quintic polynomial minimal surfaces up to similarities and linear reparametrization. We give explicit expressions for each family of polynomial minimal surfaces with parameters that can be used to adjust their shapes. We also prove that an isoparametric curve that is a planar PH curve always exists for any polynomial minimal surfaces. Finally we apply the classification to some existing minimal surfaces.
Accurate Real-time CNC Curve Interpolators Based Upon Richardson Extrapolation
2021, CAD Computer Aided Design
Real-time CNC interpolators achieving a constant or variable feedrate $V$ along a parametric curve $r (ξ)$ are usually based on truncated Taylor series expansions defining the time-dependence of the curve parameter $ξ$ . Since the feedrate should be specified as a function of a physically meaningful variable (such as time $t$ , path arc length $s$ , or curvature $κ$ ) rather than $ξ$ , successive applications of the differentiation chain rule are necessary to determine Taylor series coefficients beyond the linear term. The closed-form expressions for the higher-order coefficients are increasingly cumbersome to derive and implement, and consequently error-prone. To address this issue, the use of Richardson extrapolation as a simple means to compute rapidly convergent approximations to the higher-order coefficients is investigated herein. The methodology is demonstrated in the context of (1) an arc-length-dependent feedrate for cornering motions; (2) direct real-time offset curve interpolation; and (3) a curvature-dependent feedrate. All of these examples admit simple implementations that circumvent the need for tedious symbolic calculations of higher-order coefficients, and are compatible with real-time controllers with millisecond sampling intervals.

View all citing articles on Scopus

View full text

Analytic properties of plane offset curves

Abstract

Computer Aided Geometric Design

Computer Aided Design

Computer Aided Geometric Design

Computer Aided Design

Computer Aided Design

Computer Aided Design

The Theory of Splines and their Applications

Cusps of Gauss Mappings

Curves and Singularities