Pythagorean hodograph quintic transition between two circles with shape control

Habib, Zulfiqar; Sakai, Manabu

doi:10.1016/j.cagd.2007.03.004

Cited by 56 publications

(26 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…This approach facilitates the development of many algorithms for constructing and analyzing planar PH curves. PH quintics, for example, can be constructed as interpolants to first-order Hermite data [17]; as C 2 splines interpolating a sequence of points under prescribed end conditions [15]; or through several shape-constrained methods -e.g., see [9,12,22,33,34,35].…”

Section: Pythagorean-hodograph Curvesmentioning

confidence: 99%

See 1 more Smart Citation

Rational swept surface constructions based on differential and integral sweep curve properties

Farouki

Nittler

2015

Computer Aided Geometric Design

View full text Add to dashboard Cite

A swept surface is generated from a profile curve and a sweep curve by employing the latter to define a continuous family of transformations of the former. By using polynomial or rational curves, and specifying the homogeneous coordinates of the swept surface as bilinear forms in the profile and sweep curve homogeneous coordinates, the outcome is guaranteed to be a rational surface compatible with the prevailing data types of CAD systems. However, this approach does not accommodate many geometrically intuitive sweep operations based on differential or integral properties of the sweep curve -such as the parametric speed, tangent, normal, curvature, arc length, and offset curves -since they do not ordinarily have a rational dependence on the curve parameter. The use of Pythagorean-hodograph (PH) sweep curves surmounts this limitation, and thus makes possible a much richer spectrum of rational swept surface types. A number of representative examples are used to illustrate the diversity of these novel swept surface forms -including the oriented-translation sweep, offset-translation sweep, generalized conical sweep, and oriented-involute sweep. In many cases of practical interest, these forms also have rational offset surfaces. Considerations related to the automated CNC machining of these surfaces, using only their high-level procedural definitions, are also briefly discussed.

show abstract

Section: Pythagorean-hodograph Curvesmentioning

confidence: 99%

“…However, if d(v) has a quadratic or higher-order dependence on v, the offset (12) is not a rational surface, because of the denominator 1 + d ′2 (v) in (22). The parametric speed along the u = constant and v = constant isoparametric curves is…”

Section: Offset-translation Sweepmentioning

confidence: 99%

Rational swept surface constructions based on differential and integral sweep curve properties

Farouki

Nittler

2015

Computer Aided Geometric Design

View full text Add to dashboard Cite

show abstract

“…No simple and intuitive constraints on the geometry of the control polygon, analogous to those mentioned above for cubics, are available to guide the construction of PH quintics. Some partial results were given in [3], but their geometrical meaning is rather opaque -as is the case with the constraints in equations (17)- (20). Instead, PH quintics are usually specified as interpolants to first-order Hermite data.…”

Section: Has a Pythagorean Hodograph If And Only If It Satisfiesmentioning

confidence: 99%

“…This gives scale-free conditions in which both sides of (11)- (12) and (17)- (20) are of order unity, allowing direct comparison with the machine unit for the floating-point number system in use: for double-precision binary arithmetic with 53-bit mantissa and rounding, this is η = 2 −53 ≈ 1.11 × 10 −16 .…”

Section: Remarkmentioning

confidence: 99%

“…Many methods for the construction of planar and spatial PH curves are available [8,12,14,20,21,22,23,24,25,27,29,30]. The goal of this study is to facilitate their importation into commercial CAD systems through existing CAD data formats, by developing algorithms that (i) identify whether or not specified Bézier/B-spline data define a PH curve; and (ii) if so, reconstruct its "internal structure" variables.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Identification and “reverse engineering” of Pythagorean-hodograph curves

Farouki

Giannelli

Sestini

2015

Computer Aided Geometric Design

View full text Add to dashboard Cite

Methods are developed to identify whether or not a given polynomial curve, specified by Bézier control points, is a Pythagorean-hodograph (PH) curve -and, if so, to reconstruct the internal algebraic structure that allows one to exploit the advantageous properties of PH curves. Two approaches to identification of PH curves are proposed. The first is based on the satisfaction of a system of algebraic constraints by the control-polygon legs, and the second uses the fact that numerical quadrature rules that are exact for polynomials of a certain maximum degree generate arc length estimates for PH curves exhibiting a sharp saturation as the number of sample points is increased. These methods are equally applicable to planar and spatial PH curves, and are fully elaborated for cubic and quintic PH curves. The reverse engineering problem involves computing the complex or quaternion coefficients of the pre-image polynomials generating planar or spatial Pythagorean hodographs, respectively, from prescribed Bézier control points. In the planar case, a simple closed-form solution is possible, but for spatial PH curves the reverse engineering problem is much more involved.

show abstract

Fair cubic transition between two circles with one circle inside or tangent to the other

2008

Self Cite

View full text Add to dashboard Cite

This paper describes a method for joining two circles with a C-shaped and an S-shaped transition curve, composed of a cubic Bézier segment. As an extension of our previous work; we show that a single cubic curve can be used for blending or for a transition curve preserving G 2 continuity regardless of the distance of their centers and magnitudes of the radii which is an advantage. Our method with shape parameter provides freedom to modify the shape in a stable manner.

show abstract

Pythagorean hodograph quintic transition between two circles with shape control

Cited by 56 publications

References 12 publications

Rational swept surface constructions based on differential and integral sweep curve properties

Rational swept surface constructions based on differential and integral sweep curve properties

Identification and “reverse engineering” of Pythagorean-hodograph curves

Fair cubic transition between two circles with one circle inside or tangent to the other

Contact Info

Product

Resources

About