guarantees for the existence of solutions in all cases, although existence is demonstrated for the asymptotic case of densely-sampled data from a smooth curve. Modulation of the hodograph by a scalar ...polynomial is proposed as a means of introducing additional degrees of freedom, in cases where solutions to the end-point interpolation problem are not found. The methods proposed herein are expected to find important applications in exactly specifying rigid-body motions along curved paths, with minimized rotation, for animation, robotics, spatial path planning, and geometric sweeping operations.
A novel high-speed cornering strategy for piecewise-linear motions is proposed, based on the
G
2
continuous Pythagorean–hodograph (PH) rounding segments and continuously-variable feedrates described ...in Part I of this two-part paper. This strategy employs an acceleration-limited approach to feedrate scheduling, including smooth deceleration and acceleration profiles along linear segments entering and leaving the rounded segments. A G-code part program parsing software package has been developed, that automatically identifies toolpath corners and inserts appropriately-sized PH quintic corner rounding segments, with associated feedrate suppression ratios. The method has been tested on a three-axis CNC mill with an open-architecture controller incorporating real-time interpolators that realize smoothly varying feedrates along the PH corner curves. Tests on a representative selection of toolpaths yield substantial savings (up to 40 %) in execution times, compared to the traditional “full stop” strategy for unmodified sharp corners.
An adapted orthonormal frame (
f
1
(
ξ
),
f
2
(
ξ
),
f
3
(
ξ
)) on a space curve
r
(
ξ
),
ξ
∈ 0, 1 comprises the curve tangent
f
1
(
ξ
)
=
r
′
(
ξ
)
/
|
r
′
(
ξ
)
|
and two unit vectors
f
2
(
ξ
),
...f
3
(
ξ
) that span the normal plane. The variation of this frame is specified by its angular velocity
Ω
= Ω
1
f
1
+ Ω
2
f
2
+ Ω
3
f
3
, and the
twist
of the framed curve is the integral of the component Ω
1
with respect to arc length. A
minimal twist frame
(MTF) has the least possible twist value, subject to prescribed initial and final orientations
f
2
(0),
f
3
(0) and
f
2
(1),
f
3
(1) of the normal–plane vectors. Employing the
Euler–Rodrigues frame
(ERF) — a rational adapted frame defined on spatial Pythagorean–hodograph curves — as an intermediary, an exact expression for an MTF with Ω
1
= constant is derived. However, since this involves rather complicated transcendental terms, a construction of rational MTFs is proposed by the imposition of a rational rotation on the ERF normal–plane vectors. For spatial PH quintics, it is shown that rational MTFs compatible with the boundary conditions can be constructed, with only modest deviations of Ω
1
about the mean value, by a rational quartic normal–plane rotation of the ERF. If necessary, subdivision methods can be invoked to ensure that the rational MTF is free of inflections, or to more accurately approximate a constant Ω
1
. The procedure is summarized by an algorithm outline, and illustrated by a representative selection of computed examples.
We introduce cubic-like sparse Pythagorean hodograph curves which are possibly of high degrees though they can be handled as cubic curves. They offer more flexibility than the classical Pythagorean ...hodograph cubic curves with which they share many properties. We give an elegant geometric characterization of their control polygons. This characterization leads to many interesting computational algorithms for curve design. We show how to extend such algorithms to quintic-like sparse PH curves.
•Introduce sparse Pythagorean hodograph curves.•Characterize the control polygons of sparse cubic-like PH curves.•Solve the G1 interpolation problem with sparse cubic-like PH curves.•Solve the C1 interpolation problem with sparse quintic-like PH curves.
A rotation–minimizing frame (
f
1
,
f
2
,
f
3
) on a space curve
r
(
ξ
) defines an orthonormal basis for
ℝ
3
in which
f
1
=
r
′
/
|
r
′
|
is the curve tangent, and the normal–plane vectors
f
2
,
f
3
...exhibit no instantaneous rotation about
f
1
. Polynomial curves that admit
rational
rotation–minimizing frames (or RRMF curves) form a subset of the Pythagorean–hodograph (PH) curves, specified by integrating the form
r
′
(
ξ
)
=
A
(
ξ
)
i
A
∗
(
ξ
)
for some quaternion polynomial
A
(
ξ
)
. By introducing the notion of the
rotation indicatrix
and the
core
of the quaternion polynomial
A
(
ξ
)
, a comprehensive characterization of the complete space of RRMF curves is developed, that subsumes all previously known special cases. This novel characterization helps clarify the structure of the complete space of RRMF curves, distinguishes the spatial RRMF curves from trivial (planar) cases, and paves the way toward new construction algorithms.
Minkowski Pythagorean hodograph curves are widely studied in computer-aided geometric design, and several methods exist which construct Minkowski Pythagorean hodograph (MPH) curves by interpolating ...Hermite data in the R2,1 Minkowski space. Extending the class of MPH curves, a new class of Rational Envelope (RE) curve has been introduced. These are special curves in R2,1 that define rational boundaries for the corresponding domain. A method to use RE and MPH curves for skinning purposes, i.e., for circle-based modeling, has been developed recently. In this paper, we continue this study by proposing a new, more flexible way how these curves can be used for skinning a discrete set of circles. We give a thorough overview of our algorithm, and we show a significant advantage of using RE and MPH curves for skinning purposes: as opposed to traditional skinning methods, unintended intersections can be detected and eliminated efficiently.
The problem of specifying the two free parameters that arise in spatial Pythagorean-hodograph (PH) quintic interpolants to given first-order Hermite data is addressed. Conditions on the data that ...identify when the “ordinary” cubic interpolant becomes a PH curve are formulated, since it is desired that the selection procedure should reproduce such curves whenever possible. Moreover, it is shown that the arc length of the interpolants depends on only one of the parameters, and that four (general) helical PH quintic interpolants always exist, corresponding to extrema of the arc length. Motivated by the desire to improve the fairness of interpolants to general data at reasonable computational cost, three selection criteria are proposed. The first criterion is based on minimizing a bivariate function that measures how “close” the PH quintic interpolants are to a PH cubic. For the second criterion, one of the parameters is fixed by first selecting interpolants of extremal arc length, and the other parameter is then determined by minimizing the distance measure of the first method, considered as a univariate function. The third method employs a heuristic but efficient procedure to select one parameter, suggested by the circumstances in which the “ordinary” cubic interpolant is a PH curve, and the other parameter is then determined as in the second method. After presenting the theory underlying these three methods, a comparison of empirical results from their implementation is described, and recommendations for their use in practical design applications are made.
A polynomial Pythagorean-hodograph (PH) curve r(t)=(x1(t),…,xn(t)) in Rn is characterized by the property that its derivative components satisfy the Pythagorean condition x1′2(t)+⋯+xn′2(t)=σ2(t) for ...some polynomial σ(t), ensuring that the arc length s(t)=∫σ(t)dt is simply a polynomial in the curve parameter t. PH curves have thus far been extensively studied in R2 and R3, by means of the complex-number and the quaternion or Hopf map representations, and the basic theory and algorithms for their practical construction and analysis are currently well-developed. However, the case of PH curves in Rn for n>3 remains largely unexplored, due to difficulties with the characterization of Pythagorean (n+1)-tuples when n>3. Invoking recent results from number theory, we characterize the structure of PH curves in dimensions n=5 and n=9, and investigate some of their properties.
Algorithm 952 Dong, Bohan; Farouki, Rida T.
ACM transactions on mathematical software,
10/2015, Letnik:
41, Številka:
4
Journal Article
Recenzirano
The implementation of a library of basic functions for the construction and analysis of planar quintic Pythagorean-hodograph (PH) curves is presented using the complex representation. The special ...algebraic structure of PH curves permits exact algorithms for the computation of key properties, such as arc length, elastic bending energy, and offset (parallel) curves. Single planar PH quintic segments are constructed as interpolants to first-order Hermite data (end points and derivatives), and this construction is then extended to open or closed
C
2
PH quintic spline curves interpolating a sequence of points in the plane. The nonlinear nature of PH curves incurs a multiplicity of formal solutions to such interpolation problems, and a key aspect of the algorithms is to efficiently single out the unique “good” interpolant among them.
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