Farouki pythagorean hodographs. 3 Parametric Speed and Arc Length .
Farouki pythagorean hodographs. 0736 Sep 1, 1990 · R. Farouki and C. The interpolation problem A 4,000-year tour of algebra and geometry motivated by the investigation of Pythagorean-hodograph curves Rida T. 381 17. Symbolic Comput. (2002b, 2008) and ír and Jüttler (2005, 2007)) the study of double PH curves has relied more (Beltran and Monterde, 2007; Monterde, in press) on the Hopf map form, which appears May 28, 2025 · 1. 1016/j. Jun 22, 2016 · Pythagorean Hodograph Quintic Curves by Quaternion Representation (PHQCQR) Package :. Such curves possess rational Frenet frames and curvature/torsion functions, and encompass all helical polynomial curves as special cases. Design. , The conformal map z¿z2 of the hodograph plane. A. Pythagorean hodographs. Because of the nonlinear nature of polynomial PH curves, constructing L 2 approximants involves solving a nonlinear optimization problem. The rapid pace of this activity, encompassing diverse lines of inquiry, makes it desirable at this point to take a broad perspective of these recent developments, and assess We would like to show you a description here but the site won’t allow us. As noted by Beltran and Monterde, the Hopf map representation of spatial PH Aug 1, 2022 · In this paper, we study algebraic and geometric characteristics of a class of Algebraic-Hyperbolic curves that possess Pythagorean-Hodograph (PH) prop… Sep 19, 2019 · The past decade has witnessed sustained interest in elucidating the basic theory of Pythagorean–hodograph (PH) curves, developing construction algorithms, formulating generalizations, and investigating applications. , Zagar E. Sestini Published in Journal of symbolic…1 April 2009 Mathematics View via Publisher Save to Library Create Alert Cite Pythagorean hodograph curves were originally introduced by Farouki and Sakkalis (1990) as planar polynomial curves. Farouki and S. Helical polynomial curves and double Pythagorean hodographs I. 34, 736–752 (1990) Article MathSciNet A novel approach to constructing polynomial minimal surfaces (surfaces of zero mean curvature) with isothermal parameterization from Pythagorean triples of complex polynomials is presented, and it Hermite interpolation problem: selection of free parameters Pythagorean-hodograph (PH) curves r(ξ) = PH curve ⇐⇒ coordinate components of r0(ξ) comprise a “Pythagoreann-tuple of polynomials”in Rn PH curves incorporate special algebraic structures in their hodographs (complex number & quaternion models for planar & spatial PH curves) rational offset curves May 8, 2025 · The L 2 approximation of planar curves by Pythagorean-hodograph (PH) polynomial curves is addressed, based on the distance defined by a metric for planar curves represented as complex-valued functions of a real parameter. We investigate the properties of polynomial space curves r(t) = {x(t), y(t), z(t)} whose hodographs (derivatives) satisfy the Pythagorean condition x'Z(t) + y'Z(t) + z'2(t) - crZ(t) for some real polynomial tr(t). , have become an active research area in Pythagorean-hodograph curves provide exact solutions for computational problems in design, graphics, and robotics. Jan 19, 2022 · Pythagorean-hodograph curves : algebra and geometry inseparable. Jun 1, 2002 · The structural invariance of the four-polynomial characterization for three-dimensional Pythagorean hodographs introduced by Dietz et al. For example, it is possible to compute their arc lengths, bending energies, and offset (parallel) curves in an essentially exact manner, without recourse to approximations; and they are exceptionally well–suited to problems of real–time motion control and spatial path Farouki et al. : Hermite interpolation by rotation-invariant spatial Pythagorean-hodograph curves. " Find many great new & used options and get the best deals for Geometry and Computing Ser. Farouki Department of Mechanical & Aeronautical Engineering, University of California, Davis 17. Farouki, Carlotta Giannelli, Alessandra Sestini. The algebraic structure of the complete set of regular Pythagorean-hodograph curves in ℝ 3 is inherently more complicated than that of the corresponding set in Abstract: Pythagorean hodograph (PH) curves, defined by Farouki and Sakkalis in 1990, and their applications in offsetting, CNC machining, motion planning etc. org/10. . v34. T. We analyze their control polygons and give necessary and sufficient conditions for cubic and quintic curves to be PH. The method has the same friendly interface and properties as those for B-splines Oct 1, 2014 · Helical curves and double Pythagorean hodographs History of mathematics and PH curves Interpolation with PH curves Introduction to PH curves Minkowski geometric algebra of complex sets Quaternions and spatial rotations Rational rotation-minimizing frames Time-optimal control of CNC machines Involvement with various conferences. , have become an active research area in recent years. , have become an active research area An integrated look-ahead algorithm for parametric interpolation along NURBS curves that interpolates the sharp corners on the curve with the Pythagorean-hodograph (PH) interpolation to minimize the geometric and interpolator approximation errors simultaneously. Quaternion and Hopf map representations, Journal of Symbolic Computation, 44:2, (161-179), Online publication date: 1-Feb-2009. In this paper, the authors provide an algorithm to interactively design a cubic IPH spline curve from any given control polygon. 3 Parametric Speed and Arc Length 17. : Pythagorean hodographs. 3 Parametric Speed and Arc Length . Comput. v44 i2. Aided Geom. The package comprises a library of routine written in C++ to build a PH Quintic. (2002). Enumeration of low-degree curves Rida T. Sakkalis (1990), Pythagorean hodographs, IBM Journal of Research and Development 34, 736-752. 384 17. The hodographs of such curves correspond to certain combinations of four polynomials, given by Dietz et al. 4 Differential and Integral Properties We investigate the properties of polynomial space curves r (t) = {x (t), y (t), z (t)} whose hodographs (derivatives) satisfy the Pythagorean condition x ′ 2 (t)+ y ′ 2 (t)+ z ′ 2 (t)≡σ 2 (t) for some real polynomial σ (t). However, a simplified method that May 8, 2025 · The L 2 approximation of planar curves by Pythagorean-hodograph (PH) polynomial curves is addressed, based on the distance defined by a metric for planar curves represented as complex-valued functions of a real parameter. Sakkalis,IBM Journal of Research and Development archive,发表年份:1990 - 被引用:353. Farouki Department of Mechanical & Aerospace Engineering, University of California, Davis By virtue of their special algebraic structures, Pythagorean-hodograph (PH) curves offer unique advantages for computer-aided design and manufacturing, robotics, motion control, path planning, computer graphics, animation, and related fields. For PH cubics and Pythagorean hodographs 作者:R. Every polynomial helical curve is a spatial Pythagorean-hodograph (PH) curve, and the distinctive features of the PH curves have attracted growing interest in their use for Unmanned The quaternion representation for spatial Pythagorean–hodograph (PH) curves greatly facilitates the formulation of basic algorithms for their construction and manipulation, such as first–order Hermite interpolation, transformations between coordinate systems, and determination of rotation–minimizing frames. Farouki, S. 600 R. Farouki, The conformal map z → z2 of the hodograph plane, Computer Aided Geometric Design 11, 363–390 Jul 1, 1998 · An NC system that machines a curved shape at fixed depth of cut experiences time-varying cutting forces due to the ‘curvature effect’—the material removal rate is higher than nominal in concave regions, and lower in convex regions. Math. Farouki, The conformal map z → z2 of the hodograph plane, Computer Aided Geometric Design 11, 363–390 Nov 1, 2002 · An exact specification of the rotation-minimizing frame on a spatial Pythagorean-hodograph (PH) curve can be derived by integration of a rational function. , Sakkalis T. Apart from the independent scale factors p and q for the x- and y-directions, these curves are distinguished by one basic “shape” parameter, namely, the parameter value 0 of the x-axis intercept. : Pythagorean-Hodograph Curves : Algebra and Geometry Inseparable by Rida T. The Hopf map and quaternion models3 for spatial PH curves were introduced simultaneously by Choi et al. After reviewing the definitions and properties of these representations, and conversions between them, linear and characterization of spatial Pythagorean hodographs R. Farouki, Carlotta Giannelli, Alessandra Sestini 2009 Journal of symbolic computation The incorporation of Pythagorean–hodograph (PH) structures in parametric curves facilitates exact computation of many geometrical properties that otherwise require numerical approximations. Mar 1, 2015 · Specifically, planar and spatial Pythagorean hodographs are constructed by quadratic mappings of complex and quaternion polynomials, respectively, and the coefficients of these pre-image polynomials are required for the exact determination of various PH curve properties. Very soon, a first survey on PH curves was summarized by Farouki in Section 17 of the Handbook of Computer Aided Geometric Design (2002), see Farin et al Hermite interpolation problem: selection of free parameters Pythagorean-hodograph (PH) curves r(ξ) = PH curve ⇐⇒ coordinate components of r0(ξ) comprise a “Pythagoreann-tuple of polynomials”in Rn PH curves incorporate special algebraic structures in their hodographs (complex number & quaternion models for planar & spatial PH curves) rational offset curves Mar 15, 2025 · Farouki R. It turns out that this ERF enjoys some nice properties. Des. As indicated by the Contents, this research spans a spectrum of topics ranging from elucidation of the basic mathematical theory of PH curves, through development of practical algorithms for their construction and analysis, to the demonstration of their use in computer–aided design and manufacturing applications. (2017). T. Sakkalis, Pythagorean hodographs, IBM Journal of Research and Development 34, 736-752 (1990) R. A rational curve is an irreducible curve whose parametrisation, for example (x(q), y(q)), would generate all points on the curve for the values of q, which is a parameter. [1]. R. The proof relies on a factored-quaternion representation for Pythagorean hodographs in three-dimensional Euclidean space—a particular instance of the “PH representation map” proposed by Choi Pythagorean-hodograph curves and related topics Pythagorean-hodograph (PH) curves were introduced (Farouki and to provide exact solutions to a number of basic computational problems that arise in computer-aided design and manufacturing, computer graphics and animation, robot path planning, and related fields. Shah/Computer Aided Geometric Design 13 (1996) 583-600 R. We derive a 17 Pythagorean—hodograph Curves 17. [2] Farouki, R. In particular, the complex formulation of PH curves greatly facilitates the derivation of these expressions, yielding compact and efficient implementations Aug 1, 2021 · Ever since the Pythagorean hodograph (PH) curves first appeared in (Farouki and Sakkalis, 1990), they have been an undisputed phenomenon in geometric modeling and related disciplines to which a huge number of articles have been devoted. Study of their properties and applications influenced significantly also investigating in further theoretical or applied disciplines, and nowadays we can find a large number of papers and other contributions related to this interesting Nov 9, 2007 · Chapter Google Scholar Farouki, R. 2011. , (x′ (t), y′ (t), σ (t)) form a “Pythagorean triple Abstract In this paper, the G2 interpolation by Pythagorean-hodograph (PH) quintic curves in ℝ d, d ≥ 2, is considered. 01. 345. Nov 30, 2010 · By virtue of their special algebraic structures, Pythagorean-hodograph (PH) curves offer unique advantages for computer-aided design and manufacturing, robotics, motion control, path planning, computer graphics, animation, and related fields. Farouki, Pythagorean hodographs, IBM J. Adv. [4], that admit compact descriptions in terms of quaternions – an instance of the “PH representation map” proposed by Helical polynomial curves and double Pythagorean hodographs I. Independently of the dimension d, they supply a G2 quintic PH spline that locally interpolates two points, two tangent directions and two curvature vectors at these points. Res. This book offers a comprehensive and self-contained treatment of the mathematical theory of PH curves, including algorithms for their construction and Oct 1, 1996 · R. ABSTRACT: The geometric characteristics and the construction for cubic indirect Pythagorean-hodograph (indirect-PH) curves are presented in this study. v11. IBM J. 64 (1995), 1589-1609 Request permission Abstract: The Pythagorean hodograph (PH) curves are polynomial parametric curves {x (t), y (t)} whose hodograph (derivative) components satisfy the Pythagorean condition x ′ 2 (t) + y ′ 2 (t) ≡ σ 2 (t) for Rida T. Giannelli, A. Oct 11, 2007 · By virtue of their special algebraic structures, Pythagorean-hodograph (PH) curves offer unique advantages for computer-aided design and manufacturing, robotics, motion control, path planning, computer graphics, animation, and related fields. 1 Planar Pythagorean Hodographs 17. , al Kandari M. , Application of a metric for complex polynomials to bounded modification of planar Pythagorean–hodograph curves, 2024, preprint. Although the quaternion model has since enjoyed greater use (Farouki et al. , Vitrih V. g. Farouki, Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable, Springer, Berlin, 2008. Enumeration of low-degree curves. Farouki, The conformal map z → z2 of the hodograph plane, Computer Aided Geometric Design 11, 363–390 In our contribution we study cubic and quintic Pythagorean Hodograph (PH) curves in the Euclidean and Minkowski planes. If w = 0 the curve is symmetric about the x-axis, whereas it becomes increasingly skewed as I w I increases. These curves were introduced as a generalization of the planar PH Bézier curves defined firstly in Farouki and Sakkalis (1990). the surveys Farouki, 2002, Farouki, 2008; Kosinka and Lávička (2014). What have the authors contributed in "Identification and “reverse engineering” of pythagorean–hodograph curves" ? In this paper, two approaches to identify whether 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 May 8, 2025 · Abstract The L 2 approximation of planar curves by Pythagorean-hodograph (PH) polynomial curves is addressed, based on the distance defined by a metric for planar curves represented as complex-valued functions of a real parameter. J. The significance of this feature in diverse applications (such as path planning and motion control for robotics, manufacturing, computer animation, and unmanned aerial vehicles) has spurred intensive May 20, 2023 · Indirect Pythagorean hodographs (IPH) spline curves are a set of curves which have rational Pythagorean hodographs after reparameterization by a fractional quadratic transformation. Shah (1995), Real-time CNC interpolators for Pythagorean-hodograph curves, Report No. 386 Apr 1, 2009 · A “double” Pythagorean-hodograph (DPH) curve r (t) is characterized by the property that | r (t) | and | r (t) × r ″ (t) | are both polynomials in the curve parameter t. Nov 1, 2021 · The constructions of planar and spatial Pythagorean–hodograph (PH) curves are based on models that employ the complex–number and quaternion algebras, respectively (Farouki, 2008). Jan 1, 1994 · We investigate the properties of polynomial space curves r (t)= {x (t), y (t), z (t)} whose hodographs (derivatives) satisfy the Pythagorean condition x′2 (t)+y′2 (t)+z′2 (t)≡σ2 (t) for some real polynomial σ (t). TL;DR: The structural invariance of the four-polynomial characterization for three-dimensional Pythagorean hodographs introduced by Dietz et al. Jan 1, 2014 · Pythagorean hodograph (PH) curves, defined by Farouki and Sakkalis in 1990, and their applications in offsetting, CNC machining, motion planning etc. By introducing an auxiliary control point and a parameter respectively, two geometric characteristics in terms of quantities related to Bézier control polygon of the curve are given. Introduction This paper is devoted to planar Pythagorean-hodograph curves. 1) and polynomial and rational curves with Pythagorean hodographs in the Euclidean plane. However, a simplified Hermite interpolation by Pythagorean hodograph quintics HTML articles powered by AMS MathViewer by R. The result is an angular function θ( t ) of the curve parameter, comprising in general both Abstract: Pythagorean hodograph (PH) curves, defined by Farouki and Sakkalis in 1990, and their applications in offsetting, CNC machining, motion planning etc. K. Since their introduction in Farouki and Sakkalis (1990), these curves have been the subject of many publications, see e. Neff PDF Math. Farouki and T. In this section we briefly review the fundamentals of offsets (see Fig. Quaternion and Hopf map representations Feb 1, 2009 · For regular polynomial curves r(t) in R^3, relations between the helicity condition, existence of rational Frenet frames, and a certain ''double'' Pythagorean-hodograph (PH) structure are elucidated in terms of the quaternion and Hopf map Dec 1, 2019 · Curves with Pythagorean hodographs (PH curves), introduced by Farouki and Sakkalis in 1990, see [1], have become an intensive research topic in recent years. It is a frame that can be naturally constructed from the PH condition. Farouki :Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable at the best online prices at eBay! Free shipping for many products! Nov 1, 2002 · An exact specification of the rotation-minimizing frame on a spatial Pythagorean-hodograph (PH) curve can be derived by integration of a rational function. In particular, a close examination of its angular velocity against a rotation-minimizing frame yields a Title: Quaternions, spatial Pythagorean hodographs, and rotations in three and four dimensions Speaker: Professor Rida T. Neff, On the numerical condition of Bernstein-Bezier subdivision processes, Mathematics of Computation 55, 637-647 (1990) Feb 1, 2009 · For regular polynomial curves r (t) in R 3, relations between the helicity condition, existence of rational Frenet frames, and a certain “double” Pythagorean-hodograph (PH) structure are elucidated in terms of the quaternion and Hopf map representations of spatial PH curves. The algebraic structure of the complete set of regular Pythagorean-hodograph curves in ℝ3 is inherently more complicated than that of the corresponding set in ℝ2. By virtue of their algebraic structures, PH curves offer unique Oct 1, 2002 · We investigate the properties of a special kind of frame, which we call the Euler–Rodrigues frame (ERF), defined on the spatial Pythagorean-hodograph (PH) curves. A polynomial parametric curve is said to have a Pythagorean hodograph if there exists a polynomial σ (t) such that x′ 2 (t) + y′ 2 (t) ≡ σ 2 (t), i. 17, 369–383 (2002) Article MATH MathSciNet Google Scholar Farouki, R. Farouki,T. Farouki, C. Quaternion and Hopf map representations. Farouki's 87 research works with 4,621 citations and 3,761 reads, including: Real-time needle guidance for venipuncture based on optical coherence tomography Oct 1, 2024 · Request PDF | On Oct 1, 2024, Francesca Pelosi and others published Control point modifications that preserve the Pythagorean–hodograph nature of planar quintic curves | Find, read and cite all Mar 1, 2015 · Specifically, planar and spatial Pythagorean hodographs are constructed by quadratic mappings of complex and quaternion polynomials, respectively, and the coefficients of these pre-image polynomials are required for the exact determination of various PH curve properties. The quaternion representation for spatial Pythagorean-hodograph (PH) curves greatly facilitates the formulation of basic algorithms for their construction and manip-ulation, such as first-order Hermite interpolation, transformations between coordinate systems, and determination of rotation-minimizing frames. This book offers a comprehensive and self-contained treatment of the mathematical theory of PH curves, including algorithms for their construction and 1985 1990 1995 2000 2005 2010 2015 2020 0 20 40 Abstract. UM-MEAM-95-11, University of Michigan, Ann Arbor. [7] R. 2 Bezier Control Points of PH Curves 17. Kubota, Pythagorean triples in unique factorization domains, American Mathematical Monthly 79, 503–505 (1972) R. , Sakkalis, T. Jan 19, 2022 · This book explores the inseparable relationship between algebra and geometry through Pythagorean-hodograph curves, offering insights into their mathematical properties and applications. Sakkalis, Pythagorean hodographs, IBM Journal of Research and Development 34 736–752 (1990) R. Sestini, Helical Polynomial Curves and Double Pythagorean Hodographs I Quaternion and Hopf map representations, Journal of Symbolic Computation 44 (2) (2009) 161–179. We Nov 1, 2012 · A polynomial Pythagorean-hodograph (PH) curve in is characterized by the property that its derivative components satisfy the Pythagorean condition for some polynomial , ensuring that the arc length is simply a polynomial in the curve parameter . Neff, "Hermite Interpolation by Pythagorean-Hodograph Quintics, in preparation (1990). It was proved (Farouki and Sakkalis, 1990; Kubota, 1972) that the coordinates of hodographs of polynomial PH curves and σ(t) form the following Pythagorean triples Apr 1, 2009 · Helical polynomial curves and double Pythagorean hodographs I. We now show Apr 1, 1996 · Pythagorean-hodograph (PH) curves admit closed-form expressions for the integral of the square of the curvature with respect to arc length (the “energy” integral) involving only rational functions, arctangents, and natural logarithms. Indeed, the combination of their application potential and the intrinsic mathematical beauty made them in some sense a part Figure 4 illustrates some representative standard-form crunodal cubics. Journal of Symbolic Computation, 44 (4):307-332, 2009. cagd. In the case of Euclidean Jun 1, 2020 · 1. Comp. 75 DOI: 10. This book offers a comprehensive and self-contained treatment of the mathematical theory of PH curves, including algorithms for their construction and Pythagorean hodographs are known for their rational properties. Sakkalis, Pythagorean–hodograph space curves, Advances in Computational Mathematics 2, 41–66 (1994) Sep 1, 1990 · R. The result is an angular function θ (t) of the curve parameter, comprising in general both rational and logarithmic terms, that specifies the orientation of the rotation-minimizing frame relative to the Frenet frame. Jun 1, 2020 · Recently, a concept of Pythagorean hodograph (PH) B-spline curves was presented in Albrecht et al. 34 (5) , 736-752 https://doi. For PH cubics and Aug 1, 2017 · Abstract A helical curve, or curve of constant slope, offers a natural flight path for an aerial vehicle with a limited climb rate to achieve an increase in altitude between prescribed initial and final states. Enumeration of low-degree curves R. e. , № 34, с. 002 Farouki, Variable-feedrate CNC interpolators for constant material removal rates along Pythagorean-hodograph curves, Comput. PH curves have thus far been extensively studied in and , by means of the complex-number and the quaternion or Hopf map representations, and the basic Spatial Pythagorean hodographs, quaternions, and rotations in 3 R and 4 R — a study in the evolution of scientific ideas — Rida T. The algebraic structure ofthe complete set of regular Pythagorean-hodograph curves in R 3 isinherently more complicated than that of the corresponding set in 112. The text outlines various applications of PH curves, including motion control and rotation-minimizing frames. , Knez M. We present a sufficient-and-necessary algebraic characterization of the Pythagorean-hodograph property, analyze its geometric implications in terms of Bernstein—Bézier forms, and survey the useful attributes it entails in various applications. 736-752. "Helical polynomial curves and double Pythagorean hodographs I. (2004). 19 (2002) 395–407. Dev. 2 B ́ezier Control Points of PH Curves . Develop. (1993), under arbitrary spatial rotations, is demonstrated. , Structural invariance of spatial Pythagorean Hodographs, Comput. However, a simplified method that Nov 1, 2002 · An exact specification of the rotation-minimizing frame on a spatial Pythagorean-hodograph (PH) curve can be derived by integration of a rational function. A curvature-dependent feedrate function that automatically compensates for this effect is formulated, and it is shown that, for Pythagorean-hodograph (PH R. The hodograph of a plane parametric curve r (t) = {x (t), y (t)} is the locus described by the first parametric derivative r′ (t) = {x′ (t), y′ (t)} of that curve. Rida T. 386 17. The obtained results turn out as a useful tool in practical applications. Pythagorean(n+1)–tuples, endows the Pythagorean–hodograph (PH) curves with many computationally attractive features. Farouki (2007, Hardcover) at the best online prices at eBay! Free shipping for many products! Dec 31, 2024 · Find many great new & used options and get the best deals for Rida T. 0736 Farouki, Rational Pythagorean-hodograph space curves, Comput. , № 28, с. 363 K. 1147/rd. An early survey on 1 September 1990 journal article Published by IBM in IBM Journal of Research and Development Vol. Helical polynomial curves and double Pythagorean hodographs II. Although several authors have discussed the Nov 29, 2007 · Farouki R, Giannelli C and Sestini A (2009). Farouki Date: November 29, 2012 Series: UCD MAE Weekly Seminar Where The interpolation of first-order Hermite data by spatial Pythagorean-hodograph curves that exhibit closure under arbitrary 3-dimensional rotations is addressed. Farouki, Carlotta Giannelli, A. 1 Planar Pythagorean Hodographs . The fundamental feature of the planar PH curves lies in the fact that they admit a closed-form polynomial representation of their arc length and thus also exact The quaternion representation for spatial Pythagorean–hodograph (PH) curves greatly facilitates the formulation of basic algorithms for their construction and manipulation, such as first–order Hermite interpolation, transformations between coordinate systems, and determination of rotation–minimizing frames. These curves (and other objects with a Pythagorean property) have fascinated many researchers and one can find hundreds of papers devoted to this beautiful topic. 736 DOI: 10. 161-179. [doi] Authors BibTeX References Bibliographies Reviews Related Mar 1, 2025 · Farouki R. , al-Kandari, M. Feb 9, 2017 · Pythagorean-hodograph (PH) curves offer distinct advantages in planning curvilinear paths for unmanned or autonomous air, ground, or underwater vehicles. iqod hp58 7zypjps7m be 5cm2bnh q92l llr lfu a7h8 duch