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We present a scheme to find spatial quintic Pythagorean-hodograph (PH) curves that interpolate given first-order Hermite data and Frenet frames. The two free parameters that appear in general quintic PH interpolants are determined to adjust the orientation of binormal vectors. Using the unique cubic interpolant to the given set of Hermite data as a reference, we produce a quintic PH interpolant that shares not only the first-order Hermite data but also the Frenet frames at the endpoints with the cubic counterpart. This approach can be readily applied to a sequence of points to generate a quintic PH C 1 spline curve with Frenet-frame continuity. We also prove that for any nonlinear analytic curve, such PH Frenet interpolants uniquely exist if the Hermite data are extracted from sufficiently close points on the curve.

C 2 -continuous Hermite interpolation of streaming data with the use of cubic splinelets – building blocks for C 2 -continuous cubic splines – is presented. Two specific types of splinelets, denoted as A and B, are formally defined (in closed form for equidistant internal knots) and then comparatively evaluated in a series of numerical experiments. The presented results show that the interpolant based on the cubic spline of type B outperforms not only its quintic correspondent, but also the ( C 1 -continuous only!) cubic Hermite spline. The proposed approach can be directly applied to streams that do or do not contain data related to the second derivative of the interpolated function, and to multidimensional curves/trajectories defined parametrically. One other possible application would be processing data sets that are too large to fit in memory.

Kim and Moon (2017) have recently proposed rectifying control polygons as an alternative to Bezier control polygons and a way of controlling planar PH curves by the rectifying control polygons. While a Bezier control polygon determines a unique polynomial curve, a rectifying control polygon gives a multitude of PH curves. This multiplicity of PH curves naturally raises the selection problem of the “best” PH curves, which is the main topic of this paper. To resolve the problem, we first classify PH curves of degree 2 n + 1 into 2 n subclasses by defining the types of PH curves, and propose the absolute hodograph winding number as a topological index to characterize the topological behavior of PH curves in shape. We present a lower bound of the topological index of a PH curve which is given solely by its type, and prove the uniqueness of the best PH curve by exploiting it. The existence theorems are also proved for cubic and quintic PH curves. Finally, we propose a selection rule of the best PH curve only based on its type.

A well–known feature of the Pythagorean–hodograph (PH) curves is the multiplicity of solutions arising from their construction through the interpolation of Hermite data. In general, there are four distinct planar quintic PH curves that match first–order Hermite data, and a two–parameter family of spatial quintic PH curves compatible with such data. Under certain special circumstances, however, the number of distinct solutions is reduced. The present study characterizes these singular cases, and analyzes the properties of the resulting quintic PH curves. Specifically, in the planar case it is shown that there may be only three (but not less) distinct Hermite interpolants, of which one is a “double” solution. In the spatial case, a constant difference between the two free parameters reduces the dimension of the solution set from two to one, resulting in a family of quintic PH space curves of different shape but identical arc lengths. The values of the free parameters that result in formal specialization of the (quaternion) spatial problem to the (complex) planar problem are also identified, demonstrating that the planar PH quintics, including their degenerate cases, are subsumed as a proper subset of the spatial PH quintics.

The problem of constructing a plane polynomial curve with given end points and end tangents, and a specified arc length, is addressed. The solution employs planar quintic Pythagorean-hodograph (PH) curves with equal-magnitude end derivatives. By reduction to canonical form it is shown that, in this context, the problem can be expressed in terms of finding the real solutions to a system of three quadratic equations in three variables. This system admits further reduction to just a single univariate biquadratic equation, which always has positive roots. It is found that this construction of G 1 Hermite interpolants of specified arc length admits two formal solutions - of which one has attractive shape properties, and the other must be discarded due to undesired looping behavior. The algorithm developed herein offers a simple and efficient closed-form solution to a fundamental constructive geometry problem that avoids the need for iterative numerical methods. An algorithm to construct interpolants to planar G1 Hermite data, with exact prescribed arc lengths, is presented.The problem admits a closed-form solution, requiring little more than the solution of a quadratic equation.There exist two formal solutions, the "good" solution corresponding to the least value for the absolute rotation index.The algorithm accommodates the special cases of parallel or symmetric end tangents.

The paper investigates a quasi-interpolation framework for the construction of operators that provide approximations for bivariate functions from the space of Argyris macro-element splines on a general triangulation. An operator of such type is determined by local approximation operators that produce quintic polynomials and are naturally associated with vertices and triangles of the triangulation. The quasi-interpolant resembles the properties of the standard Argyris interpolant, especially in the determination of data corresponding to the vertices of the triangulation. The main difference is in the definition of the cross-boundary derivatives, where a novel approach with degree raising is used to eliminate the need for additional interpolation of derivatives at the edges. Despite reduction in degrees of freedom, a suitable choice of local approximation operators ensures that the quasi-interpolation operator has quintic precision and is of optimal approximation order. To demonstrate the usability of the presented approach, three approximation methods are derived based on local Hermite and Lagrange interpolation and least square fitting.

For given C 1 Hermite data, there exists a two-parameter family of Pythagorean hodograph (PH) quintic curves which interpolate the data (two end-points and end-derivatives) as observed by Farouki et al. (2002b) . As “good” candidate curves for a selection problem, we propose a special type of PH quintic interpolating curves called extremal interpolants and prove that the extremal interpolants preserve planarity, i.e., they are planar curves if the data are planar. Since there are only four distinct extremal interpolants, the selection problem, when only considering extremal interpolants as possible candidates, is reduced to picking one curve from finite candidates. Due to the preservation of planarity, extremal interpolants coincide with one of p 0 , 0 ( t ) , p 0 , π ( t ) , p π , 0 ( t ) , p π , π ( t ) if the data are planar, where p ϕ 0 , ϕ 2 ( t ) denotes the parametrization proposed by Sir and Juttler (2005) . However, any of the four extremal interpolants is generically not identical to the interpolants for non-planar data, and empirical results suggest that being compared with the unique cubic interpolant, the best curve is one of extremal interpolants among all extremal interpolants and p 0 , 0 ( t ) , p 0 , π ( t ) , p π , 0 ( t ) , p π , π ( t ) .

In this paper, the problem of C 2 Hermite interpolation by triarcs composed of Pythagorean-hodograph (PH) quintics is considered. The main idea is to join three arcs of PH quintics at two unknown points - the first curve interpolates given C 2 Hermite data at one side, the third one interpolates the same type of given data at the other side and the middle arc is joined together with C 2 continuity to the first and the third arc. For any set of C 2 planar boundary data (two points with associated first and second derivatives) we construct four possible interpolants. The best possible approximation order is 4. Analogously, for a set of C 2 spatial boundary data we find a six-dimensional family of interpolating quintic PH triarcs. The results are confirmed by several examples.

In order to construct a C 1 -quadratic spline over an arbitrary triangulation, one can split each triangle into 12 subtriangles, resulting in a finer triangulation known as the Powell-Sabin 12-split. It has been shown previously that the corresponding spline surface can be plotted quickly by means of a Hermite subdivision scheme (Dyn and Lyche, 1998). In this paper we introduce a nodal macro-element on the 12-split for the space of quintic splines that are locally C 3 and globally C 2 . For quickly evaluating any such spline, a Hermite subdivision scheme is derived, implemented, and tested in the computer algebra system Sage. Using the available first derivatives for Phong shading, visually appealing plots can be generated after just a couple of refinements.

The problem of designing smoothly rounded right-angle corners with Pythagorean-hodograph (PH) curves is addressed. A G 1 corner can be uniquely specified as a single PH cubic segment, closely approximating a circular arc. Similarly, a G 2 corner can be uniquely constructed with a single PH quintic segment having a unimodal curvature distribution. To obtain G 2 corners incorporating shape freedoms that permit a fine tuning of the curvature profile, PH curves of degree 7 are required. It is shown that degree 7 PH curves define a one-parameter family of G 2 corners, facilitating precise control over the extremum of the unimodal curvature distribution, within a certain range of the parameter. As an alternative, a G 2 corner construction based upon splicing together two PH quintic segments is proposed, that provides two free parameters for shape adjustment. The smooth corner shapes constructed through these schemes can exploit the computational advantages of PH curves, including exact computation of arc length, rational offset curves, and real-time interpolator algorithms for motion control in manufacturing, robotics, inspection, and similar applications.

Spiral segments are useful in the design of fair curves. They are important in CAD/CAM applications, the design of highway and railway routes, trajectories of mobile robots and other similar applications. The quintic Pythagorean-hodograph (PH) curve discussed in this article is polynomial; it has the attractive properties that its arc-length is a polynomial of its parameter, and the formula for its offset is a rational algebraic expression. This paper generalises earlier results on planar PH quintic spiral segments and examines techniques for designing fair curves using the new results.

A rotation-minimizing frame on a space curve r(t) is an orthonormal basis (f"1,f"2,f"3) for R^3, where f"1=r^'/|r^'| is the curve tangent, and the normal-plane vectors f"2,f"3 exhibit no instantaneous rotation about f"1. Such frames are useful in spatial path planning, swept surface design, computer animation, robotics, and related applications. The simplest curves that have rational rotation-minimizing frames (RRMF curves) comprise a subset of the quintic Pythagorean-hodograph (PH) curves, and two quite different characterizations of them are currently known: (a) through constraints on the PH curve coefficients; and (b) through a certain polynomial divisibility condition. Although (a) is better suited to the formulation of constructive algorithms, (b) has the advantage of remaining valid for curves of any degree. A proof of the equivalence of these two different criteria is presented for PH quintics, together with comments on the generalization to higher-order curves. Although (a) and (b) are both sufficient and necessary criteria for a PH quintic to be an RRMF curve, the (non-obvious) proof presented here helps to clarify the subtle relationships between them.

We construct a suitable normalized B-spline representation for C^2-continuous quintic Powell-Sabin splines. The basis functions have a local support, they are nonnegative, and they form a partition of unity. The construction is based on the determination of a set of triangles that must contain a specific set of points. We are able to define control points and cubic control polynomials which are tangent to the spline surface. We also show how to compute the Bezier control net of such a spline in a stable way.

We present a tension-controlled 2-point Hermite interpolatory subdivision scheme that is capable of reproducing circles starting from a sequence of sample points with any arbitrary spacing and appropriately chosen first and second derivatives. Whenever the tension parameter is set equal to 1, the limit curve coincides with the rational quintic Hermite interpolant to the given data and has guaranteed C^2 continuity, while for other positive tension values, continuity of curvature is conjectured and empirically shown by a wide range of experiments.

We investigate univariate C^2 quintic L"1 splines and C^3 seventh-degree L"1 splines and revisit C^1 cubic L"1 splines. We first investigate these L"1 splines when they are calculated by minimizing integrals of absolute values of expressions involving various levels of derivatives from zeroth derivatives (function values) to fourth derivatives and compare these L"1 splines with conventional ''L"2 splines'' calculated by minimizing analogous integrals involving squares of such expressions. The L"2 splines do not preserve the shape of irregular data well. The quintic and seventh-degree L"1 splines also do not preserve shape well, although they do so better than quintic and seventh-degree L"2 splines. Consistent with previously known results, the cubic L"1 splines do preserve shape well. For both L"1 and L"2 splines, the lower the level of the derivative in the minimization principle, the better the shape preservation is. Function-value-based cubic L"1 splines preserve shape well for all situations tested except the one in which an ''S-curve'' occurs when a flatter representation might be expected. A multi-step procedure for calculating the coefficients of quintic and seventh-degree L"1 splines is proposed. As a basis for this procedure, the function-value-based cubic L"1 spline is calculated. The quintic L"1 spline is calculated by fixing the first derivatives at the nodes to be those of the cubic L"1 spline and calculating the second derivatives at the nodes by minimizing a second-derivative-based quintic L"1 spline functional. The seventh-degree L"1 spline is calculated by fixing the first and second derivatives at the nodes to be those of the quintic L"1 spline and calculating the third derivatives at the nodes by minimizing a second-derivative-based seventh-degree L"1 spline functional. Computational results indicate that C^2 quintic L"1 splines and C^3 seventh-degree L"1 splines calculated in this manner preserve shape well for all situations tested except one in which an ''S-curve'' occurs when a flatter representation might be expected. To ensure that the parameters of cubic and higher-degree L"1 splines depend continuously on the positions of the data, weights that depend on the local interval length need to be used in the integrals in the minimization principles of these splines.

A topological approach to identifying the ''good'' interpolant among the four distinct solutions to the first-order Hermite interpolation problem for planar quintic Pythagorean-hodograph curves is presented. An existence theorem is proved, together with a complete analysis of uniqueness/non-uniqueness properties. A simple formula for finding the ''good'' solution, without appealing to curve fairness or energy integrals, is also presented.

The paper describes a method to join two circles with a C-shaped and an S-shaped transition curve, composed of a Pythagorean hodograph quintic segment, preserving G^2 continuity. It is considered desirable to have such a curve in satellite path planning, highway or railway route designing, or non-holonomic robot path planning. As an extension of our previous work, we show that a single quintic curve can be used for blending or for a transition curve between two circles, including the previously unsolved cases of concentric and tangential circles.

We present a new semi-topological quantity, called the absolute hodograph winding number, that measures how close the quintic PH spline interpolating a given sequence of points is to the cubic spline interpolating the same sequence. This quantity then naturally leads into a new criterion of determining the best quintic PH spline interpolant. This seems to work favorably compared with the elastic bending energy criterion developed by Farouki [Farouki, R.T., 1996. The elastic bending energy of Pythagorean-hodograph curves. Comput. Aided Geom. Design 13 (3), 227-241]. We also present a fast method that is a modification of the method of Albrecht, Farouki, Kuspa, Manni, and Sestini [Albrecht, G., Farouki, R.T., 1996. Construction of C^2 Pythagorean-hodograph interpolating splines by the homotopy method. Adv. Comput. Math. 5 (4), 417-442; Farouki, R.T., Kuspa, B.K., Manni, C., Sestini, A., 2001. Efficient solution of the complex quadratic tridiagonal system for C^2 PH quintic splines. Numer. Algorithms 27 (1), 35-60]. While the basic scheme of our approach is essentially the same as theirs, ours differs in that the underlying space in which the Newton-Raphson method is applied is the double covering space of the hodograph space, whereas theirs is the hodograph space itself. This difference, however, seems to produce more favorable results, when viewed from the above mentioned semi-topological criterion.

We introduce L p ( 1 ⩽ p ⩽ ∞ ) piecewise polynomial parametric splines of degree 3 or 5 which smoothly interpolate data points. The coefficients of these polynomial splines are calculated by minimizing the L p norm of the second derivative. It is demonstrated that the C 1 -continuous cubic (resp. G 1 -continuous cubic and C 2 -continuous quintic) L p polynomial splines are unique for 1 p ∞ . The focus is mainly on L 1 polynomial splines which preserve the shape of data even with abrupt changes in direction or spacing. When there are several candidates we introduce a tension parameter which selects from the set of solutions the smallest in norm. The optimisation uses a nonlinear functional; therefore we discretize it so as to obtain a linear problem and we give an upper bound to the error. Computational examples using a primal affine (interior point) algorithm are presented. We will also show how interesting it can be to be able to change the partition associated to the data points in order to obtain convexity properties when possible. Moreover, we will show that the solution is not unvarying by rotation in the L 1 -case and we propose an alternate approach with a local change of coordinates on each segment [ P i , P i + 1 ] . We show that this method can be applied to any kind of norm for the minimization problem.

When designing curves, it is often desirable to join two points, at which G^2 Hermite data are given, by a low degree parametric polynomial curve which has no extraneous curvature extrema. Such curves are referred to as being fair. The join can be accomplished by constructing the curve from a pair of polynomial spiral segments. The purpose may be practical, e.g., in highway design, or aesthetic, e.g., in the computer aided design of consumer products. A Pythagorean hodograph curve is polynomial and has the attractive properties that its arc-length is a polynomial of its parameter, and the formula for its offset is a rational algebraic expression. A technique for composing a fair curve from a pair of Pythagorean hodograph quintic spiral segments is examined and presented.

In the highway and rail route designs, or a car-like robot path planning it is often desirable to have a method of joining a circle to a circle with an S-shaped or a broken back C-shaped spiral transition. This paper describes a transition between two such circles. It is shown that a single Pythagorean hodograph quintic curve can be used for blending or for a transition curve preserving G^2 continuity with local shape control parameter and flexible constraints. Provision of a shape control parameter provides freedom to modify the shape in a stable manner.

A novel formulation for spatial Pythagorean hodograph (PH) curves, based on the geometric product of vectors from Clifford algebra, is proposed. Compared to the established quaternion representation, in which a hodograph is generated by a continuous sequence of scalings/rotations of a fixed unit vector [email protected]?, the new representation corresponds to a sequence of scalings/reflections of [email protected]?. The two representations are shown to be equivalent for cubic and quintic PH curves, when freedom in choosing [email protected]? is retained for the vector formulation. The latter also subsumes the original (sufficient) characterization of spatial Pythagorean hodographs, proposed by Farouki and Sakkalis, as a particular choice for [email protected]?. In the context of the spatial PH quintic Hermite interpolation problem, variation of the unit vector [email protected]? offers a geometrically more-intuitive means to explore the two-parameter space of solutions than the two free angular variables that arise in the quaternion formulation. This space is seen to have a decomposition into a product of two one-parameter spaces, in which one parameter determines the arc length and the other can be used to vary the curve shape at fixed arc length.

A scheme to approximate the trimmed surfaces defined by two tensor-product surface patches, intersecting in a smooth curve segment that extends between diametrically opposite patch corners, is formulated. The trimmed surface approximations are specified by triangular Bezier patches, whose tangent planes agree precisely with those of the tensor-product surfaces along the two sides where they coincide. Topological consistency of the two trimmed surfaces is achieved by requiring the "hypotenuse" sides of the triangular patches to be coincident. In the case of bicubic tensor-product patches and quintic triangular trimmed surface approximations, enforcing these conditions entails the solution of a linear system of 30 equations in 32 unknowns. The two remaining scalar freedoms, together with four additional free control points, are employed to enhance the accuracy and/or smoothness properties of the intersection curve and trimmed surface approximations. By means of an appropriate subdivision preprocessing, the trimmed surface scheme may be used on models described by arbitrary bicubic surface patches.

In this paper a class of C2 ∩ FC3 spline curves possessing tension properties is described. These curves can be constructed using a simple modification of the well-known geometric construction of C4 quintic splines; therefore their shape can be easily controlled using the control net. Their applications in approximation and interpolation of spatial data will be discussed.

In this paper we present methods for approximating a helix segment by quintic Bezier curves or quintic rational Bezier curves based on the geometric Hermite interpolation technique in space. The fitting curve interpolates the curvatures as well as the Frenet frames of the original helix at both ends. We achieve a high accuracy of the approximation by giving a proper parametrization of the curve, and the approximation order of the height function along the helix axis is 9 provided that the screw angle of the helix is fixed. Numerical examples are also presented to illustrate the efficiency of the new method.

This paper deals with the modeling of G1 bifurcation. A branch-blending strategy is applied to model the bifurcation such that the coherence among the individual branching segments is characterized. To achieve this, bi-cubic Bezier patches are first used to generate three half-tubular surfaces by sweeping operations. The bifurcation modeling is then converted to a problem of filling two triangular holes surrounded by the swept half-tubular surfaces. In order for the bifurcation model to be G1, the candidate surfaces for hole filling are required to have (i) an inter-patch tangential continuity along the so-called star-lines; and (ii) a cross-boundary tangential continuity with the surrounding half-tubular surfaces. For inter-patch tangential continuity, we use the method proposed by Gregory and Zhou (1994) to determine the center-point, the star-lines, and the associated vector-valued cross-boundary derivatives. The problem of ensuring the cross-boundary tangential continuity with the surrounding surfaces is more difficult. We first derive the conditions for the twist-compatibility from the requirements of cross-boundary tangential continuity. The solutions for the conditions derived are then developed. The hole boundaries, originally in cubic Bezier form, are constructively modified to the quintic form to ensure the twist-compatibility and uniqueness of the tangent planes at the hole corners. Subsequently, the half-tubular surfaces in bi-cubic Bezier form are degree-elevated to quintic form along the sweeping directions. This is followed by the modification of the second row control points with the half-tubular surfaces in order to retain a cubic form of the surfaces' cross-boundary derivatives. Vector-valued cross-boundary derivatives in quintic Bezier form are constructed for the hole patches. Using a Coons-Boolean sum approach, the derivatives are utilized to modify the three Bezier patches into the final bi-quintic form in the triangle fill area for each hole.

In general, the problem of interpolating given first-order Hermite data (end points and derivatives) by quintic Pythagorean-hodograph (PH) curves has four distinct formal solutions. Ordinarily, only one of these interpolants is of acceptable shape. Previous interpolation algorithms have relied on explicitly constructing all four solutions, and invoking a suitable measure of shape—e.g., the absolute rotation index or elastic bending energy—to select the “good” interpolant. We introduce here a new means to differentiate among the solutions, namely, the winding number of the closed loop formed by a union of the hodographs of the PH quintic and of the unique “ordinary” cubic interpolant. We also show that, for “reasonable” Hermite data, the good PH quintic can be directly constructed with certainty, obviating the need to compute and compare all four solutions. Finally, we present an algorithm based on the subdivision, degree elevation, and convex hull properties of the Bernstein form, that gives rapidly convergent curvature bounds for PH curves, using only rational arithmetic operations on their coefficients.

This paper presents a method for approximating conic sections using quintic polynomial curves. The constructed quintic polynomial curve has G 3 -continuity with the conic section at the end points and G 1 -continuity at the parametric mid-point. It is found that for any conic section, there exist three quintic polynomial curves satisfying the mentioned geometric continuity. One of them is the geometric Hermite interpolant proposed in (Floater, 1997) and one of the others is shown to have much smaller error and better shape-preserving property.

Two strategies for contour milling of free-form surfaces, using real-time CNC interpolators for Pythagorean-hodograph (PH) curves, are described. The first method, applicable to convex surfaces, employs a flat-end mill and approximates the surface section curves by planar PH quintics. The second approach, which employs a ball-end mill, approximates the tool-center trajectory by quintic PH space curves, and can accommodate nonconvex surfaces by choosing a sufficiently small tool radius. Both schemes generate compact part programs, in which numerous short linear/circular G code motions are replaced by fewer analytic path segments, and eliminate the need for explicit offset curve or surface representations to compensate for the tool radius. The surface sectioning and PH curve approximation algorithms required by these methods are presented, with appropriate tolerance analyses, and preliminary results from machining experiments performed on an open-architecture 3-axis CNC mill are described.

This paper presents methods for approximating circular arcs using quintic polynomial curves. Different boundary conditions are considered in the approximation methods, thus resulting approximation curves with G2, G3, or G4 continuities at the circular arc's ends. The resulting approximation radial errors are generally very small and converge at the eighth or tenth power of the circular arc's angular span.

Spiral segments are useful in the design of fair curves. They are important in CAD/CAM applications, the design of highway and railway routes, trajectories of mobile robots and other similar applications. A Pythagorean hodograph curve has the properties that its arc-length is a polynomial of its parameter, and the formula for its offset is a rational algebraic expression. Recent work demonstrated the composition of G 2 curves by joining circular arcs and/or straight line segments with cubic Bezier spiral segments and Pythagorean hodograph quintic spiral segments. These spiral segments are members of wider classes of spiral segments which are now examined. Selecting members from these wider classes of spiral segments allows for more flexible curve design; it is not necessary to incorporate circular arcs and straight line segments when using them.

This paper extends to C 1 quintic Bezier triangular patches the recent results obtained by Greiner using C 2 bicubic B-spline surfaces. Tensor notation is used to elucidate the invariance properties of Greiner's fairness functional and of his blending-surface functional based on the Laplace-Beltrami operator. The notation shows how to construct additional invariant functionals. A functional that minimizes the square of Gaussian curvature is introduced following the guidelines of Greiner. The paper also discusses a few test results obtained using the Laplace-Beltrami and Gaussian functionals, comparing them with a simple quadratic functional based on a special choice of parametric coordinates.

The focus of this paper is the construction of a new tetrahedral patch, which allows an explicit representation but does not require a split of the domain tetrahedra. The patch will be represented as a rational convex combination of a quintic Bezier tetrahedron, where the four inner Bezier points are duplicated. It can be regarded as generalization of the hybrid cubic triangular Bezier patch. The patch is used to represent a C 1 trivariate scattered data interpolant to data sampled in a volume.

The problem of interpolating a free form curve network with irregular topology is investigated in order to create a curvature continuous surface. The spanning curve segments are parametric quintic polynomials, the interpolating surface elements are biquintic Gregory patches. A necessary compatibility condition is formulated and proved which need to be satisfied at each node of the curve network. Constraints are derived for assuring G 2 continuity between biquintic Gregory patches, which share a common side or a common corner point. The above conditions still leave certain geometric freedom for defining the entire G 2 surface, so following some analysis a particular construction is presented, by which after computing the principle curvatures at each node the free parameters are locally set for each interpolating Gregory patch.

A piecewise G1 spline surface composed of sextic triangular Bezier patches in one-to-one correspondence with the faces of a triangular control mesh is presented. Surfaces of arbitrary topological type are created by approximating any mesh that represents a triangulated 2-manifold. The surface has local support and is affine invariant. A set of shape parameters are available for additional local control over the shape of the surface. If some local regularities are present in the structure of the control mesh, then the corresponding patches may be parametrized as quintic or even quartic Bezier triangles. In the case of a regular triangulation (with appropriate shape parameters), the surface generated by this method is equivalent to a quartic C2 triangular B-spline.

Wang, T., A C 2 -quintic spline interpolation scheme on triangulation, Computer Aided Geometric Design 9 (1992) 379-386. A C 2 piecewise quintic interpolating scheme is described for values of position, gradient and Hessian at scattered points in two variables on the triangulation with minimal split. The domain is assumed to have been triangulated. The interpolant has local support and explicit representation. Thus an open problem is solved.

We present an algorithm which determines a bivariate smooth piecewise polynomial interpolant to function values given at points scattered in R 2fs. In particular the degree of the polynomials will be as small as possible for arbitrary triangulations and a high degree of polynomial precision is achieved. We perform several experiments and numerical tests for quintic splines of class C2.

4. Hermite Interpolants . . , . . . . . . . . . . . . . 104 4.1. The Co nine parameter interpolant ...... 104 4.2. C’ quintic interpolants 104 4.3. The general case 106 5. Split Triangle Interpolants . . . . . . . . . 107 5.1. The C’ Clough-Tocher interpolant . . _ . . 108 5.2. Limitations of the Clough-Tocher split . 110 5.3. The C’ Powell-Sabin interpolants . . . . 111 5.4. C’ Split square interpolants . . . . . . . . . 112

G. Nielson [1974] gave a piecewise polynomial alternative to splines under tension, the so-called v-splines. The v-splines are the solution of a minimal norm problem and they have curvature continuity. In this paper the v-splines are generalized to geometric spline curves and the computational equations of quintic splines curves with curvature and torsion continuity are given. These curves will be referred to as @t-splines.