R. E. Barnhill, G. Farin, M. Jordan, and B. R. Piper. Surface /surface intersection. Computer Aided Geometric Design, 4(12) :3--16, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....adaptively. By splitting each rectangular into two triangles, we obtain a piecewise linear approximation of the surface. The goal is to approximate curved regions using more triangles than for the at regions, while ensuring that the approximation satis es the requirements of a given tolerance [1]; polygonize trimming curves; consists in obtaining a piecewise linear approximation of the trimming curves within a given tolerance. merge trimming polygon and rectangular grid; given a rectangle in the grid of the parameter domain, along with its status and a set of trimming polygons, the ....

....in this process. 5.1. SSI. The methods available for intersecting patches can be divided into two categories: curve following and subdivision. Firstly, the curve following method nds some points of intersection. Then, the points on the intersection curve are detected using a numerical method [1, 2]. This method is especially useful for processing singular cases of intersection [3] The subdivision method divides the problem into smaller problems by approximating the patch into simpler linear or quadratic subpatches [23, 6] The patches are intersected, resulting in curves that are then re ....

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R.E.Barnhill, G.Farin, M.Jordan, B.R.Piper, Surface Surface Intersection, Computer Aided Geometric Design, 4 (1987), pp. 3-16.

.... to knots typeder struct PR knots struct PR knots pf next; Pointer to next one Pint pf size; Size of this one Pfloat knots[i] Array of the knots PR knots; typeder struct PR pts struct PR pts Ppoint4 PR pts; pf next; Pointer to next one pf size; Size of this one pts[1]; Array of the Points , typeder enum PVudir, PVvdir PE dir; The PR nurb data structure can be used either by static or dynamic allocation The following examples serve to show this 1. Using static allocation Allocate values to each field in the NURBS data structure ....

R E Barnhill and R F Riesenfeld. Computer-Aided Geometric Design. Academic Press, 1974.

....discussions. Definition 1. Under a parallel projection, if the plane containing the space conic is not perpendicular to the projection plane, then the parallel projection is nondegenerate. Under non degenerate parallel projections, all conics are equivalent, i.e. conics are mapped to conics [5]. In addition, the class of conic curves is invariant under such projection. It follows that, under non degenerate parallel projections, ellipses, parabolas, and hyperbolas in line drawings are projections of ellipses, parabolas, and hyperbolas, respectively, in space [13] Therefore, if at least ....

G Farin, Curves and Surfaces, Computer Aided Geometric Design, 4 rd edn. Academic Press, New York, 1997.

....the surfaces into piecewise linear approximations and intersect the facets. The accuracy of these subdivision algorithms depends on how the flatness of the subpatches is defined. Small intersection loops or isolated points are very difficult, if not impossible, to find. Marchingbased algorithms [1, 2] begin by finding a starting point on the intersection curve, and proceed to march along the curve. Because of the inherent geometric complexity of high degree algebraic curves that could yield from the intersection of two regular surfaces, marching along a branch or a closed loop is also a ....

R. E. Barnhill, G. Farin, M. Jordan, and B.R.Piper. Surface/surface intersection. Computer Aided Geometric Design, 4(1/2):3--16, July 1987.

....succession of spline segments. This model is built as a succession of one or more curves defined as functional combinations of a common set of 3D control points with a set of parametric blending functions. It encompasses a great deal of classical spline models both interpolating (Hermitian splines [Barnh74] , Catmull Rom splines [Catmu74] and TCB splines [Kocha84] and approximating (Bezier splines [Bzie66,77] Farin90] B splines [Riese73] Foley89] splines [Barsk81,83] and NURBS [Piegl91] Extension of this modelling scheme to surfaces or volumes is straightforward. Hence, we model ....

Barnhill R., Riesenfeld R. Computer aided Geometric design, Academic Press, 1974.

....example, any scalar field f on S can be thought of or visualized as a surface in its own right, e.g. the surface S 0 : fs f(s)n s ; s 2 Sg; 3) where n s is the unit surface normal at the point s. This is the reason why such surfaces are also sometimes referred to as surfaces on surfaces [11]. For example, S 0 could represent the true surface of the earth, with S being the reference sphere ellipsoid and f the height above the sea level. Or, S 0 could be an offset surface to S, i.e. a surface whose distance to S is a fixed number (in which case the function f in (3) is ....

....missing on the sphere, such as the convex hull property. Typical scattered data interpolation approximation methods on the sphere start with a triangulation of the sphere, for example the so called Delaunay triangulation. We refer the reader to [68] for a survey on triangulations, and also to [11,39], for a discussion of triangulation methods on general surfaces. Methods for triangulation of scattered data points specifically over the sphere are discussed in [41,57,65] Figures 9 and 10 below show wire plots of smooth C 1 quadratic and cubic spherical splines, respectively, corresponding to ....

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R. E. Barnhill and H. S. Ou, Surfaces defined on surfaces, Computer Aided Geometric Design, 7:323--336, 1990.

....paper investigates the representation properties of spline based neurons, and analyzes the advantages of this approach in some applications, both in pattern recognition and data processing domains. In particular, we show that among the available spline curves, the cubic Catmull Rom (CR) spline [19], 20] is suitable for implementing an adaptive activation function with some interesting features, due to its local interpolation and regularization characteristics. In Section II, the new ASNN using cubic splines is presented. In Section III, a backpropagation learning algorithm is derived. ....

....span. Hence, there must be a unique mapping that allows us to calculate the local parameter , as well as the proper curve span , from the abscissa global parameter. In this way, we can represent any point lying on the spline curve as a point belonging to the single curve span. It follows (see [19]) that the th curve span can be described as follows: 2) 674 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 10, NO. 3, MAY 1999 where are the spline polynomials. As in (1) each coordinate is described by a univariate function, namely a cubic polynomial of the variable . B. The GS Neuron Let be ....

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R. E. Barnhill and R. F. Riesenfeld, Eds., Computer Aided Geometric Design. New York: Academic, 1974.

....11] The former represents the center position of the rigid body, and the latter does its orientation. Considering an articulated figure, a posture of a freely rotating limb can be represented by the rotational component. The fairing in R 3 is well established in computer aided geometric design [7, 6, 16]. However, there are few results on fairing a sequence of orientation samples. In this paper, we present a new fairing algorithm that iteratively smoothes 6 DOF rigid motion data. Section 2 discusses an underlying idea for our ap proach in motion fairing. In Section 3 we introduce a unit ....

....and Section 6 gives experimental results. Finally, Section 7 concludes this paper. 2 BACKGROUND In geometric modeling, fairing plays an important role for designing high quality curves and surfaces. There are rich results on fairing to get an aesthetically pleasing and functionally useful curve [7, 6, 16]. In most of literatures, the resulting curve minimizes the weighted sum of derivatives of the curve, which is given by E(C) Z C n X m=1 ff m kC (m) s)k 2 ds; 1) where C (m) s) is a m th order derivative for a curve C(s) which is parameterized by arc length. If we choose weights ....

G. Farin, G. Rein, N. Sapidis, and A. J. Worsey. Fairing cubic b-spline curves. Computer Aided Geometric Design, 4:91--103, 1987.

....parameter values. The biggest drawback in this approach is the lack of robustness. Small loops could easily be missed depending on the frequency with which the curves are evaluated. In the last decade, techniques based on curve tracing have been widely used to evaluate high degree curves [BFJP87, BHHL88, KPP90, MC91] The main idea is to com 13 pute at least one point on every component of the curve and use the local geometry of the curve to evaluate successive points. In this class of methods, identifying a point on every loop is significantly harder than identifying a point on open ....

R. Barnhill, G. Farin, M. Jordan, and B. Piper. Surface/surface intersection. Computer Aided Geometric Design, 4(3):3--16, 1987.

....third derivatives of the two space curves at the junction point of four patches by moving it into a better position decreases the physical spline energy. Here again, it is a global fairing method. Farin et al. s method for cubic B spline curves Based on Kjellander s fairness criterion Farin et al. [7] propose a local algorithm where in each fairing step only a small number of control points is involved. They use a knotremoval reinsertion step at the most offending inner knot t . The B spline curve becomes momentarily C 3 continuous at t . The same procedure is than applied to another ....

Farin, G., G. Rein, N. Sapidis, and A. J. Worsey, Fairing cubic B-spline curves, Computer Aided Geometric Design 4 (1987), 91--103.

....differential equations problem. At the intersection point on the surfaces, the intersection curve must be mutually orthogonal to the normals of the surfaces. Consequently, the vector field which the tracing point must follow is given by the cross product of the normals. Barnhill et al. [Barnhill87] use the isoparametric method for finding starting points, and then use the tracing method described above. Also, Barnhill and Kersey [Barnhill90] discuss improved tracing methods, and Mullenheiim [Mullenheim91] discusses a method for finding starting points for tracing. 10 5.1.3.4 Analytic ....

R. E. Barnhill, G. Farin, M. Jordan, B. R. Piper. Surface/surface intersection. ComputerAided Geometric Design, Vol. 4, No. 1, pp. 3-16, 1987. 16

....model is transformed to boundary representation. For describing surfaces implicit and parametric representations are common in CAGD. If two surfaces intersect, the result will be either a set of isolated points, a set of curves, a set of overlapping surfaces or any combination of these three cases [2]. Previous work: The SSI problem has been studied detailed in a variety of publications. Apart from some special (surface dependent) methods which provide exact solutions, two general approaches are common. Both are approximation methods, each approach can be qualified according to the attributes ....

R. Barnhill, G. Farin, M. Jordan, and B. Piper. Surface/surface intersection. Computer Aided Geometric Design, 4:3--16, 1987.

....SSI is a basic problem in CAGD, it is stated very simply: Given are two surfaces in R 3 , compute all parts of the intersection curve. If two surfaces intersect, the result will be either a set of isolated points, a set of curves, a set of overlapping surfaces or any combination of these cases [1]. Because exact solutions can be found only for some special surface classes, approximation methods are used for the general case. In [4] and [5] we introduced a fast algorithm for computing an inclusion of the intersection curve of two general parametric surfaces. Each surface is defined as s(u; ....

R. Barnhill, G. Farin, M. Jordan, and B. Piper. Surface/Surface Intersection. Computer Aided Geometric Design, 4:3--16, 1987.

....model is transformed to boundary representation. For describing surfaces implicit and parametric representations are common in CAGD. If two surfaces intersect, the result will be either a set of isolated points, a set of curves, a set of overlapping surfaces or any combination of these three cases [1]. Previous work: The SSI problem has been studied detailed in a variety of publications. Apart from some special (surface dependent) methods which provide exact solutions, two general approaches are common. Both are approximation methods, each approach can be qualified according to the attributes ....

R. Barnhill, G. Farin, M. Jordan, and B. Piper. Surface/Surface Intersection. Computer Aided Geometric Design, 4:3--16, 1987.

....The methods for solving the surface intersection problem in CAGD belong to two major classes: marching methods and decomposition methods. Marching or continuation methods compute the intersection curve in 3D object space by marching along the curve in the direction of its tangent vector [1, 2, 26, 28]. Decomposition or subdivision methods compute the trimming curves in 2D parameter space by recursively refining the solution at each step [8, 12, 16] Previous solutions to the surface intersection problem work well in many cases, but do not handle the mesh intersection problem as defined in ....

R. E. Barnhill, G. Farin, M. Jordan, and B. R. Piper. Surface/surface intersection. Computer Aided Geometric Design, 4:3--16, 1987.

....most often occurs when the surfaces are grazing or barely penetating. In some cases, lattice methods or subdivision methods are used to find starting points for use by the tracing methods. 4.2. 6 Tracing methods The tracing method begins with a given point known to be on the intersection curve [BFJP87, BHHL88,Hoh91,MC91, KM97]. Then the intersection curve is traced in sufficiently small steps until the edge of the patch is found, or until the curve returns to itself to close a loop. While it is easy to check for meetings with a patch boundary, it is difficult to know when the tracing point has returned to its starting ....

R. Barnhill, G. Farin, M. Jordan, and B. Piper. Surface/surface intersection. Computer Aided Geometric Design, 4(3):3--16, 1987.

....Relevant to this work are free form surfaces based on a triangular parameter domain. One powerful variant of triangular free form surfaces is the rational triangular B zier surface. Rational surfaces allow the exact modeling of quadratic surfaces, which are common in mechanical design. See [6] for a survey. Triangular rational B zier surfaces are defined as follows: 1 , 0 , 0 0 , 0 0 , v u v u with v u B w v u B w P v u P n j j n i n j i j i n j j n i n j i j i j i (1) with the corresponding B zier weight functions: j i n k v u ....

G. Farin, Triangular Bernstein-Bézier patches, Computer Aided Geometric Design 3(2), pp. 83-128, 1986.

....constant parameter values. The biggest drawback in this approach is the lack of robustness. Small loops could easily be missed depending on the frequency with which the curves are evaluated. In the last decade, techniques based on curve tracing have been widely used to evaluate high degree curves [BFJP87, BHHL88, KPP90, MC91] The main idea is to compute at least one point on every component of the curve and use the local geometry of the curve to evaluate successive points. In this class of methods, identifying a point on every loop is significantly harder than identifying a point on open ....

R. Barnhill, G. Farin, M. Jordan, and B. Piper. Surface/surface intersection. Computer Aided Geometric Design, 4(3):3--16, 1987.

.... typedef struct PRdir Pint pfk; order in u Pint pfn; number of vertices Pint pfnt; number of knots n k PRknots pfkk; Pointer to knots PRdir; typedef struct PRknots struct PRknots pfnext; Pointer to next one Pint pfsize; Size of this one Pfloat knots[1]; Array of the knots PRknots; typedef struct PRpts struct PRpts pfnext; Pointer to next one Pint pfsize; Size of this one Ppoint4 pts[1] Array of the Points PRpts; typedef enum PVudir, PVvdir PEdir; 1 st CGU MCC CGU The NURBS Procedure Library 15 10 Use of the ....

.... PRdir; typedef struct PRknots struct PRknots pfnext; Pointer to next one Pint pfsize; Size of this one Pfloat knots[1] Array of the knots PRknots; typedef struct PRpts struct PRpts pfnext; Pointer to next one Pint pfsize; Size of this one Ppoint4 pts[1]; Array of the Points PRpts; typedef enum PVudir, PVvdir PEdir; 1 st CGU MCC CGU The NURBS Procedure Library 15 10 Use of the Library The PR nurb data structure can be used either by static or dynamic allocation. The following examples serve to show this. 1. Using static allocation ....

R E Barnhill and R F Riesenfeld. Computer-Aided Geometric Design. Academic Press, 1974.

....an unknown terrain, a robot builds a navigational plan by combining a number of simpler courses each trough a triangular region [20,50] As is often the case, the terrain contains natural obstacles that must be excluded from the triangulation. An even more challenging problem arises in the design [1, 2, 14], analysis [15, 30, 48] and manufacture [6, 13] of vehicles from the aeronautical and automobile industries to the parts manufactured by a typical job shop. In each of these cases, surfaces must have an internal computer representation that reflects the desired geometry of the modeled region ....

R. Barnhill and R. Riesenfeld, Eds., Computer Aided Geometric Design, Academic Press, New York, 1974.

....alors que pour une meilleure stabilit e, ces algorithmes ont besoin d un troisi eme etat qui permet de g erer correctement les cas ambigus ou ind ecidables. Les approches classiques de d etection d intersection consistent en une classification par subdivision et comparaison des boites englobantes [BFJP87, BK90]. Elles op erent dans une logique a 2 etats et classifient surfaces en : 1. Non Intersectants 2. Potentiellement Intersectants Un traitement suppl ementaire est toujours n ecessaire car on a besoin d une phase de raffinement de l etat 2 pour d etailler la classification avant de calculer ....

R.E. Barnhill, G. Farin, M. Jordan, and B.R. Piper. Surface/surface intersection. Computer Aided Geometric Design, 4(1-2):3--16, 1987.

....of algorithm. Generally, divide and conquer algorithms are considered to be more robust, but also less efficient, than marching algorithms. Marching algorithms work by first finding a point on each intersection curve, then for each intersection, marching along it to discover the entire curve. See Barnhill et al. 1987) and Mullenheim (1991) for details on such algorithms. Our intersector belongs to the marching class of algorithms. Issues in implementing a marching type algorithm include what space to march in, how to find the first point on the intersection, how to find the direction to march in, how big a ....

R. E. Barnhill, G. Farin, M Jordan, and B. R. Piper. Surface/surface intersection. Computer Aided Geometric Design, 4:3--16, 1987.

....not be practical. Hence, we must resort to numerical techniques. The methods for solving the surface intersection problem in CAGD belong to two major classes: Marching or continuation methods compute the intersection curve in 3D object space by marching in the direction of its tangent vector [5 7].Decomposition or subdivision methods compute the trimming curves in 2D parameter space, by recursively refining the solution at each step [8] Previous solutions to the surface intersection problem work well in many cases, but do not handle the mesh intersection problem as defined above. An ....

R. E. Barnhill, G. Farin, M. Jordan, and B. R. Piper. Surface/surface intersection. Computer Aided Geometric Design, 4:3--16, 1987.

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G. Farin and D. Hansford. Discrete Coons patches. Computer Aided Geometric Design, 16(7):671--689, 1999.

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G. Farin. Computer Aided Geometric Design. In J. Thompson, B. Soni, and N. Weatherill, editors, Handbook for Grid Generation, pages 28--1 -- 28--22. CRC Press, 1999.

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H. Wolters and G. Farin. Geometric curve approximation. Computer Aided Geometric Design, 14(6):499--513, 1997.

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C. Baumgarten and G. Farin. Rational approximation of spirals. Computer Aided Geometric Design, 14(6):515--532, 1997.

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R. Barnhill, G. Farin, M. Jordan, and B. Piper. Surface / surface intersection. Computer Aided Geometric Design, 4(1-2):3--16, 1987.

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G. Farin, G. Rein, N. Sapidis, and A. Worsey. Fairing cubic B-spline curves. Computer Aided Geometric Design, 4(1-2):91--104, 1987.

No context found.

G. Farin. Some remarks on V 2 - splines. Computer Aided Geometric Design, 2(2):325--328, 1985.

....intersection problems. The biggest drawback in this approach is the lack of robustness. Small loops could easily be missed depending on the frequency with which the curves are evaluated. In the last decade, techniques based on curve tracing have been widely used to evaluate high degree curves [BFJP87, BHHL88, KPP90, MC91] In these class of methods, identifying a point on every loop is significantly harder than that on open components. Therefore, a number of techniques for loop detection have been proposed [SKW85, SM88, THS89, Che89, Hoh91, Kim90, KPP90, KPW90] However, most of these efforts ....

R. Barnhill, G. Farin, M. Jordan, and B. Piper. Surface /surface intersection. Computer Aided Geometric Design, 4(3):3--16, 1987.

....parameter values. The biggest drawback in this approach is the lack of robustness. Small loops could easily be missed depending on the frequency with which the curves are evaluated. In the last decade, techniques based on curve tracing have been widely used to evaluate high degree curves [BFJP87, BHHL88, KPP90, MC91, KM96, KM97] The main idea is to compute at least one point on every component of the curve and use the local geometry of the curve to evaluate successive points. In these class of methods, identifying a point on every loop is significantly harder than that on open ....

R. Barnhill, G. Farin, M. Jordan, and B. Piper. Surface/surface intersection. Computer Aided Geometric Design, 4(3):3--16, 1987.

....role: Given a surface or a set of data points of a surface, e.g. coming from a double curved area of a ship surface, approximate the data by a developable surface. Most CAGD research on developable surfaces has been focused on the construction of developable surfaces to be used in CAD systems (see [7] and the references therein) Surface approximation by developable surfaces is addressed in several con2 tributions [8, 9, 10, 11, 12, 13] but all these papers either do not allow our input data or the methods they describe could not perform very well due to limitations. In this paper we ....

H. Pottmann and G. E. Farin, Developable rational Bezier B-spline surfaces, Computer Aided Geometric Design. 12, 1995, 513-531.

....constant parameter values. The biggest drawback in this approach is the lack of robustness. Small loops could easily be missed depending on the frequency with which the curves are evaluated. In the last decade, techniques based on curve tracing have been widely used to evaluate high degree curves [BFJP87, BHHL88, KPP90, MC91] The main idea is to compute at least one point on every component of the curve and use the local geometry of the curve to evaluate successive points. In these class of methods, identifying a point on every loop is significantly harder than that on open components. As a ....

R. Barnhill, G. Farin, M. Jordan, and B. Piper. Surface/surface intersection. Computer Aided Geometric Design, 4(3):3--16, 1987.

....marching methods, use a local approach to the surface intersection problem. Starting from a point known to be on both surfaces, these methods build an approximation for the intersection curve by marching along the curve, successively computing a new point based on the previous point (or points) [6]. Continuation methods must use numerical approximations not only for marching along the curve, but also for finding starting points. Since the intersection might have several connected components, a starting point is needed on each component. Moreover, care must be taken for handling closed ....

R. E. Barnhill, G. Farin, M. Jordan, and B. R. Piper. Surface/surface intersection. Computer Aided Geometric Design, 4(1-2):3--16, July 1987.

....marching methods, use a local approach to the surface intersection problem. Starting from a point known to be on both surfaces, these methods build an approximation for the intersection curve by marching along the curve, successively computing a new point based on the previous point (or points) [6]. Continuation methods must use numerical approximations not only for marching along the curve, but also for finding starting points. Since the intersection curve may have several connected components, a starting point is needed on each component. Moreover, care must be taken for handling closed ....

R. E. Barnhill, G. Farin, M. Jordan, and B. R. Piper. Surface/surface intersection. Computer Aided Geometric Design, 4(1-2):3--16, July 1987.

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R. E. Barnhill, G. Farin, M. Jordan, and B. R. Piper. Surface /surface intersection. Computer Aided Geometric Design, 4(12) :3--16, 1987.

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R. E. Barnhill and R. F. Riesenfeld, eds., Computer AidedGeometric Design. York: Academic Press, 1974.

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Farin G., G. Rein, N. Sapidis, and A. J. Worsey, Fairing cubic B-spline curves, Computer Aided Geometric Design 4 (1987), 91--103.

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Farin, G., G. Rein, N. Sapidis, and A. J. Worsey, Fairing cubic B-spline curves, Computer Aided Geometric Design 4 (1987), 91--103.

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Barnhill, R., Farin, G., Jordan, M., and Piper, B. Surface/surface intersection. Computer Aided Geometric Design 4 (1987), 3--16.

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Farin G., Rein G., Sapidis N., Worsey A.J.: Fairing cubic B-spline curves. Computer Aided Geometric Design 4, 91--103 (1987).

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R. E. Barnhill and R. F. Riesenfeld, eds., Computer Aided Geometric Design. New York: Academic Press, 1974.

No context found.

R. E. Barnhill, G. Farin, M. Jordan, and B. R. Piper. Surface/surface intersection. Computer Aided Geometric Design, 4(1-2):3--16, July 1987.

No context found.

Farin, G., G. Rein, N. Sapidis, and A. J. Worsey,Fairing cubic B-spline curves, Computer Aided Geometric Design 4 (1987), 91--103.

No context found.

Barnhill, R. E., Farin, G., Jordan, M. and Piper, B. R., Surface/surface intersection, Computer Aided Geometric Design, 4(1-2):3--16, July 1987.

No context found.

R.E. Barnhill - G. Farin - M. Jordan - B.R. Piper, Surface Surface Intersection, Computer Aided Geometric Design, 4 (1987), pp. 3-16.

No context found.

R. E. Barnhill and R. F. Riesenfeld, Computer Aided Geometric Design, Academic Press, New York, 1974.

No context found.

Robert E. Barnhill and Richard F. Riesenfeld, editors. Computer Aided Geometric Design. Academic Press, 1974.

No context found.

R. E. Barnhill, G. Farin, M. Jordan, B. R. Piper: Surface/surface intersection, Computer Aided Geometric Design 4, 3--16, 1987

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