Sederberg, T. W., and Parry, S. R. Comparison of three curve intersection algorithms. Computer Aided Design 18, 1 (January/February 1986), 58--63.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....be tackled geometrically, that is, using subdivision techniques for the curves, that exploit the convex hull property of the NURBS curves [27, 25, 38] or numerically. The latter is done either by nding all the zeros of the function resulting from the system, when a curve has been implicitized [39] (this is pro tably applicable only for low degree curves) or by nding the zeros of the vector function in two variables (2D surface) obtained from the di erence between the curves [16] The problem of nding the zeros of a 2D surface can be solved using the same methods as those used to ....

T.W.Sederberg and Parry, Comparison on three curve intersection algorithm, Computer-Aided Design, 18 (1986), pp 58-63.

.... checked in the case that A and B are the polynomial B ezier curves fl(t) and ffi(u) In that case, the curve fpg Omega ffi Gamma1 (u) is a rational B ezier curve, and testing (14) requires an algorithm to find the intersections of polynomial or rational B ezier curves see, for example, [25]. 4.7 Computed examples To illustrate the operation of the algorithm, we now present an example of its use to compute the Minkowski product of two B ezier curves. Figure 3 shows the two B ezier curves that were chosen as operands in the Minkowski product they are both polynomial B ezier ....

T. W. Sederberg and S. R. Parry (1986), Comparison of three curve intersection algorithms, Computer Aided Design 18, 58--64. 37

....3 Curve Surface Intersection The problems of computing the intersection of curves and surfaces are fundamental in computer graphics and geometric modeling. Common applications include surfacesurface intersection, ray tracing, hidden curve removal and visibility algorithms [Hof89, EC90, NSK90, SP86] Our surface surface intersection algorithm (chapter 4) needs starting points on each component of the intersection curve. We use curve surface intersection to evaluate these starting points. Our algorithm for boundary evaluation relies on a rayshooting approach for the classification of certain ....

....and curve surface intersections. In each case, the problem is reduced to an eigenvalue problem and we compute the eigenvalues in a domain. We have performed comparisons with the algorithm based on QR iterations in [MD94] and an implementation of implicitization based algorithm described in [SP86] and B ezier clipping described in [ B ezier Clipping: B ezier Clipping is an iterative method which takes advantage of the convex hull property of B ezier curves, and iteratively clips away regions of the curve that does not intersect with the surface. B ezier clipping converges more ....

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T.W. Sederberg and S.R. Parry. Comparison of three curve intersection algorithms. Computer-Aided Design, 18(1):58--63, 1986.

....and most stable was recursive subdivision. 7 The recursive subdivision method for locating crossings exploits two properties of the Bezier representation. First, a Bezier 7 For descriptions and comparisons of different methods for finding points where two Bezier curves intersect, see Sederberg[55]. 117 Figure 4.8: Before overlap pruning. A magnified view of a portion of G non Gammaplanar (k = 3) for the Ehrenstein test figure is shown prior to overlap pruning. Boundary fragments are drawn thick, potential completions (including redundant completions) are drawn thin. Figure 4.9: After ....

Sederberg, T.W. and Parry, S., A Comparison of Three Curve Intersection Algorithms, Computer Aided Geometric Design 18, pp. 58-64, 1986.

....the graphics API in order to have more control over the accuracy. This can lead to an explosion in the amount of data that needs to be stored and then sent down the rendering pipeline. Ray tracing does not have this limitation. Intersection tests with NURBS have been done in several ways (e.g. [16, 29, 33]. Our approach computes an estimate to the intersection point and then uses a brute force approach to compute Figure 13: Left: one sample per pixel with hard shadows. Right: one sample per pixel with soft shadows. Note that the method captures the singularity near the box edge. Figure 14: Two ....

Thomas W. Sederberg and Scott R. Parry. Comparison of three curve intersection algorithms. Computer-aided Design, 18(1), January/February 1986.

....The problems of computing the intersection of parametric and algebraic curves are fundamental to geometric and solid modeling. Parametric curves, like B splines and B ezier curves, are extensively used in the modeling systems and algebraic plane curves are becoming popular as well [Hof89, MM89, SP86, Sed89] Intersection is a primitive operation in the computation of a boundary representation from a CSG (constructive solid geometry) model in a CAD system. Other applications of intersection include hidden curve removal for free form surfaces, finding complex roots of polynomials etc. EC90, ....

....endpoints. Thus, we obtain a rectangular bounding box and the subdivision amounts to evaluating the coordinate of the midpoint of the interval and defining the resulting rectangles. The rest is similar to subdivision. The relative performance and accuracy of these algorithms is highlighted in [SP86] In particular, implicitization based approaches are considered faster than other intersection algorithms for curves of degree up to four. This includes faster subdivision based algorithms [SWZ89] However, their relative performance degrades for higher degree curves. This is mainly due to ....

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T.W. Sederberg and S.R. Parry. Comparison of three curve intersection algorithms. Computer-Aided Design, 18(1):58--63, 1986.

....or algebraic surfaces has received a great deal of attention [Pra86, Hof89] Given two surfaces, the prob Ming C. Lin Stefan Gottschalk 7 lem corresponds to computing all components of the intersection curve, robustly and accurately. It includes work on curves and surface intersections [SWZ89, SP86, BHHL88, Hof89, Hof90, MD94,MD95, KM97]. All these algorithms have focussed on accurate computation of the intersection set for static models. However, for collision detection we are actually dealing with a restricted version of this problem. That is, given two surfaces we want to know whether they intersect. Furthermore, we are ....

T.W. Sederberg and S.R. Parry. Comparison of three curve intersection algorithms. Computer-Aided Design, 18(1):58--63, 1986.

....This list is in rough order of complexity, the earlier ones being fast to calculate but offering a cruder approximation, the later ones offering tighter bounds but requiring more complex intersection algorithms. Variations on these bounding box methods are commonly found in computer graphics see [4, 7, 13, 18, 20] for details. A graphical snapshot of the data structures in the middle of this algorithm is shown in figure 4. For collision detection in a dynamic environment, for example where the robot is moving amidst obstacles which are also moving, or where a system requires the coordinated motion of ....

T. W. Sederberg and S. R. Parry. Comparison of three curve intersection algorithms. Computer-Aided Design, 18(1):58--63, January/February 1986.

....before the first cusp, the region after the second cusp, and the region between the two cusps. The third part, between the cusps, must be deleted. The first two should then be intersected against each other to find the self intersection point using standard curve curve intersection algorithms [4, 12, 19], trimmed properly to the intersection point, and then merged Error Bounded Offset G. Elber and E. Cohen 9 Figure 4: Variable distance surface offset (u direction linear, v constant) Figure 5: Offset operation local loops are trimmed using a distinct characteristic. back. See Figs. 5 and 7 for ....

T. Sederberg and S. Parry. Comparison of Three Curve Intersection Algorithms. Computer Aided Design, Volume 18, Number 1, January/February 1986.

....polygons, there is no intersection between C and D. Otherwise, the curve segment C (resp. D) is subdivided into two subsegments C 1 and C 2 (resp. D 1 and D 2 ) and the intersection problem is recursively reduced to four subproblems, each for a pair of subsegments C i and D j (i; j = 1; 2) see [Gun89, SS86]. There may be many subproblems where the bounding polygons do not intersect, however, in the worst case, each pair of the bounding polygons would intersect and the time complexity becomes T (n) 4 Delta T (n=2) k = O(n 2 ) where k is a constant and n is a measure of the sizes of C and D. ....

Sederberg, T.W., and Scott, R.P., (1986), "Comparison of three curve intersection algorithms," Computer Aided Design, Vol. 18, No. 1, pp. 58--63.

....in achieving guaranteed numerical properties for cubic curves, we make no attempt to extend the analysis to curves of higher order. It may be helpful to look at the background material on the numerical properties of B ezier Bernstien curves given by Farouki and Rajan [2] and Sederberg and Parry [12]. The overall organization of the paper is as follows: Section 2 derives the implicit form for rational cubic curves and explains how to choose a rotated coordinate system. There is also an alternative form that reduces the error when the crossing point is nearby. Section 3 gives bounds on 4 ....

T. W. Sederberg and S. R. Parry. Comparison of three curve intersection algorithms. ComputerAided Design, 18(1):58--63, 1986.

....width and height. In this case, the Separating Axes Theorem (in Section 5) can be specialized to show that it is sufficient to test only three axes. Curve and Surface Intersections: Computing the intersection between curves and surfaces is a fundamental problem in geometric and solid modeling [SP86]. Current approaches are based on algebraic methods, subdivision methods and interval arithmetic. Algebraic methods are restricted to low degree intersections. For high degree curve intersections, algorithms based on interval arithmetic have been found to be the fastest [SP86] Such algorithms ....

....and solid modeling [SP86] Current approaches are based on algebraic methods, subdivision methods and interval arithmetic. Algebraic methods are restricted to low degree intersections. For high degree curve intersections, algorithms based on interval arithmetic have been found to be the fastest [SP86]. Such algorithms compute a decomposition of the curve in terms of AABB s. We plan to apply our algorithm based on OBB s to such problems. It involves subdividing the curve, computing tight fitting OBB sfor each segment, and checking them for overlaps. Deformable Models: Efficient interference ....

T.W. Sederberg and S.R. Parry. Comparison of three curve intersection algorithms. Computer-Aided Design, 18(1):58--63, 1986.

....of degree d [21, 4] Let kvk = max (jx v j ; jy v j) where v is a vector in the Euclidean plane. For d 2, the diagonal D and the length L of the control polygon of V (t) are defined as: D = max 0rd Gamma2 kV r 2 Gamma 2V r 1 V r k; L = max 0rd Gamma1 kV r 1 Gamma V r k: Wang [26, 23] gives the following result. If the de Casteljau subdivision algorithm (midpoint case) is applied down to depth k to a polynomial Bezier curve of degree d 2 with control points V r , with: k = log 4 d(d Gamma 1) 8 D ffl ; 1) then all the chords (straight line segments) joining the ....

.... interval I i and using the bound (2) it is straightforward to see that any oe 1 such that: 0 oe 2 8ffl d(d Gamma 1)D (6) will satisfy (5) Let k be the smallest integer such that oe = 2 Gammak satisfies (6) then: k = log 4 d(d Gamma 1) 8 D ffl : Equation (1) is cited in [23] as a result derived by Wang. We can now bound the length of the chords given by a priori subdivision of depth k. The mean value theorem applied to V (t) on interval I i gives: kE i 1 Gamma E i k oe max t I i kV (1) t)k: The maximum of kV (1) t)k over I i is less than or equal to its ....

[Article contains additional citation context not shown here]

Sederberg, T.W. and Parry, S.R. Comparison of three curve intersection algorithms. Computer-Aided Design, Vol. 18 (1986).

....3 Curve Surface Intersection The problems of computing the intersection of curves and surfaces are fundamental in computer graphics and geometric modeling. Common applications include surfacesurface intersection, ray tracing, hidden curve removal and visibility algorithms [Hof89, EC90, NSK90, SP86] Our surface surface intersection algorithm (chapter 4) needs starting points on each component of the intersection curve. We use curve surface intersection to evaluate these starting points. Our algorithm for boundary evaluation relies on a rayshooting approach for the classification of certain ....

....with a parametric surface and curve surface intersections. In each case, the problem is reduced to an eigenvalue problem and we compute the eigenvalues in a domain. We have performed comparisons with the QR algorithm in [MD94] and an implementation of implicitization based algorithm described in [SP86] and B ezier clipping described in [SN90] B ezier Clipping: B ezier Clipping is an iterative method which takes advantage of the convex hull property of B ezier curves, and iteratively clips away regions of the curve that does not intersect with the surface. B ezier clipping converges more ....

[Article contains additional citation context not shown here]

T.W. Sederberg and S.R. Parry. Comparison of three curve intersection algorithms. Computer-Aided Design, 18(1):58--63, 1986.

....r 2 = kC(u; w) Gamma R(u; w)k 2 Gamma r 2 = Q(u; w) 8R(w) R; 6) Surface Region Optimization G. Elber 10 and eliminating the regions in C(u) that are too close to any R(w) R. The solution of equation (6) can be manifested as a variation on a freeform curve curve intersection problem [12, 13, 14]. Herein, the zero set of the bivariate function Q(u; w) is computed, and the u domain boundaries of the zero set are used to specify the locations that C(u) must be subdivided at. R, the parabolic set of surface S is computed as the zero set of the determinant of the second fundamental form ....

T. W. Sederberg and S. Parry. Comparison of Three Curve Intersection Algorithms. Computer Aided Design, Vol. 18, No. 1, pp 58-63, January/February 1986.

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Sederberg, T. W., and Parry, S. R. Comparison of three curve intersection algorithms. Computer Aided Design 18, 1 (January/February 1986), 58--63.

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Sederberg, T. W., Parry, S. R. (1986) Comparison of three curve intersection algorithms, Computer{aided design 18, 58-63.

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T. W. Sederberg and S. Parry. Comparison of Three CurveIntersection Algorithms. Computer Aided Design, vol. 18, no. 1, pp 58-63, January/February 1986.

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T. Sederberg and S. Parry. Comparison of Three Curve Intersection Algorithms. Computer Aided Design, Volume 18, Number 1, January/February 1986.

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T.W. Sederberg - S.R. Parry, Comparison on three curve intersection algorithm, Computer-Aided Design, 18 (1986), pp. 58-63.

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T.W. Sederberg and S.R. Parry. Comparison of three curve intersection algorithms. Computer-Aided Design, 18(1):58--63, 1986.

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T. W. Sederberg and S. R. Parry. Comparison of Three Curve Intersection Algorithms. Computer Aided Design, 18(1), 1986.

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T.W. Sederberg and S.R. Parry. Comparison of three curve intersection algorithms. Computer-Aided Design, 18(1):58--63, 1986.

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Sederberg, T.W., and Scott, R.P., (1986), \Comparison of three curve intersection algorithms," Computer Aided Design, Vol. 18, No. 1, pp. 58-63.

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Thomas W. Sederberg and Scott R. Parry. Comparison of three curve intersection algorithms. Computer-Aided Design, 18(1):58-- 63, 1986.

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