C. L. Bajaj, C. M. Hoffmann, J. E. Hopcroft, R. E. Lynch. "Tracing surface intersections". Computer Aided Geometric Design, vol. 5, pp 285-307, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for algebraic and parametric surfaces. Depending on the algorithms involved in the different underlying tasks, intersection computation methods may be classified into four main categories [Boe91, Pat93, Abd96] Analytical methods [Cha87, Pat93] lattice evaluation methods [Bar87] marching methods [Baj88] and subdivision methods. In the present work, we are particularly interested in computing intersection curves of subdivision surfaces in the context of solid algebra where objects are all modeled by subdivision surfaces. Although for practical reasons our intersection computing algorithm has ....

C. L. Bajaj, C. M. Hoffmann, J. E. Hopcroft, R. E. Lynch. "Tracing surface intersections". Computer Aided Geometric Design, vol. 5, pp 285-307, 1988.

....more seriously, implicitly defined curves, such as x 2 y 2 = 1 and sin(x cos y) cos(y sin x) need to be converted to the functional form y = f(x) Sometimes this conversion is straightforward, sometimes not. Without conversion to functional form, one must resort to local methods such as [3, 13]. This is not always successful. The problem of implicitly defined relations is also addressed in the contouring of functions [12] The method of interval constraint plotting has the property that the plot shows a hull of the function or relation. This property holds independently of rounding ....

Bajaj, C., C. Hoffman, J. Hopcroft and R. Lynch, "Tracing Surface Intersections," Computer Aided Geometric Design, 5, 1988, pp. 285--307.

....Farouki et al. 2] for an exact rational parametrization of the singular quartic space curve. In the case of regular intersection curves, a starting point must be generated on each closed loop. After that, each curve component is traced using the routine: Trace Regular TSI Curve (see Bajaj et al. [1]) In Line (3) we detect the case of empty intersection. Regular intersection curves are detected and computed in Lines (4) 7) according to the conditions enumerated in Section 2.2. 4 Conclusions In this paper, we presented a simple algorithm for the construction of torus sphere intersection ....

C. Bajaj, C. Hoffmann, J. Hopcroft, and R. Lynch, Tracing Surface Intersections, in Computer Aided Geometric Design, Vol. 5, No. 4, pp. 285--307, 1988. 8 A simple algorithm for torus/sphere intersection

....which can be implemented efficiently and robustly. All degenerate conic sections (circles) can also be detected using a few additional simple geometric tests. The intersection curve itself (a quartic space curve, in general) is then approximated with a sequence of cubic curve segments [1, 3, 4]. Key Words: Surface surface intersection, torus, sphere, Configuration space. This research was supported in part by the POSTECH Information Research Laboratories under Grant 96F502, by the Korean Ministry of Science and Technology under Grant 95 S 05 A 02 A and 96 NS 01 05 A 02 A of STEP 2000, ....

....curve are detected using a few additional simple geometric tests. Singular intersections are detected based on testing tangency in certain circle circle intersections. The TSI curve itself (a quartic space curve, in general) is then numerically approximated with a sequence of cubic curve segments [1, 3, 4]. Our algorithm is based on a geometric transformation that reduces the TSI problem to a simpler problem of: i) classifying the relative position of a point with respect to the regions bounded by two tori, or (ii) intersecting a circle with two concentric spheres. The geometric transformation ....

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Bajaj, C., Hoffmann, C., Hopcroft, J., and Lynch, R., "Tracing Surface Intersections," Computer Aided Geometric Design, Vol. 5, pp. 285--307, 1988.

....in R 7 . For example, when two implicit surfaces are given in R 3 , their intersection curve can be locally approximated by a cubic parametric curve. The curvature and torsion of the intersection curve can be evaluated based on the two surface gradient vectors and their Hessian matrices [2]. Subsequently, in the local neighborhood, the nonlinear optimization problem in R 7 with m constraints can be reduced to a nonlinear optimization problem in R 7 Gammam with no constraint. When we restrict the problem to the computation of x, we have to search the vector x in the orthogonal ....

Bajaj, C., Hoffmann, C., Lynch, R., and Hopcroft, J., "Tracing Surface Intersections," Computer-Aided Geometric Design, Vol. 5, 1988, pp. 285--307.

....Curve To generate the GAP points for each monotone curve segment, we compute each GAP point while tracing along the curve. Curve tracing is trivial for parametric curves. For implicit algebraic curves, we can use the robust numerical curve tracing algorithm of Bajaj, Hoffmann, Hopcroft, and Lynch [BHHL88]. Further, we can also convert the implicit curve representations into parametric forms by using the parametrization algorithm of Abhyankar and Bajaj [AB88] To make the presentation clear, we assume the given monotone curve segment has a parametric representation. The general method of generating ....

Bajaj, C., Hoffmann, C.M., Hopcroft, J.E., and Lynch, R.E., (1988), "Tracing Surface Intersections," Computer Aided Geometric Design, Vol. 5, pp. 285--307.

....which is a non uniform rational B spline (NURBS) piecewise interpolation and homeomorphic to the real algebraic set of the intersection (in fact, a space curve) in Euclidean space. Due to its importance, there have been persistent efforts in devising algorithms for this problem (see [BFJP87] BHLH88] ACM88] Hof89] KPP90] W 91] KP91] Pat92, Pat93] Tau94] and [Tra95a] The main issue in the intersection problem is the efficient discovery and description of all features of the solution with high precision commensurate with the tasks required by the underlying geometric ....

C. L. Bajaj, C. M. Hoffmann, R. E. Lynch, and J. E. H. Hopcroft. Tracing Surface Intersections. Computer Aided Geom. Design, 5:285-- 307, 1988.

....degree algebraic surfaces [13, 12] Using analytic methods to compute the intersection of two parametric surfaces requires implicitization and inversion techniques. The complexity of the representation causes analytic methods to be impractical for high degree parametric surfaces. Marching methods [3, 1] begin by locating at least one point onto every intersection curve and use it as a starting point to step along the required curve in a direction prescribed by the curve s local geometry. The robustness of marching methods depends on two points: the detection of all intersection lines, the ....

C.L. Bajaj, C.M. Hoffmann, R.E. Lynch, and J.E.H. Hopcfart. Tracing surface intersections. C.A.G.D., 12:285--307, 1988.

....within constant factors; any of them can be the extraneous factors of the Sylvester resultant. We can approximate the convolution curve segment C numerically (with arbitrary precisions) within O(T C (d 2 ) time by using a robust space curve tracing algorithm along two surface intersections [2, 11], where T C (d 2 ) is the time required to trace the intersection curve (in the xyt space) whose projection onto the xy plane determines the convolution curve segment C of degree O(d 2 ) Even if one has constructed the defining implicit equation F = 0 of a convolution curve C, the ....

.... F x F y F yy Delta F 2 x = 0 (see [4] Thus the problem is essentially an intersection problem of two implicit curve segments, which can be solved by using the plane algebraic curve intersection method of Johnstone and Goodrich [12] which in turn depends on the numerical curve tracing [2]. The basic strategy of [12] is as follows: while tracing along the curve F = 0, one can check the sign change of F xx Delta F 2 y Gamma 2 F xy Delta F x F y F yy Delta F 2 x and detect the solution points where the simultaneous equation F = F xx Delta F 2 y ....

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Bajaj, C., Hoffmann, C., Hopcroft, J., and Lynch, R., (1988), "Tracing Surface Intersections," Computer Aided Geometric Design, Vol. 5, pp. 285--307.

....there is no known exact method for high degree algebraic curves in general. Thus, we will rely on our previous approach to monotone segmentations using the resultants and root isolations [4] To be more precise, we state the following Lemma, where T (d) is the time required to trace along C, [1]. Lemma 2.2 (I) For a parametric curve segment C of degree d, a monotone segmentation can be obtained in O(d 3 log d) time. II) For an implicit algebraic curve segment C of degree d, a monotone segmentation can be obtained in O(d 6 log d T (d) time. 3 Gaussian Approximation In this ....

Bajaj, C., Hoffmann, C.M., Hopcroft, J.E., and Lynch, R.E., (1988), "Tracing Surface Intersections, " Computer Aided Geometric Design, Vol. 5, pp. 285--307.

....they promise precision and efficiency for low degree implicit surfaces, analytic methods are slow for high degree surfaces and impracticable for parametric ones. Moreover computer accuracy impeach the robustness of roots finding in the limit cases even for degrees as low as 2. Numerical methods [BHLH88] are appropriate for parametric surfaces, and implicit surfaces are easy to parameterize. Moreover limit cases present themselves in a more intuitive way which helps to take them into account. This explains why numerical methods have received much attention in the literature as well as in this ....

C.L. Bajaj, C.M. Hoffmann, R.E. Lynch, and J.E.H. Hopcfart. Tracing surface intersections. C.A.G.D., 12:285--307, 1988.

....on parametrically described shapes. They can represent, for example, the intersection of two parametric surfaces in R 3 , or the silhouette edges of a parametric surface in R 3 with respect to a given view. The robustness of the algorithm presented here is superior to local methods such as [TIMM77,BAJA88]. Timmer s method, for example, separates implicit curve approximation into a hunting phase, where intersections of the implicit curve with a preselected grid are computed, and a tracing phase, where the curve inside each grid cell is traced to determine how to connect the intersections. The new ....

Bajaj, C., C. Hoffman, J. Hopcroft, and R. Lynch, "Tracing Surface Intersections, " Computer Aided Geometric Design, 5, 1988, pp. 285-307.

.... with hyperplanes, that we present, deals directly with the zero sets of polynomial equations (rather than just the combinatorial structure of the polynomials) Such rational maps provide a compact data structure for algebraic varieties and also yield simpler algorithms for computing intersections[10], shading, displaying and texture mapping[7] and in general solving systems of algebraic equations[11] It is based, though not entirely, on lesser known constructs of algebraic geometry, namely the multi polynomial resultant[34] and multi polynomial remainder sequences (a generalization of ....

....surface is a standard example of a rational map. Inverting a parametrization of a surface has applications in areas such as sorting points along a parametric curve[27] Birational maps have been used in resolving the singular (nonsmooth) points of algebraic curves and surfaces[1] In particular, [10] uses this idea in the robust tracing of algebraic plane curves. Moreover, 4] use birational maps in determining whether an algebraic space curve has a rational parameterization. From a mathematical point of view, current attempts to classify surfaces and higher dimensional geometric objects ....

Bajaj, C., Hoffmann, C., Hopcroft, J., and Lynch, R., (1988) "Tracing Surface Intersections", Computer Aided Geometric Design, 5, 285 - 307.

....to our rational approximation problem yields a rational B spline approximate representation with all the above advantages for arbitrary genus algebraic plane curves. For genus zero curves our method does not provide the exact parameterization[2] however a good approximation of it. Prior Work In [8, 20], power series are constructed to locally approximate plane algebraic curves and surface intersections at simple points. The method of [20] technically relies on the Implicit Function Theorem, seeking to represent a curve branch explicitly in one coordinate as function of the other coordinate(s) ....

....series are constructed to locally approximate plane algebraic curves and surface intersections at simple points. The method of [20] technically relies on the Implicit Function Theorem, seeking to represent a curve branch explicitly in one coordinate as function of the other coordinate(s) while [8] uses a Taylor series expansion. Both these methods however 2 SKETCH OF ALGORITHM 3 do not seem to have a natural extension that handles singular points. Papers [8] and [18] survey a number of techniques for generating a piecewise linear approximation of an algebraic curve. Further, 27] present ....

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Bajaj, C., Hoffmann, C., Hopcroft, J. and Lynch, R., (1988), "Tracing Surface Intersections ", Computer Aided Geometric Design, 5, 285 - 307.

....we construct a special triangulation of the domain that respects this partition. In this triangulation, a domain triangle contains pole points only on its boundary and not in its interior. We construct a piecewise linear topologically correct approximation of the plane algebraic pole curve[6, 10] and then construct a triangulation which conforms to this linear approximation i.e. the triangles abut the linear curve segments at vertices or edges. The conforming triangulation may require additional points to be inserted in the piecewise linear approximation of the curve. Bounds on the ....

....the rational map M . Our solution to the problem is a simplification of of the method of [13] and stated in the Theorem below which allows us to give an explicit formula for the parametric equations of the seam curve and a blowing up of the base point based on affine quadratic transformations [1, 6] for the special case when the tangents at the base point are equal. The affine quadratic transformation we use are of the type T : x = r; y = rs with inverse T Gamma1 : r = x; s = y=x with the base point translated to the origin. This transformation is applied to the product of the curves X = ....

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Bajaj, C., Hoffmann, C., Hopcroft, J., and Lynch, R. (1988), "Tracing Surface Intersections," Computer Aided Geometric Design, 5, 285-307.

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Bajaj, C.L., Hoffman, C.M., Lynch, R.E., and Hopcroft, J.E.H., 1988, "Tracing Surface Intersections," Computer Aided Geometric Design, Vol. 5, pp. 285-307.

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Bajaj, C., Hoffmann, C., Lynch, R., Hopcroft, J., "Tracing surface intersections," in Computer Aided Geometric Design, Vol 5, 1988, pp. 285-307.

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Bajaj, C., Hoffmann, C., Lynch, R., Hopcroft, J., "Tracing surface intersections," in Computer Aided Geometric Design, Vol 5, 1988, pp. 285-307.

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