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In this paper, we introduce the notion of rational Hausdorff divisor, we analyze the dimension and irreducibility of its associated linear system of curves, and we prove that all irreducible real curves belonging to the linear sys...
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In this paper, we introduce the notion of rational Hausdorff divisor, we analyze the dimension and irreducibility of its associated linear system of curves, and we prove that all irreducible real curves belonging to the linear system are rational and are at finite Hausdorff distance among them. As a consequence, we provide a projective linear subspace where all (irreducible) elements are solutions of the approximate parametrization problem for a given algebraic plane curve. Furthermore, we identify the linear system with a plane curve that is shown to be rational and we develop algorithms to parametrize it analyzing its fields of parametrization. Therefore, we present a generic answer to the approximate parametrization problem. In addition, we introduce the notion of Hausdorff curve, and we prove that every irreducible Hausdorff curve can always be parametrized with a generic rational parametrization having coefficients depending on as many parameters as the degree of the input curve.
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In this paper, we present an exact error analysis for circle approximation by Bezier curve. The approximation method is a special case of Floater's conic approximation method (Comput. Aided Geom. Design 8 (1991)135). Using this, w...
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In this paper, we present an exact error analysis for circle approximation by Bezier curve. The approximation method is a special case of Floater's conic approximation method (Comput. Aided Geom. Design 8 (1991)135). Using this, we propose an approximation method of the offset curve of given plane Bézier curve by Bezier curve of the same degree.
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We present an algorithm that, given a non-rational irreducible real space curve, satisfying certain conditions, computes a rational parametrization of a space curve near the input one. For a given tolerance ∈ > 0, the algorithm c...
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We present an algorithm that, given a non-rational irreducible real space curve, satisfying certain conditions, computes a rational parametrization of a space curve near the input one. For a given tolerance ∈ > 0, the algorithm checks whether a planar projection of the given space curve is ∈-rational and, in the affirmative case, generates a planar parametrization that is lifted to a space parametrization. This output rational space curve is of the same degree as the input curve, both have the same structure at infinity, and the Hausdorff distance between their real parts is finite. Moreover, in the examples we check that the distance is small.
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A construction of "nonsymmetric" plane Peano curves is described whose coordinate functions satisfy the Lipschitz conditions of orders a and 1 - alpha for some alpha. It is proved that these curves are metric isomorphisms between ...
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A construction of "nonsymmetric" plane Peano curves is described whose coordinate functions satisfy the Lipschitz conditions of orders a and 1 - alpha for some alpha. It is proved that these curves are metric isomorphisms between the interval [0, 1] and the square [0, 1](2). This fact is used to show that the graphs of their coordinate functions have the maximum possible Hausdorff dimension for a given smoothness.
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It is well known that on any Veech surface, the dynamics in any minimal direction is uniquely ergodic. In this paper it is shown that for my genus-2 translation surface which is not a Veech surface there are uncountably many minim...
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It is well known that on any Veech surface, the dynamics in any minimal direction is uniquely ergodic. In this paper it is shown that for my genus-2 translation surface which is not a Veech surface there are uncountably many minimal but not uniquely ergodic directions. The theorem can be applied to certain billiard tables.
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During the final assembly process, finding a best fitting resolution for gap and flush between assembled parts is the main object. This paper divides these fitting problems into two optimization stages, which are attitude fitting ...
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During the final assembly process, finding a best fitting resolution for gap and flush between assembled parts is the main object. This paper divides these fitting problems into two optimization stages, which are attitude fitting and position fitting, and proposes a mixed model which combines the Gaussian Imaging method (Enriched Complex Extended Gaussian Imaging) and even fitting algorithm (Hausdorff distance based criteria). It is illustrated that Gaussian Imaging method can be used to orient two similar curve objects and therefore accelerate the optimization process by the even fitting method. We also demonstrate the model by fitting a 2D classic case, an automobile door fitting into a frame, and generalize the application of the model into 3D curve fitting by fitting a simple automobile lamp into an auto-body. The best rotation and translation is found by using two curve objects' similarity in attitude and minimum width in gap and flush as criteria.
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We consider the convolution of two compatible conic segments. First, we find an exact parametric expression for the convolution curve, which is not rational in general, and then we find the conic approximation to the convolution c...
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We consider the convolution of two compatible conic segments. First, we find an exact parametric expression for the convolution curve, which is not rational in general, and then we find the conic approximation to the convolution curve with the minimum error. The error is expressed as a Hausdorff distance which measures the square of the maximal collinear normal distance between the approximation and the exact convolution curve. For this purpose, we identify the necessary and sufficient conditions for the conic approximation to have the minimum Haudorff distance from the convolution curve. Then we use an iterative process to generate a sequence of weights for the rational quadratic Bezier curves which we use to represent conic approximations. This sequence converges to the weight of the rational quadratic Bezier curve with the minimum Hausdorff distance, within a given tolerance. We verify our method with several examples.
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The Hausdorff distance is a very natural and straightforward distance measure for comparing geometric shapes like curves or other compact sets. Unfortunately, it is not an appropriate distance measure in some cases. For this reaso...
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The Hausdorff distance is a very natural and straightforward distance measure for comparing geometric shapes like curves or other compact sets. Unfortunately, it is not an appropriate distance measure in some cases. For this reason, the Frechet distance has been investigated for measuring the resemblance of geometric shapes which avoids the drawbacks of the Hausdorff distance. Unfortunately, it is much harder to compute. Here we investigate under which conditions the two distance measures approximately coincide, i.e., the pathological cases for the Hausdorff distance cannot occur. We show that for closed convex curves both distance measures are the same. Furthermore, they are within a constant factor of each other for so-called κ-straight curves, i.e., curves where the arc length between any two points on the curve is at most a constant κ times their Euclidean distance. Therefore, algorithms for computing the Hausdorff distance can be used in these cases to get exact or approximate computations of the Frechet distance, as well.
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In Pérez-Díaz et al. (2009) [5], the authors present an algorithm to parametrize approximately e -rational curves, and they show that the Hausdorff distance, w.r.t. the Euclidean distance, between the input and output curves is ...
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In Pérez-Díaz et al. (2009) [5], the authors present an algorithm to parametrize approximately e -rational curves, and they show that the Hausdorff distance, w.r.t. the Euclidean distance, between the input and output curves is finite. In this paper, we analyze this distance for a family of curves randomly generated and we empirically find a reasonable upper bound of the Hausdorff distance between each input and output curve of the family.
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