Clipping simple polygons with degenerate intersections

In general, two quadric surface intersect in a space quartic curve. However, the intersection frequently degenerates to a collection of plane curves. Degenerate cases are frequent in geometric/solid modeling because degeneracies are often required by design. Their detection is important because degenerate intersections can be computed more easily and allow simpler treatment of important problems. In this paper, we investigate this problem for natural quadrics. Algorithms are presented to detect and compute conic intersections and linear intersections. These methods reveal the relationship between the planes of the degenerate intersections and the quadrics. Using the theory developed in the paper, we present a new and simplified proof of a necessary and sufficient condition for conic intersection. Finally, we present a simple method for determining the types of conic in a degenerate intersection without actually computing the intersection, and an enumeration of all possible conic types. Since only elementary geometric routines such as line intersection are used, all of the above algorithms are intuitive and easily implementable.

A novel f-f model is developed constructively which can express any n-dimensional piecewise linear (PWL) function by a superposition of basis functions if it is defined over an nth-order degenerate intersection formed by (n-1)th-order minimal degenerate intersections. We also propose the concrete functional forms of nth-order basis functions. Being the simplest type of the minimal degenerate intersection, the basis function is the most elementary "building block" of a PWL function defined in an arbitrary-dimensional space. In addition, the model constitutes a natural continuation to Julian's canonical formulation and can bridge the lattice PWL model and the well-established canonical representation.

The single source Weber problem with limited distances (SSWPLD) is a continuous optimization problem in location theory. The SSWPLD algorithms proposed so far are based on the enumeration of all regions of [Formula: see text] defined by a given set of n intersecting circumferences. Early algorithms require [Formula: see text] time for the enumeration, but they were recently shown to be incorrect in case of degenerate intersections, that is, when three or more circumferences pass through the same intersection point. This problem was fixed by a modified enumeration algorithm with complexity [Formula: see text], based on the construction of neighborhoods of degenerate intersection points. In this paper, it is shown that the complexity for correctly dealing with degenerate intersections can be reduced to [Formula: see text] so that existing enumeration algorithms can be fixed without increasing their [Formula: see text] time complexity, which is due to some preliminary computations unrelated to intersection degeneracy. Furthermore, a new algorithm for enumerating all regions to solve the SSWPLD is described: its worst-case time complexity is [Formula: see text]. The new algorithm also guarantees that the regions are enumerated only once.

A unified method for invalid 2D loop removal in tool-path generation

The computation of exact geodesics on triangle meshes is a widely used operation in computer-aided design and computer graphics. Practical algorithms for computing such exact geodesics have been recently proposed by Surazhsky et al. [5]. By applying these geometric algorithms to real-world data, degenerate cases frequently appear. In this paper we classify and enumerate all the degenerate cases in a systematic way. Based on the classification, we present solutions to handle all the degenerate cases consistently and correctly. The common users may find the present techniques useful when they implement a robust code of computing exact geodesic paths on meshes.

<p indent="0mm">Polygon decomposition has been widely employed in the fields of computer graphics, CAD software, and path planning. The presence of self-intersections in polygons introduces errors and inaccuracies in subsequent computations and drawing operations. Decomposing self-intersecting polygons poses a common challenge in CAD applications. Traditional algorithms rely on triangulation techniques. However, this approach yields a substantial number of triangles, significantly elevating both computational and storage complexities. In response to the issue of self-intersecting polygon decomposition, a decomposition algorithm based on region partitioning is proposed. The approach begins by identifying all intersection points within the polygon. Then, a pathfinding technique is employed to traverse the self-intersecting polygon, partitioning it into non-overlapping and self-intersection-free regions. Finally, by discerning the inclusion of each region, internal areas are preserved, while external ones are discarded. This process results in a decomposition of the self-intersecting polygon into non-overlapping simple polygons. Comparative experiments with the GluTess method on large integrated circuit boards reveal that our algorithm improves time efficiency by around 60% and reduces spatial occupancy by about 20%.

We describe a simple linear algorithm for motion and structure from three weak perspective projections using Euler angles. We first determine the epipolar equation between each pair of images, which determines the first and third Euler angles for the rotation between that pair of images, leaving only the second Euler angle undetermined. In the next step, combining the three rotations results in a very simple linear algorithm to determine the second Euler angles, up to a Necker reversal. Experimental results on synthetic and real images are presented. The degenerate cases are discussed.

The response characteristics of an electronic neuron model proposed by the authors are investigated. Periodic stimulating pulse sequences with a fixed frequency are applied to the analog neuron model and the response pulse sequences are studied. In the degenerate case, the state transition of the neuron model during one period of the stimulating pulse sequense is described by a first order piecewise linear difference equation with a jump. It is shown that the periodic response pulse sequences of the neuron model belong to a special class of pulse sequences generated by a simple algorithm, and that the relation between the pulse width (or amplitude) of the stimulating pulse and the firing rate of the neuron model takes the form of an extended Cantor function.

A flipturn transforms a nonconvex simple polygon into another simple polygon by rotating a concavity 180° around the midpoint of its bounding convex hull edge. Joss and Shannon proved in 1973 that a sequence of flipturns eventually transforms any simple polygon into a convex polygon. This paper describes several new results about such flipturn sequences. We show that any orthogonal polygon is convexified after at most n-5 arbitrary flipturns, or at most $\lfloor 5(n-4)/6 \rfloor$ well-chosen flipturns, improving the previously best upper bound of (n-1)!/2 . We also show that any simple polygon can be convexified by at most n 2 -4n+1 flipturns, generalizing earlier results of Ahn et al. These bounds depend critically on how degenerate cases are handled; we carefully explore several possibilities. We prove that computing the longest flipturn sequence for a simple polygon is NP-hard. Finally, we show that although flipturn sequences for the same polygon can have significantly different lengths, the shape and position of the final convex polygon is the same for all sequences and can be computed in O(n log n) time.

In the edge contraction algorithm for surface simplification, two major steps are (i) ordering of the edges to be collapsed, and (ii) positioning of the new vertex to replace the collapsed edge. The accuracy of the simplified result greatly depends on which procedures are taken in the above steps. During the simplification process, degenerate cases can occur in which the optimization problem for the new vertex position produces multiple solutions. In the previous algorithms, such redundancy was utilized to achieve other desirable effects (e.g, increasing the regularness of the triangles). In this paper we give a complete characterization of the geometric situation for the degenerate cases, and show that performing edge contraction procedure in those cases can produce unexpected damage to the original shape. To cope with the problem, we propose a new procedure, called edge elimination, which eliminates the degenerate edge and all the adjacent ones, and then retriangulates the hole in a way the geometrical shape remains unchanged. The new algorithm proceeds like the edge contraction algorithm. Whenever it has to process a degenerate edge, however, the edge elimination procedure is performed instead of edge contraction.

We present a new approach to the problem of estimating multiple signal and parameter unknowns given noisy and incomplete data. Using cross-entropy, we fit a separable density to the given model density, then use this separable density to estimate each unknown independently. Not only does this method include all the various MAP methods as degenerate cases, but it also directly leads to a simple iterative algorithm which can solve either the cross-entropy method or any of the MAP methods. This algorithm is particularly effective for exponential families of densities. Applications include estimation using grouped or quantized data, and a wide variety of reconstruction, smoothing, interpolation, extrapolation and modeling problems involving linear Gaussian systems.

Finding the largest area axis-parallel rectangle in a polygon

Polygons appear as constructors in many applications and deciding if two polygons match under similarity transformations and noise is a fundamental problem. Solutions in the literature consider only matching pairs of polygons, implying a sequential comparison when we have a collection. In this paper, we present the first algorithm allowing indexed retrieval of polygons under similarities. We reduce the problem to searching points in the plane, exact searching in the absence of noise, and approximate searching for similar noisy polygons. The above gives a \(O(n+\log (m))\) time algorithm to find the matching polygons under noise and O(1) time for exact similar polygons. We tested our heuristic for indexed polygons in an extensive collection of convex, star-shaped, simple, and self-intersecting polygons. For small amounts of noise, we achieve perfect recall for all polygons. For large amounts of noise, the lowest recall is for convex polygons, while attaining the highest recall is for general (self-intersecting) polygons. The above is not a significant limitation. To recover convex polygons efficiently before indexing, we define a random permutation of the vertices, converting all input polygons to a general polygon and achieving the same successful recovery rates, which is a perfect recall for high noise levels.

\n \nThis paper deals with the numerical approximation of mild solutions of elliptic-parabolic equations, relying on the existence results of Bénilan and Wittbold (1996). \nWe introduce a new and simple algorithm based on Halpern's iteration for nonexpansive operators \n(Bauschke, 1996; Halpern, 1967; Lions, 1977), which is shown to be convergent in the degenerate case, and compare it with existing schemes (Jäger and Kačur, 1995; Kačur, 1999).\n\n

In this paper, we present a simple algorithm for the explicit construction of [Formula: see text]-particle harmonic oscillator states simultaneously belonging to irreducible representations of [Formula: see text] (or [Formula: see text]) and [Formula: see text]. For degenerate representations, the construction can be done using generating functions or hyperspherical harmonics. The cases with [Formula: see text] and [Formula: see text] are investigated at greater length. The analysis is extended to nondegenerate states, although the states are not so easily obtained as in the degenerate case. Finally, we briefly touch on the [Formula: see text] structure of Bell states.