UR
Ulrich Reitebuch
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6 records found
1
Point clouds and polygonal meshes are widely used when modeling real-world scenarios. Here, point clouds arise, for instance, from acquisition processes applied in various surroundings, such as reverse engineering, rapid prototyping, or cultural preservation. Based on these raw data, polygonal meshes are created to, for example, run various simulations. For such applications, the utilized meshes must be of high quality. This paper presents an algorithm to derive triangle meshes from unstructured point clouds. The occurring edges have a close to uniform length and their lengths are bounded from below. Theoretical results guarantee the output to be manifold, provided suitable input and parameter choices. Further, the paper presents several experiments establishing that the algorithms can compete with widely used competitors in terms of quality of the output and timing and the output is stable under moderate levels of noise. Additionally, we expand the algorithm to detect and respect features on point clouds as well as to remesh polyhedral surfaces, possibly with features. Supplementary material, an extended preprint, a link to a previously published version of the article, utilized models, and implementation details are made available online .
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Point clouds and polygonal meshes are widely used when modeling real-world scenarios. Here, point clouds arise, for instance, from acquisition processes applied in various surroundings, such as reverse engineering, rapid prototyping, or cultural preservation. Based on these raw data, polygonal meshes are created to, for example, run various simulations. For such applications, the utilized meshes must be of high quality. This paper presents an algorithm to derive triangle meshes from unstructured point clouds. The occurring edges have a close to uniform length and their lengths are bounded from below. Theoretical results guarantee the output to be manifold, provided suitable input and parameter choices. Further, the paper presents several experiments establishing that the algorithms can compete with widely used competitors in terms of quality of the output and timing and the output is stable under moderate levels of noise. Additionally, we expand the algorithm to detect and respect features on point clouds as well as to remesh polyhedral surfaces, possibly with features. Supplementary material, an extended preprint, a link to a previously published version of the article, utilized models, and implementation details are made available online .
Various computer simulations regarding, e.g. the weather or structural mechanics, solve complex problems on a two-dimensional domain. They mostly do so by splitting the input domain into a finite set of smaller and simpler elements on which the simulation can be run fast and efficiently. This process of splitting can be automatized by using subdivision schemes. Given the wide range of simulation problems to be tackled, an equally wide range of subdivision schemes is available. This paper illustrates a subdivision scheme that splits the input domain into pentagons. Repeated application gives rise to fractal-like structures. Furthermore, the resulting subdivided domain admits to certain weaving patterns. These patterns are subsequently generalized to several other subdivision schemes. As a final contribution, we provide paper models illustrating the weaving patterns induced by the pentagonal subdivision scheme. Furthermore, we present a jigsaw puzzle illustrating both the subdivision process and the induced weaving pattern. These transform the visual and abstract mathematical algorithms into tactile objects that offer exploration possibilities aside from the visual.
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Various computer simulations regarding, e.g. the weather or structural mechanics, solve complex problems on a two-dimensional domain. They mostly do so by splitting the input domain into a finite set of smaller and simpler elements on which the simulation can be run fast and efficiently. This process of splitting can be automatized by using subdivision schemes. Given the wide range of simulation problems to be tackled, an equally wide range of subdivision schemes is available. This paper illustrates a subdivision scheme that splits the input domain into pentagons. Repeated application gives rise to fractal-like structures. Furthermore, the resulting subdivided domain admits to certain weaving patterns. These patterns are subsequently generalized to several other subdivision schemes. As a final contribution, we provide paper models illustrating the weaving patterns induced by the pentagonal subdivision scheme. Furthermore, we present a jigsaw puzzle illustrating both the subdivision process and the induced weaving pattern. These transform the visual and abstract mathematical algorithms into tactile objects that offer exploration possibilities aside from the visual.
Chip-Firing Revisited
A Peek into the Third Dimension
Chip-firing was first introduced as a probabilistic game. Subsequently, it was generalized to arbitrary graph configurations and investigated mostly with regard to two-dimensional quad-grid layouts. In this paper, we lift chip-firing to the third dimension. Aside from the arising three-dimensional shapes, we are interested in the internal, two-dimensional structures. Furthermore, we explore the different shapes obtained by chip firing processes on various neighborhoods, such as the face-centered and the cube-centered grid as well as on a neighborhood inspired by knight moves.
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Chip-firing was first introduced as a probabilistic game. Subsequently, it was generalized to arbitrary graph configurations and investigated mostly with regard to two-dimensional quad-grid layouts. In this paper, we lift chip-firing to the third dimension. Aside from the arising three-dimensional shapes, we are interested in the internal, two-dimensional structures. Furthermore, we explore the different shapes obtained by chip firing processes on various neighborhoods, such as the face-centered and the cube-centered grid as well as on a neighborhood inspired by knight moves.
The approximation of a golden logarithmic spiral by quarter circles is well known. Starting from this, we show that any logarithmic spiral can be approximated by quarter circles in a similar way. Using our construction on a rectangle with aspect ratio √휙 and performing a coordinate reparametrization, we obtain an aesthetic partition of the plane as our main artwork.
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The approximation of a golden logarithmic spiral by quarter circles is well known. Starting from this, we show that any logarithmic spiral can be approximated by quarter circles in a similar way. Using our construction on a rectangle with aspect ratio √휙 and performing a coordinate reparametrization, we obtain an aesthetic partition of the plane as our main artwork.
In this paper, we are interested in classifying the different arising (topological) structures of three-dimensional Turing-like patterns. By providing examples for the different structures, we confirm a conjecture regarding these structures within the setup of three-dimensional Turing-like pattern. Furthermore, we investigate how these structures are distributed in the parameter space of the discrete model. We found twofold versions of so-called "zero-" and "one-dimensional" structures as well as "two-dimensional" structures and use our experimental findings to formulate several conjectures for three-dimensional Turing-like patterns and higher-dimensional cases.
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In this paper, we are interested in classifying the different arising (topological) structures of three-dimensional Turing-like patterns. By providing examples for the different structures, we confirm a conjecture regarding these structures within the setup of three-dimensional Turing-like pattern. Furthermore, we investigate how these structures are distributed in the parameter space of the discrete model. We found twofold versions of so-called "zero-" and "one-dimensional" structures as well as "two-dimensional" structures and use our experimental findings to formulate several conjectures for three-dimensional Turing-like patterns and higher-dimensional cases.
In this paper, we are interested in classifying the different arising (topological) structures of three-dimensional Turing-like patterns. By providing examples for the different structures, we confirm a conjecture regarding these structures within the setup of three-dimensional Turing-like pattern. Furthermore, we investigate how these structures are distributed in the parameter space of the discrete model. We found twofold versions of so-called "zero-" and "one-dimensional" structures as well as "two-dimensional" structures and use our experimental findings to formulate several conjectures for three-dimensional Turing-like patterns and higher-dimensional cases.
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In this paper, we are interested in classifying the different arising (topological) structures of three-dimensional Turing-like patterns. By providing examples for the different structures, we confirm a conjecture regarding these structures within the setup of three-dimensional Turing-like pattern. Furthermore, we investigate how these structures are distributed in the parameter space of the discrete model. We found twofold versions of so-called "zero-" and "one-dimensional" structures as well as "two-dimensional" structures and use our experimental findings to formulate several conjectures for three-dimensional Turing-like patterns and higher-dimensional cases.