Christoph M. Hoffmann

Purdue University, West Lafayette, Indiana, United States

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Publications (101)41.91 Total impact

  • Young Joon Ahn, Christoph Hoffmann, Paul Rosen
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    ABSTRACT: This paper derives expressions for the arc length and the bending energy of quadratic Bezier curves. The formulas are in terms of the control point coordinates. For fixed start and end points of the Bezier curve, the locus of the middle control point is analyzed for curves of fixed arc length or bending energy. In the case of arc length this locus is convex. For bending energy it is not. Given a line or a circle and fixed end points, the locus of the middle control point is determined for those curves that are tangent to a given line or circle. For line tangency, this locus is a parallel line. In the case of the circle, the locus can be classified into one of six major types. In some of these cases, the locus contains circular arcs. These results are then used to implement fast algorithms that construct quadratic Bezier curves tangent to a given line or circle, with given end points, that minimize bending energy or arc length.
    Journal of Computational and Applied Mathematics 01/2014; 255:887-897. · 0.99 Impact Factor
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    Ching-Shoei Chiang, Christoph Hoffmann, Sagar Mittal
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    ABSTRACT: We investigate experimentally the behavior of boids, simple agents following three simple rules of conduct. In crowds these agents can exhibit a collective pattern of movement that imitates flocking animal behavior as well as other types of crowds.
    Computer-Aided Design and Applications 08/2013; 6(6):865-875.
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    Ching-Shoei Chiang, Christoph M. Hoffmann, Paul Rosen
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    ABSTRACT: Malfatti's problem, first published in 1803, is commonly understood to ask fitting three circles into a given triangle such that they are tangent to each other, externally, and such that each circle is tangent to a pair of the triangle's sides. There are many solutions based on geometric constructions, as well as generalizations in which the triangle sides are assumed to be circle arcs. A generalization that asks to fit six circles into the triangle, tangent to each other and to the triangle sides, has been considered a good example of a problem that requires sophisticated numerical iteration to solve by computer. We analyze this problem and show how to solve it quickly.
    Computational Geometry 10/2012; 45(8). · 0.55 Impact Factor
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    Young Joon Ahn, Christoph M. Hoffmann
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    ABSTRACT: We consider the design of parametric curves from geometric constraints such as distance from lines or points and tangency to lines or circles. We solve the Hermite problem with such additional geometric constraints. We use a family of curves with linearly varying normals, LN curves. The nonlinear equations that arise can be of algebraic degree 60. We solve them using the GPU on commodity graphics cards and achieve interactive performance. The family of curves considered has the additional property that the convolution of two curves in the family is again a curve in the family, assuming common Gauss maps, making the class more useful to applications. Further, we consider valid ranges in which the line tangency constraint can be imposed without the curve segment becoming singular. Finally, we remark on the larger class of LN curves and how it relates to Bézier curves.
    Computer Aided Geometric Design 01/2012; 29:30-40. · 0.81 Impact Factor
  • Xiao-Shan Gao, Christoph M. Hoffmann, Robert Joan-Arinyo
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    ABSTRACT: Current trends in free form editing motivate the development of a novel editing paradigm for CAD models beyond traditional CAD editing of mechanical parts. To this end, we need robust and efficient 3D mesh deformation techniques such as 3D structural ...
    Computer Aided Geometric Design. 01/2012; 29:1.
  • Young Joon Ahn, Jian Cui, Christoph Hoffmann
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    ABSTRACT: We present an approximation method for geodesic circles on a spheroid. Our ap­proximation curve is the intersection of two spheroids whose axes are parallel, and it interpolates four points of the geodesic circle. Our approximation method has two merits. One is that the approximation curve can be obtained algebraically, and the other is that the approximation error is very small. For example, our approximation of a circle of radius 1000 km on the Earth has error 1·13 cm or less. We analyze the error of our approximation using the Hausdorff distance and confirm it by a geodesic distance computation.
    Journal of Navigation 10/2011; 64(04). · 0.62 Impact Factor
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    Young Joon Ahn, Christoph M. Hoffmann
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    ABSTRACT: In this paper we present an approximation method for the convolution of two planar curves using pairs of compatible cubic Bézier curves with linear normals (LN). We characterize the necessary and sufficient conditions for two compatible cubic Bézier LN curves with the same linear normal map to exist. Using this characterization, we obtain the cubic spline approximation of the convolution curve. As illustration, we apply our method to the approximation of a font where the letters are constructed as the Minkowski sum of two planar curves. We also present numerical results using our approximation method for offset curves and compare our method to previous results.
    Computer Aided Geometric Design. 01/2011; 28:357-367.
  • Jong-Hyuk Kim, Jung-Ju Choi, Christoph M. Hoffmann
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    ABSTRACT: We present a technique to parameterize skin deformation by skeletal motion and to transfer the deformation style from one character to another. We decompose skin deformation into time-varying signals and basis matrices by using dimension reduction techniques and then approximate the time-varying signals by using radial basis functions with respect to joint angles that define skeletal motion. This decomposition reduces the size of deformation data to a small number of time-varying signals that represent the complex role of muscle action. The subsequent parameterization yields a fast and intuitive control of characters; thus, it allows us to construct faithful skin deformations quickly as skeletal bones move. The representation of our parameterization allows us to capture and transfer a derived deformation style to another skeleton–skin structure without considering the input dimension of the deformation data. This style transfer can be used as a basis for realistically animating variants of sample characters that have the same skeletal topology. Parameterization of skin deformation and its style transfer can be performed within a small amount of error once the preprocessing time and control of the deformation is carried out in real time by our graphics processing unit implementation. Copyright © 2011 John Wiley & Sons, Ltd.
    Computer Animation and Virtual Worlds 01/2011; 22:511-518. · 0.44 Impact Factor
  • Young Joon Ahn, Christoph M. Hoffmann, Yeon Soo Kim
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    ABSTRACT: We present a method for G2 end-point interpolation of offset curves using rational Bézier curves. The method is based on a G2 end-point interpolation of circular arcs using quadratic Bézier biarcs. We also prove the invariance of the Hausdorff distance between two compatible curves under convolution. Using this result, we obtain the exact Hausdorff distance between an offset curve and its approximation by our method. We present the approximation algorithm and give numerical examples.
    Computer-Aided Design. 01/2011; 43:1011-1017.
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    Young Joon Ahn, Christoph M. Hoffmann
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    ABSTRACT: We consider the design of parametric curves from geometric constraints such as distance from lines or points and tangency to lines or circles. We solve the Hermite problem with such additional geometric constraints. We use a family of curves with linearly varying normals, LN curves, over the parameter interval [0, u]. The nonlinear equations that arise can be of algebraic degree 60. We solve them using the GPU on commodity graphics cards and achieve interactive performance. The family of curves considered has the additional property that the convolution of two curves in the family is again a curve in the family, assuming common Gauss maps, making the class more useful to applications. We also remark on the larger class of LN curves and how it relates to Bézier curves.
    Proceedings of the 2010 ACM Symposium on Applied Computing (SAC), Sierre, Switzerland, March 22-26, 2010; 01/2010
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    ABSTRACT: Analysis of mesh models is gaining prominence as it has wide range of applications from engineering to medicine. In this paper, we present a method for analyzing meshes through the abstraction of its prominent cross-sections (PCS). An algorithm that can robustly yield prominent cross-section at any point on the mesh that approximates the local sweep has been presented. Two types of prominent cross-sections, full (FPCS) and partial (PPCS) are defined and computed over the model. These cross-sections, that cover the object, can then be used in further analysis. Skeleton computation and shape matching using prominent cross-sections have been demonstrated.
    Computer-Aided Design and Applications 01/2010; 7.
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    ABSTRACT: We introduce the general pinhole camera (GPC), defined by a center of projection (i.e., the pinhole), an image plane, and a set of sampling locations in the image plane. We demonstrate the advantages of the GPC in the contexts of remote visualization, focus-plus-context visualization, and extreme antialiasing, which benefit from the GPC sampling flexibility. For remote visualization, we describe a GPC that allows zooming-in at the client without the need for transferring additional data from the server. For focus-plus-context visualization, we describe a GPC with multiple regions of interest with sampling rate continuity to the surrounding areas. For extreme antialiasing, we describe a GPC variant that allows supersampling locally with a very high number of color samples per output pixel (e.g., 1,024{\times}), supersampling levels that are out of reach for conventional approaches that supersample the entire image. The GPC supports many types of data, including surface geometry, volumetric, and image data, as well as many rendering modes, including highly view-dependent effects such as volume rendering. Finally, GPC visualization is efficient-GPC images are rendered and resampled with the help of graphics hardware at interactive rates.
    IEEE Transactions on Visualization and Computer Graphics 01/2010; 16(5):777-90. · 1.90 Impact Factor
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    Ching-Shoei Chiang, Christoph M Hoffmann, Paul Rosen
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    ABSTRACT: In geometric constraint solving, constructing circles with indeterminate radius is an important sub problem. Such constructions are both sequential, meaning that we seek a circle tangent to three known geometric entities, as well as simultaneous, when several sets of entities, among them variable-radius circles, must be determined together. In Part I of our investigation, we considered sequential constructions in which a single circle had to be constructed from required tangencies to known geometric entities. In Part II, we develop hardware-assisted techniques to solve the simultaneous construction problems of variable-radius circles. We utilize the graphics hardware to determine the solutions. For Part I, this could be accomplished relying fundamentally on constructing distance maps. Here, we need to construct and sample surfaces from configuration space.
    Computer-Aided Design & Applications CAD Solutions. 01/2010; 7:32-44.
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    Ching-Shoei Chiang, Christoph M Hoffmann, Paul Rosen
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    ABSTRACT: In geometric constraint solving, constructing circles with indeterminate radius is an important sub problem. Such constructions are both sequential, meaning that we seek a circle tangent to three known geometric entities, as well as simultaneous, when several sets of entities, among them variable-radius circles, must be determined together. In Part I, we investigate techniques to solve sequential construction problems of variable-radius circles, analyzing the case when at least one of the constraining entities is a Bézier curve. We consider first an algebraic solution in which we restrict to Pythagorean hodographs (PH). However, the polynomial degrees become very large, rendering this approach impractical. So, we develop an approach in which the needed computations are assisted by graphics hardware commonly available on PCs and laptops. Here we achieve greater generality, allowing arbitrary curves, greater numerical stability, and extreme speed-ups.
    Computer-Aided Design & Applications CAD Solutions. 01/2010; 7:17-32.
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    Jian Cui, Paul Rosen, Voicu Popescu, Christoph Hoffmann
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    ABSTRACT: Most images used in visualization are computed with the planar pinhole camera. This classic camera model has important advantages such as simplicity, which enables efficient software and hardware implementations, and similarity to the human eye, which yields images familiar to the user. However, the planar pinhole camera has only a single viewpoint, which limits images to parts of the scene to which there is direct line of sight. In this paper we introduce the curved ray camera to address the single viewpoint limitation. Rays are C1-continuous curves that bend to circumvent occluders. Our camera is designed to provide a fast 3-D point projection operation, which enables interactive visualization. The camera supports both 3-D surface and volume datasets. The camera is a powerful tool that enables seamless integration of multiple perspectives for overcoming occlusions in visualization while minimizing distortions.
    IEEE Transactions on Visualization and Computer Graphics 01/2010; 16(6):1235-42. · 1.90 Impact Factor
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    Alexandr Kuzminykh, Christoph Hoffmann
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    ABSTRACT: Product data exchange requires exchanging geometrical shape data that may have to be represented differently in the sending and in the receiving system. Since the translation process thereby entailed can lead to errors, emerging exchange standards include numeric invariants computed from the shape. Agreement of these invariants in the sending and receiving system are then used to increase confidence in the translation process. In this paper we show that there are many classes of noncongruent solids for which all the invariants of volume, surface area, moments and products of inertia agree. Thus these quantities alone are insufficient to exclude many translation errors. We also consider whether the examples are realistic and find that in today’s constraint-based shape constructions the examples are not unlikely to occur.
    Computer-Aided Design. 01/2008;
  • Peter Hertling, Christoph M. Hoffmann, Wolfram Luther, Nathalie Revol
    01/2008;
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    Paul Rosen, Voicu Popescu, Christoph Hoffmann, Ayhan Irfanoglu
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    ABSTRACT: In this application paper, we describe the efforts of a multidisciplinary team towards producing a visualization of the September 11 Attack on the North Tower of New York's World Trade Center. The visualization was designed to meet two requirements. First, the visualization had to depict the impact with high fidelity, by closely following the laws of physics. Second, the visualization had to be eloquent to a nonexpert user. This was achieved by first designing and computing a finite-element analysis (FEA) simulation of the impact between the aircraft and the top 20 stories of the building, and then by visualizing the FEA results with a state-of-the-art commercial animation system. The visualization was enabled by an automatic translator that converts the simulation data into an animation system 3D scene. We built upon a previously developed translator. The translator was substantially extended to enable and control visualization of fire and of disintegrating elements, to better scale with the number of nodes and number of states, to handle beam elements with complex profiles, and to handle smoothed particle hydrodynamics liquid representation. The resulting translator is a powerful automatic and scalable tool for high-quality visualization of FEA results.
    IEEE Transactions on Visualization and Computer Graphics 01/2008; 14(4):937-47. · 1.90 Impact Factor
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    Ayhan Irfanoglu, Christoph M. Hoffmann
    Journal of Performance of Constructed Facilities - J PERFORM CONSTR FACIL. 01/2008; 22(1).
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    Christoph Hoffmann, Zygmunt Pizlo, Voicu Popescu, Steve Price
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    ABSTRACT: We test the perception of 3D surfaces that have been rendered by a set of lines drawn on the surface. Each surface is rendered as a family of curves which are in the simplest case the intersections with a family of parallel planes. On each trial, a surface or its “distorted” version is shown in this way, in an arbitrary orientation on an LCD screen or in a volumetric 3D display. The distortion is produced by stretching the surface in the z-direction by 30%. The subject’s task is to decide whether two sequentially presented surfaces are identical or not. The subject’s performance is measured by the discriminability d′, which is a conventional dependent variable in signal detection experiments. The work investigates the question whether a surface rendered with planar and geodesic curves is easier to recognize than one where the curves are not planar or not geodesic.
    Displays. 01/2007;

Publication Stats

2k Citations
41.91 Total Impact Points

Institutions

  • 1978–2012
    • Purdue University
      • Department of Computer Science
      West Lafayette, Indiana, United States
  • 2010
    • Soochow University, Taiwan
      T’ai-pei, Taipei, Taiwan
  • 2004
    • University of Florida
      Gainesville, Florida, United States
  • 2003
    • Ewha Womans University
      • Department of Computer Science and Engineering
      Seoul, Seoul, South Korea
  • 1995
    • Polytechnic University of Catalonia
      Barcino, Catalonia, Spain
  • 1985–1988
    • Cornell University
      • Department of Computer Science
      Ithaca, New York, United States
  • 1984
    • Johns Hopkins University
      • Department of Electrical and Computer Engineering
      Baltimore, Maryland, United States