187 research outputs found

    Extending 2-D Smoothed Local Symmetries to 3-D

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    3-D Smoothed Local Symmetries (3-D SLS's) are presented as a representation for three-dimensional shapes. 3-D SLS's make explicit the perceptually salient features of 3-D objects and are especially suited to representing man-made objects. The definition of the 3-D SLS is given as a natural extension of the 2-D Smoothed Local Symmetry (2-D SLS). Analytic descriptions of the 3-D SLS are derived for objects composed of planar and spherical patches. Results of an implementation of the 3-D SLS are presented, along with suggestions for further research.MIT Artificial Intelligence Laborator

    Virtual reality based creation of concept model designs for CAD systems

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    This work introduces a novel method to overcome most of the drawbacks in traditional methods for creating design models. The main innovation is the use of virtual tools to simulate the natural physical environment in which freeform. Design models are created by experienced designers. Namely, the model is created in a virtual environment by carving a work piece with tools that simulate NC milling cutters. Algorithms have been developed to support the approach, in which the design model is created in a Virtual Reality (VR) environment and selection and manipulation of tools can be performed in the virtual space. The desianer\u27s hand movements generate the tool trajectories and they are obtained by recording the position and orientation of a hand mounted motion tracker. Swept volumes of virtual tools are generated from the geometry of the tool and its trajectories. Then Boolean operations are performed on the swept volumes and the initial virtual stock (work piece) to create the design model. Algorithms have been developed as a part of this work to integrate the VR environment with a commercial CAD/CAM system in order to demonstrate the practical applications of the research results. The integrated system provides a much more efficient and easy-to-implement process of freeform model creation than employed in current CAD/CAM software. It could prove to be the prototype for the next-generation CAD/CAM system

    The Euclidean distance degree of an algebraic variety

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    The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a generic point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.Comment: to appear in Foundations of Computational Mathematic

    Continuous collision detection for ellipsoids

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    We present an accurate and efficient algorithm for continuous collision detection between two moving ellipsoids. We start with a highly optimized implementation of interference testing between two stationary ellipsoids based on an algebraic condition described in terms of the signs of roots of the characteristic equation of two ellipsoids. Then we derive a time-dependent characteristic equation for two moving ellipsoids, which enables us to develop a real-time algorithm for computing the time intervals in which two moving ellipsoids collide. The effectiveness of our approach is demonstrated with several practical examples. © 2006 IEEE.published_or_final_versio

    Geometric Modeling of Cutter/Workpiece Engagements in Three-Axis Milling Using Polyhedral Representations

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    Modeling the milling process requires cutter/workpiece engagement (CWE

    On C0 and C1 continuity of envelopes of rotational solids and its application to 5-axis CNC machining

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    We study the smoothness of envelopes generated by motions of rotational rigid bodies in the context of 5-axis Computer Numerically Controlled (CNC) machining. A moving cutting tool, conceptualized as a rotational solid, forms a surface, called envelope, that delimits a part of 3D space where the tool engages the material block. The smoothness of the resulting envelope depends both on the smoothness of the motion and smoothness of the tool. While the motions of the tool are typically required to be at least C2, the tools are frequently only C0 continuous, which results in discontinuous envelopes. In this work, we classify a family of instantaneous motions that, in spite of only C0 continuous shape of the tool, result in C0 continuous envelopes. We show that such motions are flexible enough to follow a free-form surface, preserving tangential contact between the tool and surface along two points, therefore having applications in shape slot milling or in a semi-finishing stage of 5-axis flank machining. We also show that C1 tools and motions still can generate smooth envelopes.Juan de la Cierva - Formation [grant number FJC2019-039804-I] Ram\ón y Cajal fellowship RYC-2017-22649
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