36,921 research outputs found

    The index of gradf(x,y)grad f(x,y)

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    Let f(x,y)f(x,y) be a real polynomial of degree dd with isolated critical points, and let ii be the index of gradfgrad f around a large circle containing the critical points. An elementary argument shows that ∣iβˆ£β‰€dβˆ’1|i| \leq d-1. In this paper we show that i≀max{1,dβˆ’3} i \leq max \{1, d-3 \}. We also show that if all the level sets of ff are compact, then i=1i = 1, and otherwise |i| \leq \dr -1 where \dr is the sum of the multiplicities of the real linear factors in the homogeneous term of highest degree in ff. The technique of proof involves computing ii from information at infinity. The index ii is broken up into a sum of components ip,ci_{p,c} corresponding to points pp in the real line at infinity and limiting values c \in \realinf of the polynomial. The numbers ip,ci_{p,c} are computed in three ways: geometrically, from a resolution of f(x,y)f(x,y), and from a Morsification of f(x,y)f(x,y). The ip,ci_{p,c} also provide a lower bound for the number of vanishing cycles of f(x,y)f(x,y) at the point pp and value cc.Comment: A thoroughly revised and hopefully more readable version; the main results are the same. 35 pages with 7 figure

    Teleological computer graphics modeling

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    Summary form only give. Teleological modeling, a developing approach for creating abstractions and mathematical representations of physically realistic time-dependent objects, is described. In this approach, geometric constraint-properties, mechanical properties of objects, the parameters representing an object, and the control of the object are incorporated into a single conceptual framework. A teleological model incorporates time-dependent goals of behavior of purpose as the primary abstraction and representation of what the object is. A teleological implementation takes a geometrically incomplete specification of the motion, position, and shape of an object, and produces a geometrically complete description of the object's shape and behavior as a function of time. Teleological modeling techniques may be suitable for consideration in computer vision algorithms by extending the current notions about how to make mathematical representations of objects. Teleological descriptions can produce compact representations for many of the physically derivable quantities controlling the shapes, combining-operations, and constraints which govern the formation and motion of objects

    Superquadrics and Angle-Preserving Transformations

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    Over the past 20 years, a great deal of interest has developed in the use of computer graphics and numerical methods for three-dimensional design. Significant progress in geometric modeling is being made, predominantly for objects best represented by lists of edges, faces, and vertices. One long-term goal of this work is a unified mathematical formalism, to form the basis of an interactive and intuitive design environment in which designers can simulate three-dimensional scenes with shading and texture, produce usable design images, verify numerical machining-control commands, and set up finite-element meshwork for structural and dynamic analysis. A new collection of smooth parametric objects and a new set of three-dimensional parametric modifiers show potential for helping to achieve this goal. The superquadric primitives and angle-preserving transformations extend the traditional geometric primitives-quadric surfaces and parametric patches-used in existing design packages, producing a new spectrum of flexible forms. Their chief advantage is that they allow complex solids and surfaces to be constructed and altered easily from a few interactive parameters

    A Lawyer\u27s Ramble Down the Information Superhighway

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