Construction of Cubic Triangular Patches with C^1 Continuity around a Corner
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Abstract:
This paper presents a novel approach for constructing a piecewise triangular cubic polynomial surface with C/sup 1/ continuity around a common corner vertex. A C/sup 1/ continuity condition between two cubic triangular patches is first derived using mixed directional derivatives. An approach for constructing a surface with C/sup 1/ continuity around a corner is then developed. Our approach is easy and fast with the virtue of cubic reproduction, local shape controllability, C/sup 2/ continuous at the corner vertex. Some experimental results are presented to show the applicability and flexibility of the approach.Keywords:
Cubic function
The construction of a multilayer perceptron (MLP) as a piecewise low-order polynomial approximator using a signal processing approach is presented in this work. The constructed MLP contains one input, one intermediate and one output layers. Its construction includes the specification of neuron numbers and all filter weights. Through the construction, a one-to-one correspondence between the approximation of an MLP and that of a piecewise low-order polynomial is established. Comparison between piecewise polynomial and MLP approximations is made. Since the approximation capability of piecewise low-order polynomials is well understood, our findings shed light on the universal approximation capability of an MLP.
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This paper presents a simplified method of expressing the solution to cubic equations in terms of function evaluation only. The method eliminates the need to manipulate the original coefficients of the cubic polynomial and makes the solution free from such coefficients. In addition, the usual substitution needed to reduce the cubic equation is implicit in that the final solution is expressed in terms only of the function and derivative values of the given cubic polynomial at a single point. The proposed methodology simplifies the solution to cubic equations making them easy to remember and solve.
Cubic function
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The smoothity of piecewise polynomial appl ie d in literature is discussed, and the consistency condition is given such that the piecewise polynomial is smooth in its nodes. The method of literature is improved by signal reconstructing smoothly. Finally, a simple example i s given to show effectiveness of signal smooth reconstruction.
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Cubic function
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Cubic function
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The trend item of a long-term vibration signal is difficult to remove. This paper proposes a piecewise integration method to remove trend items. Examples of direct integration without trend item removal, global integration after piecewise polynomial fitting with trend item removal, and direct integration after piecewise polynomial fitting with trend item removal were simulated. The results showed that direct integration of the fitted piecewise polynomial provided greater acceleration and displacement precision than the other two integration methods. A vibration test was then performed on a special equipment cab. The results indicated that direct integration by piecewise polynomial fitting with trend item removal was highly consistent with the measured signal data. However, the direct integration method without trend item removal resulted in signal distortion. The proposed method can help with frequency domain analysis of vibration signals and modal parameter identification for such equipment.
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The piecewise interpolating polynomial is employed to approximate arbitrary dynamic loads in the Duhamel integration for the solution of dynamic response of multiple-degree of-freedom systems. The relavant formulas are derived. Because the Duhamel integration of piecewise polynomial is exact, the proposed solution is more accurate and computationally effort saving than the traditional numerical integration schemes.
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A class of modifiable G~1-continuous piecewise cubic polynomial curves and two local control parameters on every single piece are presented in this paper. By analyzing the relationships between the curves mentioned above and the cubic Bezier curves,the geometric significance of local control parameters is given and the shape of the curves can be flexibly and easily changed by changing the values of the local control parameters.Finally, it popularizes the curves to the case of cubic polynomial surfaces and provides us rations of figures and their usage.
Cubic function
Cubic form
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Abstract Raman spectrum, as a kind of scattering spectrum, has been widely used in many fields because it can characterize the special properties of materials. However, Raman signal is so weak that the noise distorts the real signals seriously. Polynomial fitting has been proved to be the most convenient and simplest method for baseline correction. It is hard to choose the order of polynomial because it may be so high that Runge phenomenon appears or so low that inaccuracy fitting happens. This paper proposes an improved approach for baseline correction, namely the piecewise polynomial fitting (PPF). The spectral data are segmented, and then the proper orders are fitted, respectively. The iterative optimization method is used to eliminate discontinuities between piecewise points. The experimental results demonstrate that this approach improves the fitting accuracy.
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Abstract: Error control in piecewise polynomial interpolation of a smooth univariate function f requires that the interval of approximation be subdivided into many subintervals, on each of which an interpolating polynomial is determined. The number of such subintervals is often over- estimated through the use of a high-order derivative of f . We report on a partitioning algorithm, in which we attempt to reduce the number of subintervals required, by imposing conditions on f and its relevant higher derivative. One of these conditions facilitates a distinction between the need for absolute or relative error control. Two examples demonstrate the effectiveness of this partitioning algorithm. Key Words: Piecewise Polynomial; Range Partitioning; Domain Partitioning; Error Control
Interpolation
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