Degree Polynomial Curve Research Articles

On geometric interpolation by planar parametric polynomial curves

In this paper the problem of geometric interpolation of planar data by parametric polynomial curves is revisited. The conjecture that a parametric polynomial curve of degree < n can interpolate 2n given points in R 2 is confirmed for n < 5 under certain natural restrictions. This conclusion also implies the optimal asymptotic approximation order. More generally, the optimal order 2n can be achieved as soon as the interpolating curve exists.

A characterization of quintic helices

A polynomial curve of degree 5, α , is a helix if and only if both ∥ α ′ ∥ and ∥ α ′ ∧ α ″ ∥ are polynomial functions.

Development of a One-Dimensional Modular Dynamic Model for the Simulation of Surge in Compression Systems

The paper deals with the development of a nonlinear one-dimensional modular dynamic model for the simulation of transient behavior of compression systems. The model is based on balance equations of mass, momentum, and energy, which are derived through a general approach and are written by using the finite difference method. The model also takes rotating mass dynamics into account through a lumped parameter approach. Moreover, it reproduces the behavior of the system in the presence of the surge phenomenon through steady-state performance maps, which represent the compressor operation in the inverse flow region by means of a third degree polynomial curve. The model is implemented through the Matlab Simulink tool, where the system of ordinary differential equations is solved by using a fourth- and fifth-order Runge–Kutta method. A sensitivity analysis is carried out to evaluate the influence on compressor outlet pressure oscillations of the model parameters, of the supplied torque, of ambient conditions and of the shape of the compressor characteristic curves. The results show that the model proves effective in capturing the physical essence of surge phenomenon without being computationally too heavy.

High accuracy Hermite approximation for space curves in [formula omitted

In this article, it is shown that a space curve in R d can be approximated by a piecewise polynomial curve of degree m with order ( m + 1 ) + ⌊ ( m + 1 ) / ( 2 d − 1 ) ⌋ rather than m + 1 . Moreover, we show that the optimal order ( m + 1 ) + ⌊ ( m − 1 ) / ( d − 1 ) ⌋ is possible for a particular set of curves of nonzero measure. Analogous results were shown to be true for Taylor polynomial interpolation in [A. Rababah, High order approximation method for curves, Comput. Aided Geom. Design 12 (1995) 89–102].

The complex representation of algebraic curves and its simple exploitation for pose estimation and invariant recognition

Representations are introduced for handling 2D algebraic curves (implicit polynomial curves) of arbitrary degree in the scope of computer vision applications. These representations permit fast, accurate pose-independent shape recognition under Euclidean transformations with a complete set of invariants, and fast accurate pose-estimation based on all the polynomial coefficients. The latter is accomplished by a centering of a polynomial based on its coefficients, followed by rotation estimation by decomposing polynomial coefficient space into a union of orthogonal subspaces for which rotations within two-dimensional subspaces or identity transformations within one-dimensional subspaces result from rotations in x, y measured-data space. Angles of these rotations in the two-dimensional coefficient subspaces are proportional to each other and are integer multiples of the rotation angle in the x, y data space. By recasting this approach in terms of a complex variable, i.e., x+iy=z, and complex polynomial-coefficients, further conceptual and computational simplification results. Application to shape-based indexing into databases is presented to illustrate the usefulness and the robustness of the complex representation of algebraic curves.

Geometric and densitometric standardization of intraoral radiography through use of a modified XCP system

Objective. The purposes of this study were to examine the density correction afforded by curve-fitting algorithms and to investigate whether the device we developed significantly improves the reliability of longitudinal alveolar process bone radiographic density measurements. Study Design. Stepwedges were radiographed over a range of impulse settings, and curve-fitting algorithms were fitted to sets of step images on each digitized film. Differences between the actual thicknesses of an alternate set of steps and their corresponding thickness estimates were calculated. Next, clinicians made periapical radiographs from interproximal bony sites on a dry skull using our imaging device. Differences in bone densities between corresponding regions of interest taken 1 week apart were calculated. Results. Analysis of variance and Duncan’s Multiple Range test demonstrated that piecewise linear, third-degree polynomial, and fourth-degree polynomial curves provided significantly better estimates of stepwedge thickness than did sigmoid or first degree polynomial-curves ( P < .05) and that the differences between repeat bone density measurements made with density correction were significantly less than those made without density correction ( P < .05). Conclusions. Piecewise linear, third-degree polynomial, and fourth-degree polynomial curve-fitting algorithms provided the best densitometric correction. The use of our imaging device increased the reliability of longitudinal bone density measurements. (Oral Surg Oral Med Oral Pathol Oral Radiol Endod 1999;87;253-7)

Practical reliable Bayesian recognition of 2D and 3D objects using implicit polynomials and algebraic invariants

We treat the use of more complex higher degree polynomial curves and surfaces of degree higher than 2, which have many desirable properties for object recognition and position estimation, and attack the instability problem arising in their use with partial and noisy data. The scenario discussed in this paper is one where we have a set of objects that are modeled as implicit polynomial functions, or a set of representations of classes of objects with each object in a class modeled as an implicit polynomial function, stored in the database. Then, given partial data from one of the objects, we want to recognize the object (or the object class) or collect more data in order to get better parameter estimates for more reliable recognition. Two problems arising in this scenario are discussed: 1) the problem of recognizing these polynomials by comparing them in terms of their coefficients; and 2) the problem of where to collect data so as to improve the parameter estimates as quickly as possible. We use an asymptotic Bayesian approximation for solving the two problems. The intrinsic dimensionality of polynomials and the use of the Mahalanobis distance are discussed.

New Algorithm of Intra Aortic Balloon Pumping in Patients With Atrial Fibrillation

Intra aortic balloon pumps have been operated inaccurately when used in patients with atrial fibrillation. In 36 pre-operative cardiac surgical patients with atrial fibrillation, the interval between the R wave of the electrocardiogram and the dicrotic notch (R-DCN interval) of the aortic pressure waveform showed significant correlation with the preceding R-R interval by second degree polynomial curves (0.53 < r2 < 0.74, p < 0.01). Based on these results a new algorithm for atrial fibrillation was designed. During brief off periods of the intra-aortic balloon pump in atrial fibrillation, electrocardiograms and aortic pressure waveforms were simultaneously recorded and analyzed to obtain a second degree polynomial curve between the preceding R-R and R-DCN intervals. The precise timing of balloon inflation was controlled using the equation in response to the preceding R-R interval. This new algorithm showed the smallest time lags (SD = 8.5 msec) when compared to other existing algorithms (the Kontron model-10, 53.49 msec; the Aries Medical model-700, 43.69 msec; the Datascope system-90, 25.98 msec; the Mansfield 3000, 19.65 msec; and the Kontron K-2000, 14.89 msec) in mock simulation driving using data obtained from clinically recorded waveforms.

Reference Values for Respiratory Rate in the First 3 Years of Life

Raised respiratory rate is a useful sign to diagnose lower respiratory infections in childhood. However, the normal range for respiratory rate has not been defined in a proper, large sample. To assess the respiratory rate in a large number of infants and young children in order to construct percentile curves by age; to determine the repeatability to the assessment using a stethoscope and compare it with observation. Respiratory rate was recorded for 1 minute with a stethoscope in 618 infants and children, aged 15 days to 3 years old, without respiratory infections or any other severe disease when awake and calm and when asleep. In 50 subjects we compared respiratory rate taken 30 to 60 minutes apart to assess repeatability, and in 50 others we compared simultaneous counts obtained by stethoscope versus observation. Repeatability was good as the standard deviation of differences was 2.5 breaths/minute in awake and 1.7 breaths/minute in asleep children. Respiratory rate obtained with a stethoscope was systematically higher than that obtained by observation (mean difference 2.6 breaths/minute in awake and 1.8 breaths/minute in asleep children; P = .015 and P < .001, respectively). A decrease in respiratory rate with age was seen for both states, and it was faster in the first few months of life when also a greater dispersion of values was observed. A second degree polynomial curve accurately fitted the data. Reference percentile values were developed from these data. The repeatability of respiratory rate measured with a stethoscope was good. Percentile curves would be particularly helpful in the first months of life when the decline in respiratory rate is very rapid and prevents to use cut off values for defining "normality."

Derivation of Shear Rates From Near-Wall LDA Measurements Under Steady and Pulsatile Flow Conditions

Atherosclerosis, thrombosis, and intimal hyperplasia are major forms of cardiovascular diseases in the United States. Previous studies indicate a significant correlation between hemodynamics, in particular, wall shear rate, and pathology of the arterial walls. While results of these studies implicate morphologic and functional changes related to wall shear rate magnitude, a standard technique for wall shear rate measurement has not been established. In this study, theoretical and in-vitro experimental fully developed steady and physiologic pulsatile flow waveforms have been used to obtain velocity profiles in the near-wall region. The estimated wall shear rates from these results are compared to the theoretical value to assess the accuracy of the approximating technique. Experimentally obtained results from LDA suggest that in order to minimize the error in velocity data, and subsequently, the wall shear rate, the first measured velocity has to be 500 microns away from the wall. While a linear approximation did not produce errors larger than 16.4 percent at peak systole, these errors substantially increased as the velocity magnitudes decreased during late systole and diastole. Overall, a third degree polynomial curve fit using four points produced the most accurate estimation of wall shear rate through out the cardiac cycle. Results of higher degree curve-fitting functions can be unpredictable due to potential oscillations of the function near the wall. Hence, based on the results of this study, use of a linear approximation is not recommended; a third degree curve-fitting polynomial, using four points provided the most accurate approximation for these flow waveforms.

Detecting cusps and inflection points in curves

In many applications it is desirable to analyze parametric curves for undesirable features like cusps and inflection points. Previously known algorithms to analyze such features are limited to cubics and in many cases are for planar curves only. We present a general purpose method to detect cusps in polynomial or rational space curves of arbitrary degree. If a curve has no cusp in its defining interval, it has a regular parametrization and our algorithm computes that. In particular, we show that if a curve has a proper parametrization then the necessary and sufficient condition for the existence of cusps is given by the vanishing of the first derivative vector. We present a simple algorithm to compute the proper parametrization of a polynomial curve and reduce the problem of detecting cusps in a rational curve to that of a polynomial curve. Finally, we use the regular parametrizations to analyze for inflection points.

On the attenuation of direct solar irradiance in Athens due to aerosols

In this study the trend of the time sequence of the “integral aerosol optical depth” (ka), as proposed by Unsworth and Monteith, was determined for clear days in summer for the period 1962–1988 in Athens. The trend was found by fitting a third degree polynomial curve and it was concluded that (ka) showed a considerable increase (i.e. from the value of 0.18 to 0.31) in the period 1962–1976 and remained approximately constant until 1979, after which it started decreasing again slowly until 1988. The increase of (ka) in the period 1962–1976 is likely attributable to the rapid development of the city in this period, while the decrease of (ka) after 1979 probably reflects the efficiency of some restrictions which were imposed on the pollutant emissions during this period. In addition, an analysis of the percentage frequency distribution found that while 95% of the values of (ka) ranged from 0.100 to 0.400 in the beginning of the period (1964–1967), in recent years (1984–1987) the same percentage of the values of (ka) ranged from 0.100 to 0.500.

Effects of statistical quality, sampling rate and temporal filtering techniques on the extraction of functional parameters from the left ventricular time-activity curves.

Several techniques for the processing of a series of curves derived from two left ventricular time-activity curves acquired at rest and during exercise with a nuclear stethoscope were evaluated. They were three and five point time smoothing, Fourier filtering preserving one to four harmonics (H), truncated curve Fourier filtering, and third degree polynomial curve fitting. Each filter's ability to recover, with fidelity, systolic and diastolic function parameters was evaluated under increasingly 'noisy' conditions and at several sampling rates. Third degree polynomial curve fittings and truncated Fourier filters exhibited very high sensitivity to noise. Three and five point time smoothing had moderate sensitivity to noise, but were highly affected by sampling rate. Fourier filtering preserving 2H or 3H produced the best compromise with high resilience to noise and independence of sampling rate as far as the recovery of these functional parameters is concerned.

312 SERUM FREE TOCOPHEROL LEVELS IN PREMATURE INFANTS (PI) RECEIVING TOTAL PARENTERAL ALIMENTATION (TPN)

Documentation (J Pediatr 95:869, 1979) that PI are biochemically deficient in Vitamin E (V-E) led to the recommendation of supplementing V-E during the first 2 mos. for PI less than 1500g. However, exact V-E requirements for PI have not been documented. The table shows the predicted level (mg/dl) of V-E on PI (n = 10) who were receiving TPN only without lipid source. The TPN contains 1 ml of M.V.I. (USV Pharmaceutical Corp.) providing about 1 IU/kg/day of V-E. The mean gestational age was 30.8 wks with a mean birth weight of 1401 g. Serum free tocopherol levels were measured using a fluorometric micro method (Amer J of Clin Path 36:133, 1966) on the PI's cord blood and on 8 weekly samples. The change in V-E levels over 8 wks was done by fitting a second degree polynomial curve through the points of plot of V-E versus time in wks. The curve was described by an equation: V-E level = 3.65 + 0.365 day - 0.00336 day. This shows that free tocopherol levels above 0.5 mg/dl could be attained after approximately 7 days of TPN administration.

Field Estimation of the Functional Response of Itoplectis behrensii1 , a Parasite of the California Oakworm, Phryganidia californica2

The functional response of ltoplectis behrensii (Cresson), an epidemic pupal parasite of Phryganidia californica Packard, was estimated from field data involving various host densities on the trunks of coast live oak, Quercus agrifolia Nee. Pupal hosts were detected as uniformly distributed on the trunk surface at high host densities using a nearest neighbor index of dispersion. The number of parasitized pupae were recorded at 3-4 day intervals throughout host pupation for 4 consecutive P. californica generations. A second degree polynomial curve was fitted to observed % parasitization at different host densities at 14 day intervals. Most generated curves were significant (at 5% level) to observed data. Minimum thresholds of parasite response were between 1 and 2 pupae/m2 of trunk surface. Optimum host densities ranged between 6.6-13.2 pupae/m2. Maximum parasitization ranged between 20.3 and 55.2%. Host densities beyond the optium range showed a decreasing rate of parasitization indicating inverse density dependence. Based on the estimated functional response, I. behrensii is more efficient at lower host densities between the minimum threshold and the optimim range.

Acceleration-Time Relationships in an Ice Skating Start

Abstract The purpose of the study was to assess the fundamental characteristics of the ice skating start through an analysis of the acceleration patterns exhibited during the first 20 feet of skating. A Locam 16 mm camera, operated at 100 frames per second, was used to film the skating starts of four subjects. The subjects possessed varying levels of skating ability ranging from moderately skilled to expert. The segmental method was used to determine the centers of gravity of each subject at every fourth frame of film throughout the first 20 feet of the starting task. The horizontal coordinates were plotted to form displacement curves. A fifth degree polynomial curve fit was used to smooth the displacement and velocity curves in order to generate acceleration-time functions. Analysis of the results indicated that similar patterns existed for all subjects. The acceleration curves typically revealed high acceleration initially and this lasted for approximately 1.25 seconds. The curves then tapered off and a...

ウォームの歯形測定

This article describes new methods to measure the tooth profile of the axial section of a worm accurately to a high degree with a ball shaped or a disk shaped probe. When the probe is moved along the tooth flank keeping its center on the axial plane of the worm, the trace of the center does not represent the tooth profile to be determined. In this paper, three methods of determining the tooth profile from the measured trace are presented.(1) A method based on the nominal tooth profile.(2) A method using the tooth profile approximated by a polynomial curve of degree 6.(3) A simplified method using an approximate formula.It is proved concerning two types of worm that the measured profile which is obtained by using one of these methods agrees with the other two.