Bezier curve

Curves resulting from the choice of Bezier functions blend the values of the known boundary potentials to produce interior potential values and have the appropriate smoothness properties desired in the final solution. Further, the 5, have maximum values that distribute evenly through the mesh regions. For instance, for u between 0 and j in Eq. (15.20), the value of 5q,4 is greatest, and all other B variables approach minimum values. Thus, Bq,4 serves to sample that particular range of u values. 

Thus the behaviour at the ends of the curve gets more and more similar to the control at the end of a Bezier curve. 

The end-conditions described above cover two distinct cases, those of interpolating schemes, which are likened to Lagrange interpolation, and those of B-splines, likened to the Bezier end-conditions. The schemes which interpolate when the data lies on a cubic or higher polynomial do not really fit either of these cases. They are almost interpolating (when the data is really smooth) but not quite. Somebody needs to play with these schemes to find out how they currently misbehave at the ends and what kinds of control are required to make them do what the curve designer wants. 

G.Aumann Corner cutting curves and a new characterization of Bezier and B-spline curves. CAGD 14(5), pp449-474, 1997... 

R.Ait-Haddou and W.Herzog Convex subdivision of a Bezier curve. CAGD 19(8), pp663-672, 2002... 

Surface-oriented systems are able to generate objects, known as free-form surfaces, whose surfaces are made up of numerous curved and analytically indescribable surfaces. One feature visible in an internal computer representation of free-form surfaces is their interpolated or approximated nature. Therefore, various processes have been developed, such as the Bezier approximation, Coon s surfaces, the NURBS representations, and the B-spUne interpolation (see Section 4.1). 

EnPlot software ASM s analytical engineering graphics software that is used to transform raw data into meaningful, presentation-ready plots and curves. It offers users a wide array of mathematical functions that are used to fit data to known curves and includes quadratic Bezier spline, straight-line polynomial, Legendre polynomial, nth order, and exponential splines. See computer software mathematics. 

A similar curve description was achieved by Philip De Casteljeau at the French firm Citroen. Nevertheless, the method of approximation of the control polygon was linked to the name of Bezier in the literature even if the description of the curves uses functions other than Bernstein polynomials. 

As a solution to the problems of Bezier curves, the B-spline curve gained a leading position during the 1990s. The B-spline curve is inherently segmented, in other words piecewise. The... 

It is easy to recognize that the cubic curve in the above example is a Bezier curve. A Bezier curve can be considered as a special case of a non-periodic B-spline. 

Deletion of the analytical shape constraint from a rational B-spline curve enables the B-spline to be modified as a free form curve. The effect of a shape constraint can be deleted for the entire curve or only for one or more of its segments. The initial shape of a free form curve is free to change. This freedom is restricted by the representation capabilities of the applied mathematics background. For example, a Bezier curve cannot be modified locally whereas a rational B-spline curve ensures excellent local modification. The shape of a curve is locally restricted by, among others, the shape of the available and economical cutting tools. 

Equations (4) and (5) are not evaluated explicitly in the minimization program, but are fit using a combination of spline [17] methods, which provide stability, the ability to filter noise easily, and the flexibility to describe an arbitrarily shaped potential curve. Moreover, the final functional form is inexpensive to evaluate, making it amenable to global minimization. The initial step in our methodology is to fit the statistical pair data for each amino acid and for the density profile to Bezier splines [17]. In contrast to local representations such as cubic splines, the Bezier spline imposes global as well as local smoothness and hence effectively eliminates the random oscillatory behavior observed in our data. 

Systematic data smoothing by using Bezier curves... 

Bezier curve (of degree 4), constructed with five control points Pq, Pi,..., P4- All points B are located at f = 0.4 on the respective legs. 

This is the representation of the famous cubic Bezier curves which are basic elements of cubic Bezier splines. 

The example in Fig. 4.4 shows a Bezier curve, which was constructed by using five control points. Therefore, it is of degree 4 and parametrized by... 

We observe that a Bezier curve of degree k is always represented by a Ath-order polynomial. Even more interesting is that, for t fixed, the expansion coefficients can be interpreted as probabilities with which each of the control points contributes to the location of the point of the Bezier curve that is parametrized by t. In fact, the prefactors remind us of the binomial distribution and the compact representation of any Bezier curve of degree n reads... 

Equation (4.51) is the most general expression for a Bezier curve, as it is parametrized by the local curve parameter. This allows to create any kind of curve, even such that include loops. However, in this form it is not very suitable for the representation of functions y x) in two-dimensional space. Fortunately, for control points yt equally spaced in x space on the interval [xq,x ], x, =xq - - i x - xo)/n, it is simple to re-express Eq. (4.51) in a more convenient, scalar form. First, we note that x and t are trivially related with each other ... 

Top) Function y (x) (dark gray) as defined in Eg. (4.56), with random noise added,/rand W (light gray), and Bezier reconstruction /bez (if) (black curve) for fluctuation widths (a) tmax = 0, (b) Zmax = 0.5, and (c) tmax = 1-0- (Bottom) Exact errors Srand,bez (if) ofyrand,bez (if) fot the same noise amplitudes. 

We first assume that we have performed M experiments to obtain statistically uncorrelated data setsy =y xi), k=, ..., K,fox eachx, f = 0,..., n. Each data set possesses the unique Bezier curve (4,55)... 

Since the exact expectation value Y) is unknown, we introduce the average of the Bezier curves at x by... 

The unknown exact variance of the Bezier curves obtained in cx) experiments is Oyix) = (T )(x) — (Y) (x). The finite-A estimator for this variance,... 

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