Bezier spline

Trosset J-Y, Scheraga HA (1999) Flexible docking simulations scaled collective variable Monte Carlo minimization approach using Bezier splines, and comparison with a standard Monte Carlo algorithm. J Comp Chem 20 244-252... 

Global Minimum in Docking Simulations A Monte Carlo Energy Minimization Approach Using Bezier Splines. 

Trosset, J. Y. Scheraga, H. A. (1998). Reaching the global minimum in docking simulations a Monte Carlo energy minimization approach using Bezier splines. Proc Natl Acad Sci U S A 95(14), 8011-5. 

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. 

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. 

Figure 1. Data smoothing via Bezier and cubic splines. Bezier splines are shown as circular data points which approximate a typical noisy density profile (black line). Cubic splines (dashed line) are then fit to the Bezier data points (at a higfier resolution than is own here).

Another type of MC method is the scaled collective variable Monte Carlo method used in the software package PRODOCK [219]. This method performs energy minimizations after each MC step, which helps to distinguish native conformations from low-energy non-native conformations. Bezier splines and other techniques have been incorporated into the method to improve its... 

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

In the 1950s, to develop a mathematical representation for the autobody surface, Pierre Bezier, at Renault in France, first published his work on spline that is represented with control points on the curve, which is now commonly referred to as the Bezier spline. Figure 2.9a illustrates a Bezier curve in solid line with four control points, 1,2,3, and 4, and its control polygon in dashed line, and Figure 2.9b illustrates two B-spline curves, each with multiple Bezier arcs, in solid, dash, or dot line, with a unified mechanism defining continuity at the joints. 

For shape design purposes it is usually far more convenient to use a basis where the basis functions sum to f. All the coefficients then transform as points, and we call them control points. For polynomials this can be achieved by using the Bezier basis, which is a special case of the B-spline basis which will be encountered shortly. 

Higher degree B-splines can be constructed explicitly (using, for example, the Bezier basis for each span of the function) by recursion on degree, applying the same simple recipe ... 

For approximating schemes we can design end-conditions by analogy with the B-splines, where the most widely used end-condition is that called Bezier end-conditions. 

In fact all three approaches are equivalent, and we shall illustrate the first with Bezier end-conditions for box-splines, and the second with Lagrange conditions for the four-point scheme. 

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... 

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. 

Prautzsch, H., Boehm, W., and Paluszny, M. Bezier and B-Spline Techniques, Springer, Berlin, 2002. 

The schema provides for regular conic curves (e.g. circle, ellipse, etc.), and quadric surfaces (e.g. cylinder, cone, etc.). Each of these has a specified parameterisation which must be adhered to when providing parameter values for other entities (e.g. trimmed curves). This is an attempt to exclude the idea of default parameterisation, the interpretation of which may differ from one implementation to another. In addition, parametric curves and surfaces are provided for by B-spline entities. These are specified by their control points, and may be rational or non-rational, uniform or non-uniform. Bezier representation of curves and surfaces is provided as a... 

If Bezier =. T. the curve represented is a single segment Bezier curve. The knot set will not be included explicitly. The curve can be represented either as a B-spline curve with multiple knots (0,0,...,0,1,1,. .. 1) each value 0 or 1 being repeated degree + 1 times, or directly as a Bezier curve ... 

It should be noted that every Bezier curve has an equivalent representation as a B-spline curve but not every B-spline curve can be represented as a single Bezier curve. 

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