**Bezier Curves and Surfaces **

We had discussed in the previously that we can create complicated geometries along with the aid of polygon meshes that are further constituted of standard polygonal objects as triangle, square, rectangle etc. although apart from this method to draw complicated geometries, we are having several more advanced methods to do the similar job like we can utilize mathematical equations as parametric equations and polynomial equations or splines, Bezier, fractals curves and so no. In this topic we will see some of these techniques, although to have the flavor of the detailed investigation of these methods you can refer to books specified in suggested readings. Before discuss the tour of Bezier curves we have a brief discussion on another method, by using mathematical equations as polynomial equations and parametric equations to realize the complicated natural scenes is not all the times successful as it needs an enormous number of calculations that consumes an immense amount of processing time. The high technique to generate complicate natural scenes is to utilize fractals. Fractals are geometry methods that utilize procedures and not mathematical equations to model objects as mountains, waves in sea and so on. There are different kinds of fractals like self-similar, self-affined, etc. This topic is quite interesting but is out of the scope of this unit. Currently, let us discuss Bezier curves that are a Spline approximation method developed through the French engineer Pierre Bezier for utilize in the implementation of Renault automobile bodies. Bezier splines have a number of properties which make them highly helpful and convenient for curve and surface design. They are also simple to implement. For these purpose, Bezier splines are broadly available in different CAD systems, in common graphics packages as GL on Silicon Graphics systems, and in assorted drawing and painting packages as Aldus Super Paint and Cricket Draw.

Before discuss in details of the Bezier curves, we must learn something about the idea of Spline and their representation. In fact Spline is a flexible strip utilized to produce a smooth curve via a designated set of points termed as Control points that is the style of fitting of the curve among these two points that gives rise to Interpolation and Approximation Splines. We can mathematically explain any curve along with a piecewise cubic Polynomial function whose first and second derivatives are incessant across the diverse curve sections. In computer graphics, the term spline curve here involves to any composite curve formed along with polynomial sections satisfying given continuity conditions as parametric continuity and Geometric continuity circumstances at the boundary of the pieces, without fulfilling such circumstances two curves can be joined smoothly. A spline surface can be explained with two sets of orthogonal spline curves. There are several different kinds of spline specifications that are utilized in graphics applications. Each one specification simply involve to an individual type of polynomial with exact specified boundary conditions. Splines are utilized in graphics applications to design surface and curve shapes, for computer storage digitize the drawings and for the objects or the camera in a scene, to specify animation paths. Classical CAD applications for splines comprise the design of automobile bodies, spacecraft and aircraft surfaces and also ship hulls.

We have mentioned above here its style of fitting of the curve among two control points that gives rise to Interpolation and estimation Splines, that we can identify a spline curve through giving a set of coordinate positions, termed as control points, that indicates the common shape of the curve. Such control points are subsequently fitted along with piecewise continuous parametric polynomial functions in one of two manners. While polynomial sections are fitted hence the curve passes via each control point, the resulting curve is consider as interpolate the set of control points. Conversely, when the polynomials are fitted to the common control-point without essentially passing via any control point, the resulting curve is consider as approximate the set of control points. Interpolation curves are commonly utilized to digitize drawings or to identify animation paths. Approximation curves are primarily utilized as design tools to structure object surfaces.

A spline curve is explained, modified and manipulated along with operations by the control points. Through interactively selecting spatial places for the control points, a designer can sketch an initial curve. After the polynomial fit is exhibited for a specified set of control points, the designer can after that repositions some or all of the control points to again structure the shape of the curve. As well, the curve can be translated, or scaled rotated along with transformations practical onto the control points. CAD packages can also insert more control points to assist a designer in adjusting the curve shapes.