<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" > <head> <title>Contents</title> <link rel="Stylesheet" type="text/css" href="../styles/pagestyle.css") /> </head> <body> <div id="contents"> <h3> BEZIER CURVES</h3> <hr /> This spline approximation method was developed by the French engineer Pierre Bezier for use in the design of Renault automobile bodies. <strong>Bezier splines</strong> have a number of properties that make them highly useful and convenient for curve and surface design. They are also easy to implement. For these reasons, Bezier splines are widely available in various CAD system.<br /><br /> <strong> Bezier Curves</strong><br /> In general, a Bezier curve section can be fitted to any number of control points. The number of control points to be approximated and their relative position determine the degree of the Bezier polynomial. As with a charactering matrix, or with blending functions. For general Bezier curves, the blending-function specification is the most convenient.<br /> Suppose we are given n + 1 control-point positions: P<sub>k</sub> = (x<sub>k</sub>, y<sub>k</sub>, z<sub>k</sub>), with k varying from 0 to n. These coordinate points can be blended to produce the following position vector P(u), which describes the path of an approximating Bezier polynomial function between P<sub>0</sub> and P<sub>n</sub>.<br /> <img class="inlinesymbol" src="pics/eqn09_1.gif" alt="" width="223" height="48" /><br /> <br /> The Bezier blending functions B<sub>k,n</sub>(u) are the Bernstein Polynomials:<br /> <img class="inlinesymbol" src="pics/eqn09_2.gif" alt="" width="190" height="28" /><br /> <br /> Where the C (n,k) are the binomial coefficients:<br /> <img class="inlinesymbol" src="pics/eqn09_3.gif" alt="" width="136" height="46" /><br /> <br /> Equivalently, we can define Bezier blending functions with the recursive calculation<br /> <img class="inlinesymbol" src="pics/eqn09_4.gif" alt="" width="326" height="26" /><br /> <br /> With <img class="inlinesymbol" src="pics/eqn09_5.gif" alt="" width="196" height="28" /> Vector equation below represents a set of three parametric equations for the individual curve coordinates.<br /> <img class="inlinesymbol" src="pics/eqn09_6.gif" alt="" width="135" height="145" /><br /> <br /> <img src="img/pic023.gif" alt="" width="550" height="400"/>&nbsp; <br /> &nbsp;<p>Figure 1: Bezier curves for varying number of control points - (a) n=2 (b, c and d) n=3 with varying location of points (e) n=4 </p> <p>As a rule, a Bezier curve is a polynomial of degree one less than the number of control points used: Three points generate a parabola, four points a cube curve, and so forth. Figure 1 demonstrates the appearance of some Bezier curves for various selections of control points in the xy plane (z = 0). With certain control-point placements, however, we obtain degenerate Bezier polynomials. For example, a Bezier  curve that is a single point.<br /> <br /> Bezier curves are commonly found in painting and drawing packages, as well as CAD system, since they are easy to implement and they are reasonably powerful in curve design. Efficient methods for determining coordinate positions along a Bezier curve can be set up using recursive calculations. For example, successive binomial coefficients can be calculates as<br /> </p> <hr /> <h3> Related Links </h3> <hr /> </div> <a href="10.htm">Properties of Bezier Curves<br /> <br /> </a><a href="11.htm">Design Techniques Using Bezier Curves<br /> <br /> </a><a href="12.htm">Cubic Bezier Curves<br /> </a> <script src="http://www.google-analytics.com/urchin.js" type="text/javascript"> </script> <script type="text/javascript"> _uacct = "UA-2741197-1"; urchinTracker(); </script> </body> </html>