<!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> PARAMETRIC CONTINUTY CONDITIONS</h3> <hr /> To ensure a smooth transition from one section of a piecewise parametric curve to the next, we can impose various continuity conditions at the connection points. If each section of a spline is described with a set of parametric coordinate functions of the form<br /> <img class="inlinesymbol" src="pics/eqn03_1.gif" alt="" width="378" height="25" /><p> <strong> Zero- order parametric continuity </strong>is described as C<sup>0</sup> continuity. Here the two curves sections must have the same coordinate position at the boundary point. </p> <p><img border="0" src="pics/eqn03_2.gif" width="121" height="30">&nbsp; </p> <p>where P<sub>1</sub> and P<sub>2</sub> are the curves defined between parameters u<sub>1</sub> and u<sub>2</sub> and u<sub>3</sub> and u<sub>4</sub> respectively.</p> <p> <strong>First order parametric continuity</strong>, or C<sup>1</sup> continuity, means that the parametric first derivatives are the same at the intersection on two successive sections. If we denote the parametric position on the curve as P(u), the direction as well as magnitude of the tangent vector P'(u) will be the same for two successive curve sections at their joining point under C<sup>1</sup> continuity.</p> <p>Thus for C<sup>1</sup> continuity </p> <p><img border="0" src="pics/eqn03_3.gif" width="286" height="36"></p> <p>&nbsp;<strong>Second-order parametric continuity</strong>, or C<sup>2</sup> continuity, means that both the first and second parametric derivatives of the two curve sections are the same at their boundary. </p> <p>Thus for C<sup>2</sup> continuity</p> <p><img border="0" src="pics/eqn03_4.gif" width="458" height="36"></p> <p> <img src="img/pic010.gif" alt="" width="318" height="186" /></p> <p>Figure 1: Discontinuous curves</p> <p> <img src="img/pic011.gif" alt="" id="IMG1" width="224" height="377" /></p> <p>Figure 2: (a) Curves having C<sup>0</sup> continuity (b) Curves having C<sup>1</sup> continuity and (c) Curves having C<sup>2</sup> continuity<br /> </div> <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>